Research ArticleDynamic Currency Futures and Options Hedging Model
Xing Yu Yanyin Li and Zhongkai Wan
School of Economics and Business Administration Central China Normal University Wuhan 430079 China
Correspondence should be addressed to Xing Yu yuxingmailccnueducn
Received 21 November 2018 Revised 18 June 2019 Accepted 20 June 2019 Published 1 July 2019
Academic Editor Emilio Gomez-Deniz
Copyright copy 2019 Xing Yu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In this paper we consider a risk averse competitive firm that adopts currency futures and options for hedging purpose Basedon the assumption of unbiased markets of currency futures and options we propose the optimal hedging model in dynamicsetting By using two-stage optimization method we prove that it is desirable for the prudent enterprise to buy exchange rateoptions to hedge currency risk Furthermore we derive the closed-form solutions of the multiperiod hedging problem with thequadratic utility function We investigate an empirical study incorporated into GARCH-t prediction on the efficiency of hedgingwith currency futures and options The empirical results demonstrate that hedging with currency futures and options can reducethe silver export firmrsquos risk exposure Profits and the effective boundaries are compared in three cases hedging with futures andoptions synchronously only with futures and without any hedge The results of multiple comparisons among different hedgingstrategies show that hedging with linear and nonlinear derivatives is advisable for the export firm
1 Introduction
Since 2005 China has begun to carry out exchange ratereform from a single fixed exchange rate system pegged totheUSdollar to a floating exchange rate systemwith referenceto a package of currencies Since the reform of the exchangerate the pressure of RMB exchange rate appreciation in fluc-tuation has been constantly increasing resulting in increasingexchange rate risk faced by foreign-related enterprises andbringing tremendous impacts on Chinarsquos import and exportenterprises Under this background what strategies shouldbe used for the import or export enterprises to cope with theexchange rate risk is a practical problem that has to be solvedurgently
With the exchange rate fluctuation increasing manyenterprises begin to transfer the exchange rate risk throughfinancial derivatives As we know foreign exchange futuresand options are two commonly used derivatives in interna-tional trade for exchange rate risk management Numerousscholars have verified the influences of financial derivativeshedging on production by empirical research Allayannis andWeston [1] used data from 720 large nonfinancial firms in theUnited States from 1990 to 1995 and found that firms usingfinancial hedging tools increased their corporate value by 487
percent Clark and Mefteh [2] also found that using financialhedging tools could improve enterprisesrsquo profits But differentfinancial hedging tools have different hedge performancesThey concluded that in short term tools such as options andforwards can significantly help enterprises ease exchange raterisk While in long run only the combination of forwardoption and swap can help enterprises avoid exchange raterisk Barjaktarovi et al (2011) explained how currency optioncontracts were used to speculate or hedge based on antici-pated foreign exchange ratemovement Although the positiverole of derivatives in the exchange rate risk managementfor import and export enterprises has been verified to acertain extent by empirical researches it lacks theoreticalsupport to the generality and universal applicability of theconclusion
Many scholars applied the foreign exchange futures andoptions to reduce the exchange rate risk enterprises facedIn terms of foreign exchange futures hedging Wong [3]studied the decisions on production operation and foreigncurrency futures hedging of the export enterprises Wong[4] examined the behaviors of competitive exporters underprice and exchange rate uncertainties and found that the keyto the optimal production and hedging decisions dependedon the degree of imperfection of the forward market (either
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8074384 11 pageshttpsdoiorg10115520198074384
2 Mathematical Problems in Engineering
commodity futures or foreign exchange futures exist) thedependence structure of price and exchange rate risk BrollandWong [5] investigated the exchange rate hedging problemin that a competitive exporter exported products to twocountries They established a hedging model of exchangerate futures to study the impact of exchange rate changeson the export decisions Exchange rate futures hedgingrelated researches can refer to Lence [6] Lien and Wong[7] Broll et al [8] Wong [9] and so on As for exchangerate options hedging Machnes [10] studied the decision-making problems of the competitive companies using com-modity options to hedge price uncertainty Wong [11] madea contrastive study for an export firm with regard to therisk and found that the variance of options hedging waslower than that of no options hedging Lien and Wong[12] derived an optimal hedging strategy for the risk aversehedgers under delivery risk through futures options hedgingTheir study thus provides a theoretical basis for optionshedging with multiple delivery prices Since both exchangerate futures and options can be used as risk hedging toolssome scholars have conducted comparative studies on thehedging effects between the two derivatives Battermann etal [13] assumed that the export price was known enterprisesonly needed to deal with the random exchange rate riskThey compared the hedging effects of exchange rate futuresand exchange rate options and showed that exchange ratefutures havemore advantages than the exchange rate optionsBut it does not mean to say that options are useless Somescholars point out that futures have an advantage in hedginglinear risk and options are more suitable to hedge nonlinearrisk (Bajo [14 15] and Wong [16]) As we all know linearrisk and nonlinear risk are generally prevailing in the realinvestment Lapan et al [17] established the classical LMH(Lapan Moschini and Hanson) model and pointed out thatoptions played a role due to the convexity utility functionFrechette [18] presented various optimal hedging portfoliomodels including futures and options when the marginalcost of hedging was nonzero Wong [19] studied the optimalhedging and decision-making problem of the competitiveexport enterprises that faced the exchange rate risk Theyproved that there were two different sources of nonlinearrisk wherein one was the multiplicative property of theprice and exchange rate and another was the nonlinearmarginal utility function Wong [20] examined the problemwherein state-dependent preference and commodity futuresand options hedging were considered for the competitivefirms There was only a linear risk stemming from stochasticprice but a nonlinear relationship between preference andthe state made the enterprises face nonlinear risk For thestudies of joint hedge with exchange rate futures and optionssee Sakong et al [21] Moschini and Lapan [22] Despiteliteratures mentioned above have studied futures hedgingoptions hedging and futures and options joint hedging thesituation they assume is static While in actual hedgingpractice investors have to adjust the positions of futuresand options dynamically to minimize risks or maximizereturns
Dynamic hedge is what needs to adjust the strategies asthe price or other characteristics of the portfolio or security
changes Moreover risks of some securities cannot be hedgedwith static positions For example option price is not linearwith the underlying assetrsquos price which means that risk ofoption can only be hedged dynamically There are manyresearches on dynamic futures hedging The used methodsmainly involve time-varying GARCH models and dynamicprogramming method Kotkatvuorirnberg [23] used CopulaDCC-EGARCHmodel to estimate bivariate error correctionand study the effectiveness of currency futures hedgingZhang et al [24] employedGARCH-Copulamodels to exam-ine the options hedging problem Dynamic programmingmethod is another widely used method to solve dynamicinvestment decisions Change and Wong [25] developed aquadratic utility model of a multinational firm that facedexchange rate risk exposure to a foreign currency cash flowChi et al [26] analyzed the hedger positionrsquos value alterationand used the dynamic programming method to set up themultiperiod futures dynamic hedging optimal model Li etal [27] constructed the minimum variance model for theestimation of the optimal hedge ratio based on the stochas-tic differential equation However the application of thedynamic programming method in risk hedging especially inoptions hedging is rare
To sum up this paper takes account of the following threeaspects First with the development of economic globaliza-tion trade across countries is increasingly frequent How touse financial derivatives to manage exchange rate risk is apractical problem Previous studies have shown that futuresand options have advantages in hedging linear and nonlinearrisks respectively and in the actual hedging practice linearrisk and nonlinear risk coexist generallyTherefore this paperestablishes a hedging model of exchange rate futures andoptions to hedge the exchange rate risk Second a largenumber of literatures have studied the dynamic hedgingof futures but few study the dynamic hedging model withoptions In this paper a dynamic hedging model of exchangerate futures andoptions is constructed by combiningGARCHmodel with dynamic programming method Third we verifythe effects of futures hedging option hedging and futures andoption combined hedging and then further to provide riskmanagement ways for the import and export enterprises toavoid foreign exchange risk
The rest of this paper is organized as follows In Section 2assumptions and notations are presented Section 3 providesthe one-stage hedging model with exchange rate options andfutures We show that for the prudent firm it is necessaryto buy options for hedging Under the quadratic utilitywe demonstrate the explicit positions of the exchange ratefutures and options Section 4 studies the multistage hedgingproblem We deduce the optimal positions of the exchangerate futures and options by using dynamic programmingmethod An empirical analysis illustrated the hedging effec-tiveness in Section 4The performance of the derived hedgingstrategy is compared in three cases of hedging with futuresand options only with futures and without hedging Thecomparison is performed in terms of the terminal wealththe terminal wealth based on utility and also the vari-ance of wealth accumulation path Section 5 concludes thepaper
Mathematical Problems in Engineering 3
2 Assumptions and Notations
Suppose a competitive export enterprise exports a singleproduct to the foreignmarket It may obtain a predeterminedforeign order in advance That is demand or output is givenas 119876 We consider the price risk and exchange rate risk facedby the enterprise The price of the product sold abroad is in foreign currency The exchange rate is 119878 Referring to thestudy of Lapan and Moschini [22] we suppose that
= 120572 + 120573119878 + 120576 120573 lt 0 (1)
where 120576 is a zero mean and independent random variablewith 119878 Furthermore according to Chang andWong [25] weassume that
119878 = 119878 + 120579 (2)
Here 119864(119878) = 119878The imported or exported products in China involve a
wide range such as clothing toys and electromechanicaland pharmaceutical products while the derivative marketscorresponding to these small commodities usually do notexist Generally speaking exchange rate derivatives are com-mon Since the relationship between commodity price andexchange rate is assumed to be shown in (1) ie hedgingexchange rate risk also eliminates some commodity price risksynchronously we assume that the exporter uses divisible andtradable exchange rate rather than commodity futures andoption for hedging Suppose that the futures price is 119865 atthe current time For the sake of simplicity we also assumethat the strike price of the exchange rate option is 119870 and thecorresponding premium is recorded as 119881 Since Wong [16]points out that the assumption of futures and optionsmarketswhich are unbiased is because firms aim to hedge rather thanarbitrage futures and options markets are assumed to beunbiased for studying the pure hedging roles of exchange ratefutures and options ie 119864(119878) = 119865 and 119864(119870 minus 119878)+ = 1198813 One-Stage Hedging Model with ExchangeRate Options and Futures
To begin with we consider one-stage static hedging problemwith exchange rate options and futures The firm determinesthe exchange rate futures position 119883 and the exchange rateoption position 119884 at 0 (current time) to maximize its utilityAccording to the assumptions above the enterprisersquos profit attime 1 is
Π = 119878119876 + (119865 minus 119878)119883 + [119881 minus (119870 minus 119878)+] 119884 (3)
where 119883 and 119884 are the exchange rate futures and optionspositions that are sold (negative when be bought) and (119870 minus119878)+ = max (119870 minus 119878 0)
The firmrsquos utility function is 119880(Π) Assume that thedecision-maker of the firm is risk averse then1198801015840(Π) gt 0 and11988010158401015840(Π) lt 0 Based on the discussion above the problem facedby the firm can be written as (1198751)
max119883119884
119864 [119880 (Π)] (4)
where Π is described in (3)
Proposition 1 Suppose the exchange rate futures market andoption market are unbiased When the risk aversion exportenterprisersquos utility function satisfies 119880101584010158401015840(Π) ge 0 then it isoptimal for the enterprise to buy options for hedging
Proof Since the enterprise is risk aversion then its utilityfunction satisfies 1198801015840(Π) gt 0 and 1198801015840(Π) lt 0 Thereforethe relation of the optimal positions of futures and options119883lowast 119884lowast is map1-to-1 ie119883lowast = 119883(119884lowast)We apply the two-stageoptimizationmethod to prove Proposition 1 In the first stagelet
119867(119884) = arg max119883
119864 [119880 (Π)] (5)
When 119884 = 0 based on (5) we have
119864 [1198801015840 (Π0) (119865 minus 119878)] = 0 (6)
According to the assumption of unbiased market (6) canbe written as follows
That is 119864[1198801015840(Π0) | 119878] is concave with regard to 119878 On theother hand since 119864119864[1198801015840(Π0) | 119878] = 119864[1198801015840(Π0)] then theequation of
has at least one solution and at most two solutionsWhen (11) has only one solution 1198781 isin (119878 119878) then (7) can
be expressed by
119864 1198801015840 (Π0 |) 119878 minus 119864 [1198801015840 (Π0)] [119864 (119878) minus 119878] = 0 (12)
4 Mathematical Problems in Engineering
Equation (12) is further expressed in an integral form
int119878119878
119864 [1198801015840 (Π0) | 119878] minus 119864 [1198801015840 (Π0)] (119891 minus 119878) 119891 (119904) 119889119904= 0
(13)
If 119878 isin (119878 1198781) the first term of the integrand in (13) isnegative and the second is positive When 119878 isin (1198781 119878) thefirst term of the integrand in (13) is positive and the secondis negative Therefore the left hand of (13) is negative whichleads a contradictory to (13)
If (11) has two solutions ie 1198781 1198782 119878 lt 1198781 lt 1198782 lt 119878 thenwhen 119878 = 1198781 and 119878 = 1198782 we have
10038161003816100381610038161003816100381610038161003816100381610038161003816119884=0= 119864 1198801015840 (Π0) [119881 minus (119870 minus 119878)+]
= cov (119864 [1198801015840 (Π0) | 119878] 119881 minus (119870 minus 119878)+)= 119864 [1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
= int119878119878
[1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
then proof that 119877(119870) lt 0 according to the functional featuresof 119877(119870) In fact it is evident that 119877(119878) = 0 From (6) we have119877(119878) = 0 Since
then when 119878 isin (1198781 1198782) we have 11987710158401015840(119870) gt 0 Similarly weobtain that when 119878 isin (119878 1198781) cup (1198782 119878) we have 11987710158401015840(119870) lt 0Based on the discussion above we have 119877(119870) lt 0 That is(120597119864[119880(Π)]120597119884)|119884=0 lt 0 and 119884lowast lt 0
Referring to the definition in Kim [28] the risk aversioninvestor whose utility function satisfies 119880101584010158401015840(Π) ge 0 is calleda prudent investor The results in Proposition 1 show thatit is wise for the prudent investor to buy unbiased optionsfor hedging which is consistent with the real intention ofenterprises who adopt hedging with the purpose of apprecia-tion and preservationWe know that although selling options
can obtain option premiums investors have to face the dailymarking risk caused by the additional margin which maylead to greater liquidity risk for investors Jorion [29] dividedliquidity risk into asset liquidity risk and capital liquidity riskHe pointed out that the capital liquidity risk is the potentiallyfatal risks faced by investors For option sellers the liquidityrisk they face mainly includes capital liquidity risk As afailure case of hedging China SouthernAirlines desire to lockin the cost of raw materials by buying call options whichcan be regarded as a hedge but selling put options based onbull market judgment creates a new margin risk exposureTherefore the conclusion of Proposition 1 coincides withthe original intention of options hedging to preserve andincrease appreciation We note the utility function of theprudent enterprise in Proposition 1 without a specific formNo matter what the utility function of the enterprise is aslong as it satisfies 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0 and 119880101584010158401015840(Π) ge0 it is optimal to use options for hedging We can findmany utility functions that satisfy 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0and 119880101584010158401015840(Π) ge 0 such as the negative exponential utilityfunction and the quadratic utility function Many scholarshave established portfolio or futures hedgingmodel assumingthat the investorrsquos utility is a quadratic utility function Steil[30] obtained the hedging position of exchange rate equitywith quadratic utility Lien [31] studied futures hedging underthe negative exponential utility and the quadratic utilityBodnar [32] gave the strategies ofmultiperiod portfolio undernegative exponential utility and the quadratic utility Becauseof the complexity of the multistage dynamic programmingmethod under the negative exponential utility function itis difficult for us to obtain the explicit solutions We thenstudy the dynamic hedging problem in the framework of thequadratic utility
Proposition 2 Assume that exchange rate futures andexchange rate option markets are unbiased If the exportenterprisersquos utility function is the quadratic utility function119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimal positions ofexchange rate futures and options are
119883lowast = 1198601311986021 minus 1198601111986023119860213
minus 1198601211986023 119876
119884lowast = 1198601111986013 minus 1198601211986021119860213
minus 1198601211986023 119876(18)
where 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+)Proof Thefirst-order conditions of the objective function are
cov (Πlowast 120579) = 0 (19)
cov (Πlowast (minus120579)+) = 0 (20)
Mathematical Problems in Engineering 5
Due to the operational rules of covariance (19) and (20)can be rewritten as
Let 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+) then we have
11986011 minus 11986012119883lowast minus 11986013119884lowast = 011986021 minus 11986013119883lowast minus 11986023119884lowast = 0 (23)
By solving equations we can obtain the optimal positionsof the exchange rate futures and options presented in Propo-sition 1
Corollary 3 Suppose that 120579119905 has a symmetric distributionfunction of 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Let 119864(120579119905) = 0119864(1205792119905 ) = 1205902 119881 = int+infin
0120579119905119889119866(120579119905) and 120593 = int+infin
01205793119905 119889119866(120579119905)
In one-stage hedging the optimal positions of exchange ratefutures and options are
119883lowast = [119875 + 120573119878 + 120573 120593 minus 119881120590212059022 minus 21198812]119876
119884lowast = 2120573119876 120593 minus 119881120590212059022 minus 21198812
(24)
The study above is to establish the optimal hedgingmodelof exchange rate futures and options in one-stage whilein actual hedging practice the enterprise has to adjust thepositions dynamically according to the market conditionsso as to maximize its utility based on the final wealth Wethen extend the one-stage hedging problem to a multiperioddynamic case and derive the dynamic positions of exchangerate futures and options
4 Multistage Hedging Model withFutures and Options
Proposition 4 Suppose that the markets of exchange ratefutures and option are unbiased If the firmrsquos utility functionis quadratic of 119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimalpositions of the exchange rate futures and options at stage 119905 are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(25)
where 11986011119905 = (119875119905minus1 + 120573119905119878119905minus1) cov (120579119905 120579119905) + 120573119905 cov (1205792119905 120579119905)11986012119905 = cov (120579119905 120579119905) 11986013119905 = cov ((minus120579119905)+ 120579119905) 11986021119905 = (119875119905minus1 +120573119905119878119905minus1) cov (120579119905 (minus120579119905)+) + 120573119905 cov (1205792119905 (minus120579119905)+)11986022 = 11986013 =cov (1205792119905 (minus120579119905)+) and 11986023119905 = cov ((minus120579119905)+ (minus120579119905)+)Proof Let 119878119905 = 1198781119905minus1 +120579119905 119905 = 1198751119905minus1 +120573119905120579119905 +120576119905 and 119865119905 = 119878119905minus1Then we have the firmrsquos profit at stage 119905 isΠ119905 = 119905119878119905119876119879 + (119878119905minus1 minus 119878119905)119883119905 + [119881119905 minus (119878119905minus1 minus 119878119905)+] 119884119905
= (119875119905minus1 + 120573119905120579119905 + 120576119905) (119878119905minus1 + 120579119905)119876119905 minus 120579119905119883119905+ [119881119905 minus (minus120579119905)+] 119884119905
(26)
At the beginning of stage 119905 the firm decides the optimalpositions 119883lowast119905 119884lowast119905 to maximize its terminal utility Let 119879 =sum119879120591=1 Π120591 Then the optimal hedging model can be describedas follows
(1198752) max119883119905 119884119905
119864 [119880 (119879)] (27)
We use the dynamic programming method to solve theproblem (1198752) To begin with at stage 119879 the firm decides theoptimal position 119883lowast119879 119884lowast119879 to maximize its utility Let
and 120579119879minus1 is independent on 120579119879 and 120576119879 we havecov [119875119879minus1119878119879minus1 (119876119879minus1 + 119876119879) minus 120579119879minus1119883119879minus1
minus (minus120579119879minus1)+ 119884119879minus1 120579119879minus1] = 0 (36)
In thisway we can deduce the optimal positions of futuresand options on the basis of mathematical induction at stage 119905are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(39)
Corollary 5 Suppose that 120579119905 has a symmetric distributionfunction 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Assume the firmexport products to the foreignmarket at the terminal timeThatis 119876119905 = 0 (119905 = 1 2 119879 minus 1) 119876119879 = 119876 In the case ofmultistage the optimal positions of exchange rate futures andoptions are
119883lowast119905 = [119875119905minus1 + 120573119905119878119905minus1 + 120573119905 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905 ]119876
119884lowast119905 = 2120573119905119876 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905(40)
China has become a major producer and consumer of silveralso a big importer and exporter According to the accessibledata we assume there is a firm in China exports silver tothe US dollar region countries The firm intends to buy theexchange rate futures and options for hedging Data resourcesare fromWind database Figure 1 shows the daily yield of thesilver exported price and the exchange rate (RMBUS dollar)
Mathematical Problems in Engineering 7
Table 1 Basic statistical characteristics of returns
Objective Maximum Minimum Mean Median Standard Deviation Skewness KurtosisSilver price 01736 -01556 -00005 -00006 00199 -04517 146842CNYUSD 00287 -00176 00000 0 00021 18943 406029
Table 2 Normality test of returns
Objective P-value J-B statistics Critical valueSilver price 10000e-03 81429e+03 59490CNYUSD 10000e-03 84688e+04 59490Note 119886119890plusmn c denote that a times119890plusmn119888 the same after
From Figure 1 we can observe that the returns ofthe silver price and the exchange rate show cluster effect(that is large fluctuations are often accompanied by largefluctuations and small fluctuations are often accompaniedby small fluctuations) We further test the ARCH effectswherein the statistics of the returns are presented in Table 1
Table 1 gives the statistical description of ARCH effectstest results with regard to the silver price and the exchangerate We find that the returns of the silver price and theexchange rate CNYUSD have nonzero skewness of minus04517and 18943 respectively The returns also have peak thicktails Based on LM statistic the ARCH effects correspondingto the first 20 lags of the two exchange rate returns are furtherconfirmed (In order to save space the specific numericalvalue is omitted here) The main objective of this paper isto examine the hedging roles of the currency futures andoptions We further to test the nonnormal of the returns
Table 2 shows that the J-B statistic is larger than the criti-cal value at the 5 significance level and the P-value is smallerthan the significant level That is the hypothesis of normaldistribution under J-B test is rejected Six typical modelsof GARCH-n GARCH-t GIR-n GIR-t EGARCH-n andEGARCH-t are tested to depict the marginal distributions ofthe silver price and the exchange rate of CNYUSD Threefitting criteria measured by LLF (optimized log-likelihoodobjective function value) AIC (Akaike information criteria)and BIC (Bayesian information criteria) are used to test howthe outlined models fit the in-sample data The values of LLFAIC and BIC for different models are in Tables 3 and 4
Since the bigger the LLF the better is and the smaller theAIC and BIC the better is from Tables 3 and 4 we find thatmodels of GARCH-t model capture the fatness tails of theunconditional distributions both silver price and CNYUSDbetter Then we use GARCH-t model to fit the return seriesfor the returns of silver price and the exchange rateThe detaildescription of GARCH-t model can be referred to Huang etal [33] if the yield sequence 119903119905 has the forms
where 119905(119889) obeys Student-t distribution with degree offreedom 119889 K is the conditional variance constant GARCHmeans the coefficients related to lagged conditional variancesARCH is the coefficients related to lagged innovations (resid-uals) and C denotes the conditional mean constant Themethod to estimate for the parameters could be artificialalgorithm In this paper we use MLE method to estimatethe parameters in GARCH model We let the informationset Ωtminus1 = 1198860 1198861 119886119905minus1 The joint density function canthen be written as 119891(1198860 1198861 119886119905) = 119891(119886119905 | Ω119905minus1)119891(119886119905minus1 |Ω119905minus2) sdot sdot sdot 119891(1198861 | Ω0)119891(1198860) Given data 1198861 119886119905 the log-likelihood is as follows
This can be evaluated using the model volatility equationfor any assumed distribution for the error term Here119871119871119865 can be maximized numerically to obtain 119872119871119864 Theestimation results can be seen in Table 5
Table 5 shows the maximum likelihood estimation ofparameters the validity of ARCH and Ljungbox tests forGARCH-t models and the fitted values of AIC (Akaikeinformation criterion) and BIC (Bayesian information cri-terion) It can be seen from Table 5 that the Ljung-Boxtest for GARCH-t model residuals at 5 confidence levelcannot reject the original hypothesis of order 1 3 5 and 7autocorrelation and the original hypothesis of ARCH effectin orders 4 6 8 and 10 cannot be rejected The values ofAIC and BIC are smaller but the LLF is larger According tothe P-value of parameter estimation it shows that GARCH-t model can fit the residual sequence of silver price and theexchange rate well We then give the expression of GARCH-t distributions for the silver price and the exchange rate ofCNYUSD as follows
119903119905 = minus52123119890 minus 04 minus 92936119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 32615119890 minus 06 + 961261205902119905minus1 + 27763119890 minus 021205762119905minus1119911119905 sim 119905 (49082)
(43)
119903119905 = minus37917119890 minus 05 + 18163119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 20000119890 minus 07 + 73552119890 minus 011205902119905minus1 + 26448119890minus 01119890 minus 021205762119905minus1
119911119905 sim 119905 (34282)
(44)
where (43) and (44) are the expressions for the silver priceand the exchange rate respectively
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
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2 Mathematical Problems in Engineering
commodity futures or foreign exchange futures exist) thedependence structure of price and exchange rate risk BrollandWong [5] investigated the exchange rate hedging problemin that a competitive exporter exported products to twocountries They established a hedging model of exchangerate futures to study the impact of exchange rate changeson the export decisions Exchange rate futures hedgingrelated researches can refer to Lence [6] Lien and Wong[7] Broll et al [8] Wong [9] and so on As for exchangerate options hedging Machnes [10] studied the decision-making problems of the competitive companies using com-modity options to hedge price uncertainty Wong [11] madea contrastive study for an export firm with regard to therisk and found that the variance of options hedging waslower than that of no options hedging Lien and Wong[12] derived an optimal hedging strategy for the risk aversehedgers under delivery risk through futures options hedgingTheir study thus provides a theoretical basis for optionshedging with multiple delivery prices Since both exchangerate futures and options can be used as risk hedging toolssome scholars have conducted comparative studies on thehedging effects between the two derivatives Battermann etal [13] assumed that the export price was known enterprisesonly needed to deal with the random exchange rate riskThey compared the hedging effects of exchange rate futuresand exchange rate options and showed that exchange ratefutures havemore advantages than the exchange rate optionsBut it does not mean to say that options are useless Somescholars point out that futures have an advantage in hedginglinear risk and options are more suitable to hedge nonlinearrisk (Bajo [14 15] and Wong [16]) As we all know linearrisk and nonlinear risk are generally prevailing in the realinvestment Lapan et al [17] established the classical LMH(Lapan Moschini and Hanson) model and pointed out thatoptions played a role due to the convexity utility functionFrechette [18] presented various optimal hedging portfoliomodels including futures and options when the marginalcost of hedging was nonzero Wong [19] studied the optimalhedging and decision-making problem of the competitiveexport enterprises that faced the exchange rate risk Theyproved that there were two different sources of nonlinearrisk wherein one was the multiplicative property of theprice and exchange rate and another was the nonlinearmarginal utility function Wong [20] examined the problemwherein state-dependent preference and commodity futuresand options hedging were considered for the competitivefirms There was only a linear risk stemming from stochasticprice but a nonlinear relationship between preference andthe state made the enterprises face nonlinear risk For thestudies of joint hedge with exchange rate futures and optionssee Sakong et al [21] Moschini and Lapan [22] Despiteliteratures mentioned above have studied futures hedgingoptions hedging and futures and options joint hedging thesituation they assume is static While in actual hedgingpractice investors have to adjust the positions of futuresand options dynamically to minimize risks or maximizereturns
Dynamic hedge is what needs to adjust the strategies asthe price or other characteristics of the portfolio or security
changes Moreover risks of some securities cannot be hedgedwith static positions For example option price is not linearwith the underlying assetrsquos price which means that risk ofoption can only be hedged dynamically There are manyresearches on dynamic futures hedging The used methodsmainly involve time-varying GARCH models and dynamicprogramming method Kotkatvuorirnberg [23] used CopulaDCC-EGARCHmodel to estimate bivariate error correctionand study the effectiveness of currency futures hedgingZhang et al [24] employedGARCH-Copulamodels to exam-ine the options hedging problem Dynamic programmingmethod is another widely used method to solve dynamicinvestment decisions Change and Wong [25] developed aquadratic utility model of a multinational firm that facedexchange rate risk exposure to a foreign currency cash flowChi et al [26] analyzed the hedger positionrsquos value alterationand used the dynamic programming method to set up themultiperiod futures dynamic hedging optimal model Li etal [27] constructed the minimum variance model for theestimation of the optimal hedge ratio based on the stochas-tic differential equation However the application of thedynamic programming method in risk hedging especially inoptions hedging is rare
To sum up this paper takes account of the following threeaspects First with the development of economic globaliza-tion trade across countries is increasingly frequent How touse financial derivatives to manage exchange rate risk is apractical problem Previous studies have shown that futuresand options have advantages in hedging linear and nonlinearrisks respectively and in the actual hedging practice linearrisk and nonlinear risk coexist generallyTherefore this paperestablishes a hedging model of exchange rate futures andoptions to hedge the exchange rate risk Second a largenumber of literatures have studied the dynamic hedgingof futures but few study the dynamic hedging model withoptions In this paper a dynamic hedging model of exchangerate futures andoptions is constructed by combiningGARCHmodel with dynamic programming method Third we verifythe effects of futures hedging option hedging and futures andoption combined hedging and then further to provide riskmanagement ways for the import and export enterprises toavoid foreign exchange risk
The rest of this paper is organized as follows In Section 2assumptions and notations are presented Section 3 providesthe one-stage hedging model with exchange rate options andfutures We show that for the prudent firm it is necessaryto buy options for hedging Under the quadratic utilitywe demonstrate the explicit positions of the exchange ratefutures and options Section 4 studies the multistage hedgingproblem We deduce the optimal positions of the exchangerate futures and options by using dynamic programmingmethod An empirical analysis illustrated the hedging effec-tiveness in Section 4The performance of the derived hedgingstrategy is compared in three cases of hedging with futuresand options only with futures and without hedging Thecomparison is performed in terms of the terminal wealththe terminal wealth based on utility and also the vari-ance of wealth accumulation path Section 5 concludes thepaper
Mathematical Problems in Engineering 3
2 Assumptions and Notations
Suppose a competitive export enterprise exports a singleproduct to the foreignmarket It may obtain a predeterminedforeign order in advance That is demand or output is givenas 119876 We consider the price risk and exchange rate risk facedby the enterprise The price of the product sold abroad is in foreign currency The exchange rate is 119878 Referring to thestudy of Lapan and Moschini [22] we suppose that
= 120572 + 120573119878 + 120576 120573 lt 0 (1)
where 120576 is a zero mean and independent random variablewith 119878 Furthermore according to Chang andWong [25] weassume that
119878 = 119878 + 120579 (2)
Here 119864(119878) = 119878The imported or exported products in China involve a
wide range such as clothing toys and electromechanicaland pharmaceutical products while the derivative marketscorresponding to these small commodities usually do notexist Generally speaking exchange rate derivatives are com-mon Since the relationship between commodity price andexchange rate is assumed to be shown in (1) ie hedgingexchange rate risk also eliminates some commodity price risksynchronously we assume that the exporter uses divisible andtradable exchange rate rather than commodity futures andoption for hedging Suppose that the futures price is 119865 atthe current time For the sake of simplicity we also assumethat the strike price of the exchange rate option is 119870 and thecorresponding premium is recorded as 119881 Since Wong [16]points out that the assumption of futures and optionsmarketswhich are unbiased is because firms aim to hedge rather thanarbitrage futures and options markets are assumed to beunbiased for studying the pure hedging roles of exchange ratefutures and options ie 119864(119878) = 119865 and 119864(119870 minus 119878)+ = 1198813 One-Stage Hedging Model with ExchangeRate Options and Futures
To begin with we consider one-stage static hedging problemwith exchange rate options and futures The firm determinesthe exchange rate futures position 119883 and the exchange rateoption position 119884 at 0 (current time) to maximize its utilityAccording to the assumptions above the enterprisersquos profit attime 1 is
Π = 119878119876 + (119865 minus 119878)119883 + [119881 minus (119870 minus 119878)+] 119884 (3)
where 119883 and 119884 are the exchange rate futures and optionspositions that are sold (negative when be bought) and (119870 minus119878)+ = max (119870 minus 119878 0)
The firmrsquos utility function is 119880(Π) Assume that thedecision-maker of the firm is risk averse then1198801015840(Π) gt 0 and11988010158401015840(Π) lt 0 Based on the discussion above the problem facedby the firm can be written as (1198751)
max119883119884
119864 [119880 (Π)] (4)
where Π is described in (3)
Proposition 1 Suppose the exchange rate futures market andoption market are unbiased When the risk aversion exportenterprisersquos utility function satisfies 119880101584010158401015840(Π) ge 0 then it isoptimal for the enterprise to buy options for hedging
Proof Since the enterprise is risk aversion then its utilityfunction satisfies 1198801015840(Π) gt 0 and 1198801015840(Π) lt 0 Thereforethe relation of the optimal positions of futures and options119883lowast 119884lowast is map1-to-1 ie119883lowast = 119883(119884lowast)We apply the two-stageoptimizationmethod to prove Proposition 1 In the first stagelet
119867(119884) = arg max119883
119864 [119880 (Π)] (5)
When 119884 = 0 based on (5) we have
119864 [1198801015840 (Π0) (119865 minus 119878)] = 0 (6)
According to the assumption of unbiased market (6) canbe written as follows
That is 119864[1198801015840(Π0) | 119878] is concave with regard to 119878 On theother hand since 119864119864[1198801015840(Π0) | 119878] = 119864[1198801015840(Π0)] then theequation of
has at least one solution and at most two solutionsWhen (11) has only one solution 1198781 isin (119878 119878) then (7) can
be expressed by
119864 1198801015840 (Π0 |) 119878 minus 119864 [1198801015840 (Π0)] [119864 (119878) minus 119878] = 0 (12)
4 Mathematical Problems in Engineering
Equation (12) is further expressed in an integral form
int119878119878
119864 [1198801015840 (Π0) | 119878] minus 119864 [1198801015840 (Π0)] (119891 minus 119878) 119891 (119904) 119889119904= 0
(13)
If 119878 isin (119878 1198781) the first term of the integrand in (13) isnegative and the second is positive When 119878 isin (1198781 119878) thefirst term of the integrand in (13) is positive and the secondis negative Therefore the left hand of (13) is negative whichleads a contradictory to (13)
If (11) has two solutions ie 1198781 1198782 119878 lt 1198781 lt 1198782 lt 119878 thenwhen 119878 = 1198781 and 119878 = 1198782 we have
10038161003816100381610038161003816100381610038161003816100381610038161003816119884=0= 119864 1198801015840 (Π0) [119881 minus (119870 minus 119878)+]
= cov (119864 [1198801015840 (Π0) | 119878] 119881 minus (119870 minus 119878)+)= 119864 [1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
= int119878119878
[1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
then proof that 119877(119870) lt 0 according to the functional featuresof 119877(119870) In fact it is evident that 119877(119878) = 0 From (6) we have119877(119878) = 0 Since
then when 119878 isin (1198781 1198782) we have 11987710158401015840(119870) gt 0 Similarly weobtain that when 119878 isin (119878 1198781) cup (1198782 119878) we have 11987710158401015840(119870) lt 0Based on the discussion above we have 119877(119870) lt 0 That is(120597119864[119880(Π)]120597119884)|119884=0 lt 0 and 119884lowast lt 0
Referring to the definition in Kim [28] the risk aversioninvestor whose utility function satisfies 119880101584010158401015840(Π) ge 0 is calleda prudent investor The results in Proposition 1 show thatit is wise for the prudent investor to buy unbiased optionsfor hedging which is consistent with the real intention ofenterprises who adopt hedging with the purpose of apprecia-tion and preservationWe know that although selling options
can obtain option premiums investors have to face the dailymarking risk caused by the additional margin which maylead to greater liquidity risk for investors Jorion [29] dividedliquidity risk into asset liquidity risk and capital liquidity riskHe pointed out that the capital liquidity risk is the potentiallyfatal risks faced by investors For option sellers the liquidityrisk they face mainly includes capital liquidity risk As afailure case of hedging China SouthernAirlines desire to lockin the cost of raw materials by buying call options whichcan be regarded as a hedge but selling put options based onbull market judgment creates a new margin risk exposureTherefore the conclusion of Proposition 1 coincides withthe original intention of options hedging to preserve andincrease appreciation We note the utility function of theprudent enterprise in Proposition 1 without a specific formNo matter what the utility function of the enterprise is aslong as it satisfies 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0 and 119880101584010158401015840(Π) ge0 it is optimal to use options for hedging We can findmany utility functions that satisfy 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0and 119880101584010158401015840(Π) ge 0 such as the negative exponential utilityfunction and the quadratic utility function Many scholarshave established portfolio or futures hedgingmodel assumingthat the investorrsquos utility is a quadratic utility function Steil[30] obtained the hedging position of exchange rate equitywith quadratic utility Lien [31] studied futures hedging underthe negative exponential utility and the quadratic utilityBodnar [32] gave the strategies ofmultiperiod portfolio undernegative exponential utility and the quadratic utility Becauseof the complexity of the multistage dynamic programmingmethod under the negative exponential utility function itis difficult for us to obtain the explicit solutions We thenstudy the dynamic hedging problem in the framework of thequadratic utility
Proposition 2 Assume that exchange rate futures andexchange rate option markets are unbiased If the exportenterprisersquos utility function is the quadratic utility function119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimal positions ofexchange rate futures and options are
119883lowast = 1198601311986021 minus 1198601111986023119860213
minus 1198601211986023 119876
119884lowast = 1198601111986013 minus 1198601211986021119860213
minus 1198601211986023 119876(18)
where 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+)Proof Thefirst-order conditions of the objective function are
cov (Πlowast 120579) = 0 (19)
cov (Πlowast (minus120579)+) = 0 (20)
Mathematical Problems in Engineering 5
Due to the operational rules of covariance (19) and (20)can be rewritten as
Let 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+) then we have
11986011 minus 11986012119883lowast minus 11986013119884lowast = 011986021 minus 11986013119883lowast minus 11986023119884lowast = 0 (23)
By solving equations we can obtain the optimal positionsof the exchange rate futures and options presented in Propo-sition 1
Corollary 3 Suppose that 120579119905 has a symmetric distributionfunction of 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Let 119864(120579119905) = 0119864(1205792119905 ) = 1205902 119881 = int+infin
0120579119905119889119866(120579119905) and 120593 = int+infin
01205793119905 119889119866(120579119905)
In one-stage hedging the optimal positions of exchange ratefutures and options are
119883lowast = [119875 + 120573119878 + 120573 120593 minus 119881120590212059022 minus 21198812]119876
119884lowast = 2120573119876 120593 minus 119881120590212059022 minus 21198812
(24)
The study above is to establish the optimal hedgingmodelof exchange rate futures and options in one-stage whilein actual hedging practice the enterprise has to adjust thepositions dynamically according to the market conditionsso as to maximize its utility based on the final wealth Wethen extend the one-stage hedging problem to a multiperioddynamic case and derive the dynamic positions of exchangerate futures and options
4 Multistage Hedging Model withFutures and Options
Proposition 4 Suppose that the markets of exchange ratefutures and option are unbiased If the firmrsquos utility functionis quadratic of 119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimalpositions of the exchange rate futures and options at stage 119905 are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(25)
where 11986011119905 = (119875119905minus1 + 120573119905119878119905minus1) cov (120579119905 120579119905) + 120573119905 cov (1205792119905 120579119905)11986012119905 = cov (120579119905 120579119905) 11986013119905 = cov ((minus120579119905)+ 120579119905) 11986021119905 = (119875119905minus1 +120573119905119878119905minus1) cov (120579119905 (minus120579119905)+) + 120573119905 cov (1205792119905 (minus120579119905)+)11986022 = 11986013 =cov (1205792119905 (minus120579119905)+) and 11986023119905 = cov ((minus120579119905)+ (minus120579119905)+)Proof Let 119878119905 = 1198781119905minus1 +120579119905 119905 = 1198751119905minus1 +120573119905120579119905 +120576119905 and 119865119905 = 119878119905minus1Then we have the firmrsquos profit at stage 119905 isΠ119905 = 119905119878119905119876119879 + (119878119905minus1 minus 119878119905)119883119905 + [119881119905 minus (119878119905minus1 minus 119878119905)+] 119884119905
= (119875119905minus1 + 120573119905120579119905 + 120576119905) (119878119905minus1 + 120579119905)119876119905 minus 120579119905119883119905+ [119881119905 minus (minus120579119905)+] 119884119905
(26)
At the beginning of stage 119905 the firm decides the optimalpositions 119883lowast119905 119884lowast119905 to maximize its terminal utility Let 119879 =sum119879120591=1 Π120591 Then the optimal hedging model can be describedas follows
(1198752) max119883119905 119884119905
119864 [119880 (119879)] (27)
We use the dynamic programming method to solve theproblem (1198752) To begin with at stage 119879 the firm decides theoptimal position 119883lowast119879 119884lowast119879 to maximize its utility Let
and 120579119879minus1 is independent on 120579119879 and 120576119879 we havecov [119875119879minus1119878119879minus1 (119876119879minus1 + 119876119879) minus 120579119879minus1119883119879minus1
minus (minus120579119879minus1)+ 119884119879minus1 120579119879minus1] = 0 (36)
In thisway we can deduce the optimal positions of futuresand options on the basis of mathematical induction at stage 119905are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(39)
Corollary 5 Suppose that 120579119905 has a symmetric distributionfunction 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Assume the firmexport products to the foreignmarket at the terminal timeThatis 119876119905 = 0 (119905 = 1 2 119879 minus 1) 119876119879 = 119876 In the case ofmultistage the optimal positions of exchange rate futures andoptions are
119883lowast119905 = [119875119905minus1 + 120573119905119878119905minus1 + 120573119905 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905 ]119876
119884lowast119905 = 2120573119905119876 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905(40)
China has become a major producer and consumer of silveralso a big importer and exporter According to the accessibledata we assume there is a firm in China exports silver tothe US dollar region countries The firm intends to buy theexchange rate futures and options for hedging Data resourcesare fromWind database Figure 1 shows the daily yield of thesilver exported price and the exchange rate (RMBUS dollar)
Mathematical Problems in Engineering 7
Table 1 Basic statistical characteristics of returns
Objective Maximum Minimum Mean Median Standard Deviation Skewness KurtosisSilver price 01736 -01556 -00005 -00006 00199 -04517 146842CNYUSD 00287 -00176 00000 0 00021 18943 406029
Table 2 Normality test of returns
Objective P-value J-B statistics Critical valueSilver price 10000e-03 81429e+03 59490CNYUSD 10000e-03 84688e+04 59490Note 119886119890plusmn c denote that a times119890plusmn119888 the same after
From Figure 1 we can observe that the returns ofthe silver price and the exchange rate show cluster effect(that is large fluctuations are often accompanied by largefluctuations and small fluctuations are often accompaniedby small fluctuations) We further test the ARCH effectswherein the statistics of the returns are presented in Table 1
Table 1 gives the statistical description of ARCH effectstest results with regard to the silver price and the exchangerate We find that the returns of the silver price and theexchange rate CNYUSD have nonzero skewness of minus04517and 18943 respectively The returns also have peak thicktails Based on LM statistic the ARCH effects correspondingto the first 20 lags of the two exchange rate returns are furtherconfirmed (In order to save space the specific numericalvalue is omitted here) The main objective of this paper isto examine the hedging roles of the currency futures andoptions We further to test the nonnormal of the returns
Table 2 shows that the J-B statistic is larger than the criti-cal value at the 5 significance level and the P-value is smallerthan the significant level That is the hypothesis of normaldistribution under J-B test is rejected Six typical modelsof GARCH-n GARCH-t GIR-n GIR-t EGARCH-n andEGARCH-t are tested to depict the marginal distributions ofthe silver price and the exchange rate of CNYUSD Threefitting criteria measured by LLF (optimized log-likelihoodobjective function value) AIC (Akaike information criteria)and BIC (Bayesian information criteria) are used to test howthe outlined models fit the in-sample data The values of LLFAIC and BIC for different models are in Tables 3 and 4
Since the bigger the LLF the better is and the smaller theAIC and BIC the better is from Tables 3 and 4 we find thatmodels of GARCH-t model capture the fatness tails of theunconditional distributions both silver price and CNYUSDbetter Then we use GARCH-t model to fit the return seriesfor the returns of silver price and the exchange rateThe detaildescription of GARCH-t model can be referred to Huang etal [33] if the yield sequence 119903119905 has the forms
where 119905(119889) obeys Student-t distribution with degree offreedom 119889 K is the conditional variance constant GARCHmeans the coefficients related to lagged conditional variancesARCH is the coefficients related to lagged innovations (resid-uals) and C denotes the conditional mean constant Themethod to estimate for the parameters could be artificialalgorithm In this paper we use MLE method to estimatethe parameters in GARCH model We let the informationset Ωtminus1 = 1198860 1198861 119886119905minus1 The joint density function canthen be written as 119891(1198860 1198861 119886119905) = 119891(119886119905 | Ω119905minus1)119891(119886119905minus1 |Ω119905minus2) sdot sdot sdot 119891(1198861 | Ω0)119891(1198860) Given data 1198861 119886119905 the log-likelihood is as follows
This can be evaluated using the model volatility equationfor any assumed distribution for the error term Here119871119871119865 can be maximized numerically to obtain 119872119871119864 Theestimation results can be seen in Table 5
Table 5 shows the maximum likelihood estimation ofparameters the validity of ARCH and Ljungbox tests forGARCH-t models and the fitted values of AIC (Akaikeinformation criterion) and BIC (Bayesian information cri-terion) It can be seen from Table 5 that the Ljung-Boxtest for GARCH-t model residuals at 5 confidence levelcannot reject the original hypothesis of order 1 3 5 and 7autocorrelation and the original hypothesis of ARCH effectin orders 4 6 8 and 10 cannot be rejected The values ofAIC and BIC are smaller but the LLF is larger According tothe P-value of parameter estimation it shows that GARCH-t model can fit the residual sequence of silver price and theexchange rate well We then give the expression of GARCH-t distributions for the silver price and the exchange rate ofCNYUSD as follows
119903119905 = minus52123119890 minus 04 minus 92936119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 32615119890 minus 06 + 961261205902119905minus1 + 27763119890 minus 021205762119905minus1119911119905 sim 119905 (49082)
(43)
119903119905 = minus37917119890 minus 05 + 18163119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 20000119890 minus 07 + 73552119890 minus 011205902119905minus1 + 26448119890minus 01119890 minus 021205762119905minus1
119911119905 sim 119905 (34282)
(44)
where (43) and (44) are the expressions for the silver priceand the exchange rate respectively
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
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Mathematical Problems in Engineering 3
2 Assumptions and Notations
Suppose a competitive export enterprise exports a singleproduct to the foreignmarket It may obtain a predeterminedforeign order in advance That is demand or output is givenas 119876 We consider the price risk and exchange rate risk facedby the enterprise The price of the product sold abroad is in foreign currency The exchange rate is 119878 Referring to thestudy of Lapan and Moschini [22] we suppose that
= 120572 + 120573119878 + 120576 120573 lt 0 (1)
where 120576 is a zero mean and independent random variablewith 119878 Furthermore according to Chang andWong [25] weassume that
119878 = 119878 + 120579 (2)
Here 119864(119878) = 119878The imported or exported products in China involve a
wide range such as clothing toys and electromechanicaland pharmaceutical products while the derivative marketscorresponding to these small commodities usually do notexist Generally speaking exchange rate derivatives are com-mon Since the relationship between commodity price andexchange rate is assumed to be shown in (1) ie hedgingexchange rate risk also eliminates some commodity price risksynchronously we assume that the exporter uses divisible andtradable exchange rate rather than commodity futures andoption for hedging Suppose that the futures price is 119865 atthe current time For the sake of simplicity we also assumethat the strike price of the exchange rate option is 119870 and thecorresponding premium is recorded as 119881 Since Wong [16]points out that the assumption of futures and optionsmarketswhich are unbiased is because firms aim to hedge rather thanarbitrage futures and options markets are assumed to beunbiased for studying the pure hedging roles of exchange ratefutures and options ie 119864(119878) = 119865 and 119864(119870 minus 119878)+ = 1198813 One-Stage Hedging Model with ExchangeRate Options and Futures
To begin with we consider one-stage static hedging problemwith exchange rate options and futures The firm determinesthe exchange rate futures position 119883 and the exchange rateoption position 119884 at 0 (current time) to maximize its utilityAccording to the assumptions above the enterprisersquos profit attime 1 is
Π = 119878119876 + (119865 minus 119878)119883 + [119881 minus (119870 minus 119878)+] 119884 (3)
where 119883 and 119884 are the exchange rate futures and optionspositions that are sold (negative when be bought) and (119870 minus119878)+ = max (119870 minus 119878 0)
The firmrsquos utility function is 119880(Π) Assume that thedecision-maker of the firm is risk averse then1198801015840(Π) gt 0 and11988010158401015840(Π) lt 0 Based on the discussion above the problem facedby the firm can be written as (1198751)
max119883119884
119864 [119880 (Π)] (4)
where Π is described in (3)
Proposition 1 Suppose the exchange rate futures market andoption market are unbiased When the risk aversion exportenterprisersquos utility function satisfies 119880101584010158401015840(Π) ge 0 then it isoptimal for the enterprise to buy options for hedging
Proof Since the enterprise is risk aversion then its utilityfunction satisfies 1198801015840(Π) gt 0 and 1198801015840(Π) lt 0 Thereforethe relation of the optimal positions of futures and options119883lowast 119884lowast is map1-to-1 ie119883lowast = 119883(119884lowast)We apply the two-stageoptimizationmethod to prove Proposition 1 In the first stagelet
119867(119884) = arg max119883
119864 [119880 (Π)] (5)
When 119884 = 0 based on (5) we have
119864 [1198801015840 (Π0) (119865 minus 119878)] = 0 (6)
According to the assumption of unbiased market (6) canbe written as follows
That is 119864[1198801015840(Π0) | 119878] is concave with regard to 119878 On theother hand since 119864119864[1198801015840(Π0) | 119878] = 119864[1198801015840(Π0)] then theequation of
has at least one solution and at most two solutionsWhen (11) has only one solution 1198781 isin (119878 119878) then (7) can
be expressed by
119864 1198801015840 (Π0 |) 119878 minus 119864 [1198801015840 (Π0)] [119864 (119878) minus 119878] = 0 (12)
4 Mathematical Problems in Engineering
Equation (12) is further expressed in an integral form
int119878119878
119864 [1198801015840 (Π0) | 119878] minus 119864 [1198801015840 (Π0)] (119891 minus 119878) 119891 (119904) 119889119904= 0
(13)
If 119878 isin (119878 1198781) the first term of the integrand in (13) isnegative and the second is positive When 119878 isin (1198781 119878) thefirst term of the integrand in (13) is positive and the secondis negative Therefore the left hand of (13) is negative whichleads a contradictory to (13)
If (11) has two solutions ie 1198781 1198782 119878 lt 1198781 lt 1198782 lt 119878 thenwhen 119878 = 1198781 and 119878 = 1198782 we have
10038161003816100381610038161003816100381610038161003816100381610038161003816119884=0= 119864 1198801015840 (Π0) [119881 minus (119870 minus 119878)+]
= cov (119864 [1198801015840 (Π0) | 119878] 119881 minus (119870 minus 119878)+)= 119864 [1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
= int119878119878
[1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
then proof that 119877(119870) lt 0 according to the functional featuresof 119877(119870) In fact it is evident that 119877(119878) = 0 From (6) we have119877(119878) = 0 Since
then when 119878 isin (1198781 1198782) we have 11987710158401015840(119870) gt 0 Similarly weobtain that when 119878 isin (119878 1198781) cup (1198782 119878) we have 11987710158401015840(119870) lt 0Based on the discussion above we have 119877(119870) lt 0 That is(120597119864[119880(Π)]120597119884)|119884=0 lt 0 and 119884lowast lt 0
Referring to the definition in Kim [28] the risk aversioninvestor whose utility function satisfies 119880101584010158401015840(Π) ge 0 is calleda prudent investor The results in Proposition 1 show thatit is wise for the prudent investor to buy unbiased optionsfor hedging which is consistent with the real intention ofenterprises who adopt hedging with the purpose of apprecia-tion and preservationWe know that although selling options
can obtain option premiums investors have to face the dailymarking risk caused by the additional margin which maylead to greater liquidity risk for investors Jorion [29] dividedliquidity risk into asset liquidity risk and capital liquidity riskHe pointed out that the capital liquidity risk is the potentiallyfatal risks faced by investors For option sellers the liquidityrisk they face mainly includes capital liquidity risk As afailure case of hedging China SouthernAirlines desire to lockin the cost of raw materials by buying call options whichcan be regarded as a hedge but selling put options based onbull market judgment creates a new margin risk exposureTherefore the conclusion of Proposition 1 coincides withthe original intention of options hedging to preserve andincrease appreciation We note the utility function of theprudent enterprise in Proposition 1 without a specific formNo matter what the utility function of the enterprise is aslong as it satisfies 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0 and 119880101584010158401015840(Π) ge0 it is optimal to use options for hedging We can findmany utility functions that satisfy 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0and 119880101584010158401015840(Π) ge 0 such as the negative exponential utilityfunction and the quadratic utility function Many scholarshave established portfolio or futures hedgingmodel assumingthat the investorrsquos utility is a quadratic utility function Steil[30] obtained the hedging position of exchange rate equitywith quadratic utility Lien [31] studied futures hedging underthe negative exponential utility and the quadratic utilityBodnar [32] gave the strategies ofmultiperiod portfolio undernegative exponential utility and the quadratic utility Becauseof the complexity of the multistage dynamic programmingmethod under the negative exponential utility function itis difficult for us to obtain the explicit solutions We thenstudy the dynamic hedging problem in the framework of thequadratic utility
Proposition 2 Assume that exchange rate futures andexchange rate option markets are unbiased If the exportenterprisersquos utility function is the quadratic utility function119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimal positions ofexchange rate futures and options are
119883lowast = 1198601311986021 minus 1198601111986023119860213
minus 1198601211986023 119876
119884lowast = 1198601111986013 minus 1198601211986021119860213
minus 1198601211986023 119876(18)
where 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+)Proof Thefirst-order conditions of the objective function are
cov (Πlowast 120579) = 0 (19)
cov (Πlowast (minus120579)+) = 0 (20)
Mathematical Problems in Engineering 5
Due to the operational rules of covariance (19) and (20)can be rewritten as
Let 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+) then we have
11986011 minus 11986012119883lowast minus 11986013119884lowast = 011986021 minus 11986013119883lowast minus 11986023119884lowast = 0 (23)
By solving equations we can obtain the optimal positionsof the exchange rate futures and options presented in Propo-sition 1
Corollary 3 Suppose that 120579119905 has a symmetric distributionfunction of 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Let 119864(120579119905) = 0119864(1205792119905 ) = 1205902 119881 = int+infin
0120579119905119889119866(120579119905) and 120593 = int+infin
01205793119905 119889119866(120579119905)
In one-stage hedging the optimal positions of exchange ratefutures and options are
119883lowast = [119875 + 120573119878 + 120573 120593 minus 119881120590212059022 minus 21198812]119876
119884lowast = 2120573119876 120593 minus 119881120590212059022 minus 21198812
(24)
The study above is to establish the optimal hedgingmodelof exchange rate futures and options in one-stage whilein actual hedging practice the enterprise has to adjust thepositions dynamically according to the market conditionsso as to maximize its utility based on the final wealth Wethen extend the one-stage hedging problem to a multiperioddynamic case and derive the dynamic positions of exchangerate futures and options
4 Multistage Hedging Model withFutures and Options
Proposition 4 Suppose that the markets of exchange ratefutures and option are unbiased If the firmrsquos utility functionis quadratic of 119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimalpositions of the exchange rate futures and options at stage 119905 are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(25)
where 11986011119905 = (119875119905minus1 + 120573119905119878119905minus1) cov (120579119905 120579119905) + 120573119905 cov (1205792119905 120579119905)11986012119905 = cov (120579119905 120579119905) 11986013119905 = cov ((minus120579119905)+ 120579119905) 11986021119905 = (119875119905minus1 +120573119905119878119905minus1) cov (120579119905 (minus120579119905)+) + 120573119905 cov (1205792119905 (minus120579119905)+)11986022 = 11986013 =cov (1205792119905 (minus120579119905)+) and 11986023119905 = cov ((minus120579119905)+ (minus120579119905)+)Proof Let 119878119905 = 1198781119905minus1 +120579119905 119905 = 1198751119905minus1 +120573119905120579119905 +120576119905 and 119865119905 = 119878119905minus1Then we have the firmrsquos profit at stage 119905 isΠ119905 = 119905119878119905119876119879 + (119878119905minus1 minus 119878119905)119883119905 + [119881119905 minus (119878119905minus1 minus 119878119905)+] 119884119905
= (119875119905minus1 + 120573119905120579119905 + 120576119905) (119878119905minus1 + 120579119905)119876119905 minus 120579119905119883119905+ [119881119905 minus (minus120579119905)+] 119884119905
(26)
At the beginning of stage 119905 the firm decides the optimalpositions 119883lowast119905 119884lowast119905 to maximize its terminal utility Let 119879 =sum119879120591=1 Π120591 Then the optimal hedging model can be describedas follows
(1198752) max119883119905 119884119905
119864 [119880 (119879)] (27)
We use the dynamic programming method to solve theproblem (1198752) To begin with at stage 119879 the firm decides theoptimal position 119883lowast119879 119884lowast119879 to maximize its utility Let
and 120579119879minus1 is independent on 120579119879 and 120576119879 we havecov [119875119879minus1119878119879minus1 (119876119879minus1 + 119876119879) minus 120579119879minus1119883119879minus1
minus (minus120579119879minus1)+ 119884119879minus1 120579119879minus1] = 0 (36)
In thisway we can deduce the optimal positions of futuresand options on the basis of mathematical induction at stage 119905are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(39)
Corollary 5 Suppose that 120579119905 has a symmetric distributionfunction 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Assume the firmexport products to the foreignmarket at the terminal timeThatis 119876119905 = 0 (119905 = 1 2 119879 minus 1) 119876119879 = 119876 In the case ofmultistage the optimal positions of exchange rate futures andoptions are
119883lowast119905 = [119875119905minus1 + 120573119905119878119905minus1 + 120573119905 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905 ]119876
119884lowast119905 = 2120573119905119876 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905(40)
China has become a major producer and consumer of silveralso a big importer and exporter According to the accessibledata we assume there is a firm in China exports silver tothe US dollar region countries The firm intends to buy theexchange rate futures and options for hedging Data resourcesare fromWind database Figure 1 shows the daily yield of thesilver exported price and the exchange rate (RMBUS dollar)
Mathematical Problems in Engineering 7
Table 1 Basic statistical characteristics of returns
Objective Maximum Minimum Mean Median Standard Deviation Skewness KurtosisSilver price 01736 -01556 -00005 -00006 00199 -04517 146842CNYUSD 00287 -00176 00000 0 00021 18943 406029
Table 2 Normality test of returns
Objective P-value J-B statistics Critical valueSilver price 10000e-03 81429e+03 59490CNYUSD 10000e-03 84688e+04 59490Note 119886119890plusmn c denote that a times119890plusmn119888 the same after
From Figure 1 we can observe that the returns ofthe silver price and the exchange rate show cluster effect(that is large fluctuations are often accompanied by largefluctuations and small fluctuations are often accompaniedby small fluctuations) We further test the ARCH effectswherein the statistics of the returns are presented in Table 1
Table 1 gives the statistical description of ARCH effectstest results with regard to the silver price and the exchangerate We find that the returns of the silver price and theexchange rate CNYUSD have nonzero skewness of minus04517and 18943 respectively The returns also have peak thicktails Based on LM statistic the ARCH effects correspondingto the first 20 lags of the two exchange rate returns are furtherconfirmed (In order to save space the specific numericalvalue is omitted here) The main objective of this paper isto examine the hedging roles of the currency futures andoptions We further to test the nonnormal of the returns
Table 2 shows that the J-B statistic is larger than the criti-cal value at the 5 significance level and the P-value is smallerthan the significant level That is the hypothesis of normaldistribution under J-B test is rejected Six typical modelsof GARCH-n GARCH-t GIR-n GIR-t EGARCH-n andEGARCH-t are tested to depict the marginal distributions ofthe silver price and the exchange rate of CNYUSD Threefitting criteria measured by LLF (optimized log-likelihoodobjective function value) AIC (Akaike information criteria)and BIC (Bayesian information criteria) are used to test howthe outlined models fit the in-sample data The values of LLFAIC and BIC for different models are in Tables 3 and 4
Since the bigger the LLF the better is and the smaller theAIC and BIC the better is from Tables 3 and 4 we find thatmodels of GARCH-t model capture the fatness tails of theunconditional distributions both silver price and CNYUSDbetter Then we use GARCH-t model to fit the return seriesfor the returns of silver price and the exchange rateThe detaildescription of GARCH-t model can be referred to Huang etal [33] if the yield sequence 119903119905 has the forms
where 119905(119889) obeys Student-t distribution with degree offreedom 119889 K is the conditional variance constant GARCHmeans the coefficients related to lagged conditional variancesARCH is the coefficients related to lagged innovations (resid-uals) and C denotes the conditional mean constant Themethod to estimate for the parameters could be artificialalgorithm In this paper we use MLE method to estimatethe parameters in GARCH model We let the informationset Ωtminus1 = 1198860 1198861 119886119905minus1 The joint density function canthen be written as 119891(1198860 1198861 119886119905) = 119891(119886119905 | Ω119905minus1)119891(119886119905minus1 |Ω119905minus2) sdot sdot sdot 119891(1198861 | Ω0)119891(1198860) Given data 1198861 119886119905 the log-likelihood is as follows
This can be evaluated using the model volatility equationfor any assumed distribution for the error term Here119871119871119865 can be maximized numerically to obtain 119872119871119864 Theestimation results can be seen in Table 5
Table 5 shows the maximum likelihood estimation ofparameters the validity of ARCH and Ljungbox tests forGARCH-t models and the fitted values of AIC (Akaikeinformation criterion) and BIC (Bayesian information cri-terion) It can be seen from Table 5 that the Ljung-Boxtest for GARCH-t model residuals at 5 confidence levelcannot reject the original hypothesis of order 1 3 5 and 7autocorrelation and the original hypothesis of ARCH effectin orders 4 6 8 and 10 cannot be rejected The values ofAIC and BIC are smaller but the LLF is larger According tothe P-value of parameter estimation it shows that GARCH-t model can fit the residual sequence of silver price and theexchange rate well We then give the expression of GARCH-t distributions for the silver price and the exchange rate ofCNYUSD as follows
119903119905 = minus52123119890 minus 04 minus 92936119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 32615119890 minus 06 + 961261205902119905minus1 + 27763119890 minus 021205762119905minus1119911119905 sim 119905 (49082)
(43)
119903119905 = minus37917119890 minus 05 + 18163119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 20000119890 minus 07 + 73552119890 minus 011205902119905minus1 + 26448119890minus 01119890 minus 021205762119905minus1
119911119905 sim 119905 (34282)
(44)
where (43) and (44) are the expressions for the silver priceand the exchange rate respectively
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
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4 Mathematical Problems in Engineering
Equation (12) is further expressed in an integral form
int119878119878
119864 [1198801015840 (Π0) | 119878] minus 119864 [1198801015840 (Π0)] (119891 minus 119878) 119891 (119904) 119889119904= 0
(13)
If 119878 isin (119878 1198781) the first term of the integrand in (13) isnegative and the second is positive When 119878 isin (1198781 119878) thefirst term of the integrand in (13) is positive and the secondis negative Therefore the left hand of (13) is negative whichleads a contradictory to (13)
If (11) has two solutions ie 1198781 1198782 119878 lt 1198781 lt 1198782 lt 119878 thenwhen 119878 = 1198781 and 119878 = 1198782 we have
10038161003816100381610038161003816100381610038161003816100381610038161003816119884=0= 119864 1198801015840 (Π0) [119881 minus (119870 minus 119878)+]
= cov (119864 [1198801015840 (Π0) | 119878] 119881 minus (119870 minus 119878)+)= 119864 [1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
= int119878119878
[1198641198801015840 (Π0) minus 1198641198801015840 (Π0) | 119878] [(119870 minus 119878)+ minus 119881]
then proof that 119877(119870) lt 0 according to the functional featuresof 119877(119870) In fact it is evident that 119877(119878) = 0 From (6) we have119877(119878) = 0 Since
then when 119878 isin (1198781 1198782) we have 11987710158401015840(119870) gt 0 Similarly weobtain that when 119878 isin (119878 1198781) cup (1198782 119878) we have 11987710158401015840(119870) lt 0Based on the discussion above we have 119877(119870) lt 0 That is(120597119864[119880(Π)]120597119884)|119884=0 lt 0 and 119884lowast lt 0
Referring to the definition in Kim [28] the risk aversioninvestor whose utility function satisfies 119880101584010158401015840(Π) ge 0 is calleda prudent investor The results in Proposition 1 show thatit is wise for the prudent investor to buy unbiased optionsfor hedging which is consistent with the real intention ofenterprises who adopt hedging with the purpose of apprecia-tion and preservationWe know that although selling options
can obtain option premiums investors have to face the dailymarking risk caused by the additional margin which maylead to greater liquidity risk for investors Jorion [29] dividedliquidity risk into asset liquidity risk and capital liquidity riskHe pointed out that the capital liquidity risk is the potentiallyfatal risks faced by investors For option sellers the liquidityrisk they face mainly includes capital liquidity risk As afailure case of hedging China SouthernAirlines desire to lockin the cost of raw materials by buying call options whichcan be regarded as a hedge but selling put options based onbull market judgment creates a new margin risk exposureTherefore the conclusion of Proposition 1 coincides withthe original intention of options hedging to preserve andincrease appreciation We note the utility function of theprudent enterprise in Proposition 1 without a specific formNo matter what the utility function of the enterprise is aslong as it satisfies 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0 and 119880101584010158401015840(Π) ge0 it is optimal to use options for hedging We can findmany utility functions that satisfy 1198801015840(Π) gt 0 11988010158401015840(Π) lt 0and 119880101584010158401015840(Π) ge 0 such as the negative exponential utilityfunction and the quadratic utility function Many scholarshave established portfolio or futures hedgingmodel assumingthat the investorrsquos utility is a quadratic utility function Steil[30] obtained the hedging position of exchange rate equitywith quadratic utility Lien [31] studied futures hedging underthe negative exponential utility and the quadratic utilityBodnar [32] gave the strategies ofmultiperiod portfolio undernegative exponential utility and the quadratic utility Becauseof the complexity of the multistage dynamic programmingmethod under the negative exponential utility function itis difficult for us to obtain the explicit solutions We thenstudy the dynamic hedging problem in the framework of thequadratic utility
Proposition 2 Assume that exchange rate futures andexchange rate option markets are unbiased If the exportenterprisersquos utility function is the quadratic utility function119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimal positions ofexchange rate futures and options are
119883lowast = 1198601311986021 minus 1198601111986023119860213
minus 1198601211986023 119876
119884lowast = 1198601111986013 minus 1198601211986021119860213
minus 1198601211986023 119876(18)
where 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+)Proof Thefirst-order conditions of the objective function are
cov (Πlowast 120579) = 0 (19)
cov (Πlowast (minus120579)+) = 0 (20)
Mathematical Problems in Engineering 5
Due to the operational rules of covariance (19) and (20)can be rewritten as
Let 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+) then we have
11986011 minus 11986012119883lowast minus 11986013119884lowast = 011986021 minus 11986013119883lowast minus 11986023119884lowast = 0 (23)
By solving equations we can obtain the optimal positionsof the exchange rate futures and options presented in Propo-sition 1
Corollary 3 Suppose that 120579119905 has a symmetric distributionfunction of 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Let 119864(120579119905) = 0119864(1205792119905 ) = 1205902 119881 = int+infin
0120579119905119889119866(120579119905) and 120593 = int+infin
01205793119905 119889119866(120579119905)
In one-stage hedging the optimal positions of exchange ratefutures and options are
119883lowast = [119875 + 120573119878 + 120573 120593 minus 119881120590212059022 minus 21198812]119876
119884lowast = 2120573119876 120593 minus 119881120590212059022 minus 21198812
(24)
The study above is to establish the optimal hedgingmodelof exchange rate futures and options in one-stage whilein actual hedging practice the enterprise has to adjust thepositions dynamically according to the market conditionsso as to maximize its utility based on the final wealth Wethen extend the one-stage hedging problem to a multiperioddynamic case and derive the dynamic positions of exchangerate futures and options
4 Multistage Hedging Model withFutures and Options
Proposition 4 Suppose that the markets of exchange ratefutures and option are unbiased If the firmrsquos utility functionis quadratic of 119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimalpositions of the exchange rate futures and options at stage 119905 are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(25)
where 11986011119905 = (119875119905minus1 + 120573119905119878119905minus1) cov (120579119905 120579119905) + 120573119905 cov (1205792119905 120579119905)11986012119905 = cov (120579119905 120579119905) 11986013119905 = cov ((minus120579119905)+ 120579119905) 11986021119905 = (119875119905minus1 +120573119905119878119905minus1) cov (120579119905 (minus120579119905)+) + 120573119905 cov (1205792119905 (minus120579119905)+)11986022 = 11986013 =cov (1205792119905 (minus120579119905)+) and 11986023119905 = cov ((minus120579119905)+ (minus120579119905)+)Proof Let 119878119905 = 1198781119905minus1 +120579119905 119905 = 1198751119905minus1 +120573119905120579119905 +120576119905 and 119865119905 = 119878119905minus1Then we have the firmrsquos profit at stage 119905 isΠ119905 = 119905119878119905119876119879 + (119878119905minus1 minus 119878119905)119883119905 + [119881119905 minus (119878119905minus1 minus 119878119905)+] 119884119905
= (119875119905minus1 + 120573119905120579119905 + 120576119905) (119878119905minus1 + 120579119905)119876119905 minus 120579119905119883119905+ [119881119905 minus (minus120579119905)+] 119884119905
(26)
At the beginning of stage 119905 the firm decides the optimalpositions 119883lowast119905 119884lowast119905 to maximize its terminal utility Let 119879 =sum119879120591=1 Π120591 Then the optimal hedging model can be describedas follows
(1198752) max119883119905 119884119905
119864 [119880 (119879)] (27)
We use the dynamic programming method to solve theproblem (1198752) To begin with at stage 119879 the firm decides theoptimal position 119883lowast119879 119884lowast119879 to maximize its utility Let
and 120579119879minus1 is independent on 120579119879 and 120576119879 we havecov [119875119879minus1119878119879minus1 (119876119879minus1 + 119876119879) minus 120579119879minus1119883119879minus1
minus (minus120579119879minus1)+ 119884119879minus1 120579119879minus1] = 0 (36)
In thisway we can deduce the optimal positions of futuresand options on the basis of mathematical induction at stage 119905are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(39)
Corollary 5 Suppose that 120579119905 has a symmetric distributionfunction 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Assume the firmexport products to the foreignmarket at the terminal timeThatis 119876119905 = 0 (119905 = 1 2 119879 minus 1) 119876119879 = 119876 In the case ofmultistage the optimal positions of exchange rate futures andoptions are
119883lowast119905 = [119875119905minus1 + 120573119905119878119905minus1 + 120573119905 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905 ]119876
119884lowast119905 = 2120573119905119876 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905(40)
China has become a major producer and consumer of silveralso a big importer and exporter According to the accessibledata we assume there is a firm in China exports silver tothe US dollar region countries The firm intends to buy theexchange rate futures and options for hedging Data resourcesare fromWind database Figure 1 shows the daily yield of thesilver exported price and the exchange rate (RMBUS dollar)
Mathematical Problems in Engineering 7
Table 1 Basic statistical characteristics of returns
Objective Maximum Minimum Mean Median Standard Deviation Skewness KurtosisSilver price 01736 -01556 -00005 -00006 00199 -04517 146842CNYUSD 00287 -00176 00000 0 00021 18943 406029
Table 2 Normality test of returns
Objective P-value J-B statistics Critical valueSilver price 10000e-03 81429e+03 59490CNYUSD 10000e-03 84688e+04 59490Note 119886119890plusmn c denote that a times119890plusmn119888 the same after
From Figure 1 we can observe that the returns ofthe silver price and the exchange rate show cluster effect(that is large fluctuations are often accompanied by largefluctuations and small fluctuations are often accompaniedby small fluctuations) We further test the ARCH effectswherein the statistics of the returns are presented in Table 1
Table 1 gives the statistical description of ARCH effectstest results with regard to the silver price and the exchangerate We find that the returns of the silver price and theexchange rate CNYUSD have nonzero skewness of minus04517and 18943 respectively The returns also have peak thicktails Based on LM statistic the ARCH effects correspondingto the first 20 lags of the two exchange rate returns are furtherconfirmed (In order to save space the specific numericalvalue is omitted here) The main objective of this paper isto examine the hedging roles of the currency futures andoptions We further to test the nonnormal of the returns
Table 2 shows that the J-B statistic is larger than the criti-cal value at the 5 significance level and the P-value is smallerthan the significant level That is the hypothesis of normaldistribution under J-B test is rejected Six typical modelsof GARCH-n GARCH-t GIR-n GIR-t EGARCH-n andEGARCH-t are tested to depict the marginal distributions ofthe silver price and the exchange rate of CNYUSD Threefitting criteria measured by LLF (optimized log-likelihoodobjective function value) AIC (Akaike information criteria)and BIC (Bayesian information criteria) are used to test howthe outlined models fit the in-sample data The values of LLFAIC and BIC for different models are in Tables 3 and 4
Since the bigger the LLF the better is and the smaller theAIC and BIC the better is from Tables 3 and 4 we find thatmodels of GARCH-t model capture the fatness tails of theunconditional distributions both silver price and CNYUSDbetter Then we use GARCH-t model to fit the return seriesfor the returns of silver price and the exchange rateThe detaildescription of GARCH-t model can be referred to Huang etal [33] if the yield sequence 119903119905 has the forms
where 119905(119889) obeys Student-t distribution with degree offreedom 119889 K is the conditional variance constant GARCHmeans the coefficients related to lagged conditional variancesARCH is the coefficients related to lagged innovations (resid-uals) and C denotes the conditional mean constant Themethod to estimate for the parameters could be artificialalgorithm In this paper we use MLE method to estimatethe parameters in GARCH model We let the informationset Ωtminus1 = 1198860 1198861 119886119905minus1 The joint density function canthen be written as 119891(1198860 1198861 119886119905) = 119891(119886119905 | Ω119905minus1)119891(119886119905minus1 |Ω119905minus2) sdot sdot sdot 119891(1198861 | Ω0)119891(1198860) Given data 1198861 119886119905 the log-likelihood is as follows
This can be evaluated using the model volatility equationfor any assumed distribution for the error term Here119871119871119865 can be maximized numerically to obtain 119872119871119864 Theestimation results can be seen in Table 5
Table 5 shows the maximum likelihood estimation ofparameters the validity of ARCH and Ljungbox tests forGARCH-t models and the fitted values of AIC (Akaikeinformation criterion) and BIC (Bayesian information cri-terion) It can be seen from Table 5 that the Ljung-Boxtest for GARCH-t model residuals at 5 confidence levelcannot reject the original hypothesis of order 1 3 5 and 7autocorrelation and the original hypothesis of ARCH effectin orders 4 6 8 and 10 cannot be rejected The values ofAIC and BIC are smaller but the LLF is larger According tothe P-value of parameter estimation it shows that GARCH-t model can fit the residual sequence of silver price and theexchange rate well We then give the expression of GARCH-t distributions for the silver price and the exchange rate ofCNYUSD as follows
119903119905 = minus52123119890 minus 04 minus 92936119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 32615119890 minus 06 + 961261205902119905minus1 + 27763119890 minus 021205762119905minus1119911119905 sim 119905 (49082)
(43)
119903119905 = minus37917119890 minus 05 + 18163119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 20000119890 minus 07 + 73552119890 minus 011205902119905minus1 + 26448119890minus 01119890 minus 021205762119905minus1
119911119905 sim 119905 (34282)
(44)
where (43) and (44) are the expressions for the silver priceand the exchange rate respectively
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Let 11986011 = (119875 + 120573119878) cov (120579 120579) + 120573 cov (1205792 120579)11986012 = cov (120579 120579) 11986013 = cov ((minus120579)+ 120579) 11986021 =(119875 + 120573119878) cov (120579 (minus120579)+) + 120573 cov (1205792 (minus120579)+) 11986022 = 11986013 =cov (120579 (minus120579)+) and 11986023 = cov ((minus120579)+ (minus120579)+) then we have
11986011 minus 11986012119883lowast minus 11986013119884lowast = 011986021 minus 11986013119883lowast minus 11986023119884lowast = 0 (23)
By solving equations we can obtain the optimal positionsof the exchange rate futures and options presented in Propo-sition 1
Corollary 3 Suppose that 120579119905 has a symmetric distributionfunction of 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Let 119864(120579119905) = 0119864(1205792119905 ) = 1205902 119881 = int+infin
0120579119905119889119866(120579119905) and 120593 = int+infin
01205793119905 119889119866(120579119905)
In one-stage hedging the optimal positions of exchange ratefutures and options are
119883lowast = [119875 + 120573119878 + 120573 120593 minus 119881120590212059022 minus 21198812]119876
119884lowast = 2120573119876 120593 minus 119881120590212059022 minus 21198812
(24)
The study above is to establish the optimal hedgingmodelof exchange rate futures and options in one-stage whilein actual hedging practice the enterprise has to adjust thepositions dynamically according to the market conditionsso as to maximize its utility based on the final wealth Wethen extend the one-stage hedging problem to a multiperioddynamic case and derive the dynamic positions of exchangerate futures and options
4 Multistage Hedging Model withFutures and Options
Proposition 4 Suppose that the markets of exchange ratefutures and option are unbiased If the firmrsquos utility functionis quadratic of 119880(Π) = Π minus 119887Π2 (119887 gt 0) then the optimalpositions of the exchange rate futures and options at stage 119905 are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(25)
where 11986011119905 = (119875119905minus1 + 120573119905119878119905minus1) cov (120579119905 120579119905) + 120573119905 cov (1205792119905 120579119905)11986012119905 = cov (120579119905 120579119905) 11986013119905 = cov ((minus120579119905)+ 120579119905) 11986021119905 = (119875119905minus1 +120573119905119878119905minus1) cov (120579119905 (minus120579119905)+) + 120573119905 cov (1205792119905 (minus120579119905)+)11986022 = 11986013 =cov (1205792119905 (minus120579119905)+) and 11986023119905 = cov ((minus120579119905)+ (minus120579119905)+)Proof Let 119878119905 = 1198781119905minus1 +120579119905 119905 = 1198751119905minus1 +120573119905120579119905 +120576119905 and 119865119905 = 119878119905minus1Then we have the firmrsquos profit at stage 119905 isΠ119905 = 119905119878119905119876119879 + (119878119905minus1 minus 119878119905)119883119905 + [119881119905 minus (119878119905minus1 minus 119878119905)+] 119884119905
= (119875119905minus1 + 120573119905120579119905 + 120576119905) (119878119905minus1 + 120579119905)119876119905 minus 120579119905119883119905+ [119881119905 minus (minus120579119905)+] 119884119905
(26)
At the beginning of stage 119905 the firm decides the optimalpositions 119883lowast119905 119884lowast119905 to maximize its terminal utility Let 119879 =sum119879120591=1 Π120591 Then the optimal hedging model can be describedas follows
(1198752) max119883119905 119884119905
119864 [119880 (119879)] (27)
We use the dynamic programming method to solve theproblem (1198752) To begin with at stage 119879 the firm decides theoptimal position 119883lowast119879 119884lowast119879 to maximize its utility Let
and 120579119879minus1 is independent on 120579119879 and 120576119879 we havecov [119875119879minus1119878119879minus1 (119876119879minus1 + 119876119879) minus 120579119879minus1119883119879minus1
minus (minus120579119879minus1)+ 119884119879minus1 120579119879minus1] = 0 (36)
In thisway we can deduce the optimal positions of futuresand options on the basis of mathematical induction at stage 119905are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(39)
Corollary 5 Suppose that 120579119905 has a symmetric distributionfunction 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Assume the firmexport products to the foreignmarket at the terminal timeThatis 119876119905 = 0 (119905 = 1 2 119879 minus 1) 119876119879 = 119876 In the case ofmultistage the optimal positions of exchange rate futures andoptions are
119883lowast119905 = [119875119905minus1 + 120573119905119878119905minus1 + 120573119905 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905 ]119876
119884lowast119905 = 2120573119905119876 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905(40)
China has become a major producer and consumer of silveralso a big importer and exporter According to the accessibledata we assume there is a firm in China exports silver tothe US dollar region countries The firm intends to buy theexchange rate futures and options for hedging Data resourcesare fromWind database Figure 1 shows the daily yield of thesilver exported price and the exchange rate (RMBUS dollar)
Mathematical Problems in Engineering 7
Table 1 Basic statistical characteristics of returns
Objective Maximum Minimum Mean Median Standard Deviation Skewness KurtosisSilver price 01736 -01556 -00005 -00006 00199 -04517 146842CNYUSD 00287 -00176 00000 0 00021 18943 406029
Table 2 Normality test of returns
Objective P-value J-B statistics Critical valueSilver price 10000e-03 81429e+03 59490CNYUSD 10000e-03 84688e+04 59490Note 119886119890plusmn c denote that a times119890plusmn119888 the same after
From Figure 1 we can observe that the returns ofthe silver price and the exchange rate show cluster effect(that is large fluctuations are often accompanied by largefluctuations and small fluctuations are often accompaniedby small fluctuations) We further test the ARCH effectswherein the statistics of the returns are presented in Table 1
Table 1 gives the statistical description of ARCH effectstest results with regard to the silver price and the exchangerate We find that the returns of the silver price and theexchange rate CNYUSD have nonzero skewness of minus04517and 18943 respectively The returns also have peak thicktails Based on LM statistic the ARCH effects correspondingto the first 20 lags of the two exchange rate returns are furtherconfirmed (In order to save space the specific numericalvalue is omitted here) The main objective of this paper isto examine the hedging roles of the currency futures andoptions We further to test the nonnormal of the returns
Table 2 shows that the J-B statistic is larger than the criti-cal value at the 5 significance level and the P-value is smallerthan the significant level That is the hypothesis of normaldistribution under J-B test is rejected Six typical modelsof GARCH-n GARCH-t GIR-n GIR-t EGARCH-n andEGARCH-t are tested to depict the marginal distributions ofthe silver price and the exchange rate of CNYUSD Threefitting criteria measured by LLF (optimized log-likelihoodobjective function value) AIC (Akaike information criteria)and BIC (Bayesian information criteria) are used to test howthe outlined models fit the in-sample data The values of LLFAIC and BIC for different models are in Tables 3 and 4
Since the bigger the LLF the better is and the smaller theAIC and BIC the better is from Tables 3 and 4 we find thatmodels of GARCH-t model capture the fatness tails of theunconditional distributions both silver price and CNYUSDbetter Then we use GARCH-t model to fit the return seriesfor the returns of silver price and the exchange rateThe detaildescription of GARCH-t model can be referred to Huang etal [33] if the yield sequence 119903119905 has the forms
where 119905(119889) obeys Student-t distribution with degree offreedom 119889 K is the conditional variance constant GARCHmeans the coefficients related to lagged conditional variancesARCH is the coefficients related to lagged innovations (resid-uals) and C denotes the conditional mean constant Themethod to estimate for the parameters could be artificialalgorithm In this paper we use MLE method to estimatethe parameters in GARCH model We let the informationset Ωtminus1 = 1198860 1198861 119886119905minus1 The joint density function canthen be written as 119891(1198860 1198861 119886119905) = 119891(119886119905 | Ω119905minus1)119891(119886119905minus1 |Ω119905minus2) sdot sdot sdot 119891(1198861 | Ω0)119891(1198860) Given data 1198861 119886119905 the log-likelihood is as follows
This can be evaluated using the model volatility equationfor any assumed distribution for the error term Here119871119871119865 can be maximized numerically to obtain 119872119871119864 Theestimation results can be seen in Table 5
Table 5 shows the maximum likelihood estimation ofparameters the validity of ARCH and Ljungbox tests forGARCH-t models and the fitted values of AIC (Akaikeinformation criterion) and BIC (Bayesian information cri-terion) It can be seen from Table 5 that the Ljung-Boxtest for GARCH-t model residuals at 5 confidence levelcannot reject the original hypothesis of order 1 3 5 and 7autocorrelation and the original hypothesis of ARCH effectin orders 4 6 8 and 10 cannot be rejected The values ofAIC and BIC are smaller but the LLF is larger According tothe P-value of parameter estimation it shows that GARCH-t model can fit the residual sequence of silver price and theexchange rate well We then give the expression of GARCH-t distributions for the silver price and the exchange rate ofCNYUSD as follows
119903119905 = minus52123119890 minus 04 minus 92936119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 32615119890 minus 06 + 961261205902119905minus1 + 27763119890 minus 021205762119905minus1119911119905 sim 119905 (49082)
(43)
119903119905 = minus37917119890 minus 05 + 18163119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 20000119890 minus 07 + 73552119890 minus 011205902119905minus1 + 26448119890minus 01119890 minus 021205762119905minus1
119911119905 sim 119905 (34282)
(44)
where (43) and (44) are the expressions for the silver priceand the exchange rate respectively
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
and 120579119879minus1 is independent on 120579119879 and 120576119879 we havecov [119875119879minus1119878119879minus1 (119876119879minus1 + 119876119879) minus 120579119879minus1119883119879minus1
minus (minus120579119879minus1)+ 119884119879minus1 120579119879minus1] = 0 (36)
In thisway we can deduce the optimal positions of futuresand options on the basis of mathematical induction at stage 119905are
119883lowast119905 = 1198601311990511986021119905 minus 1198601111990511986023119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591
119884lowast119905 = 1198601111990511986013119905 minus 1198601211990511986021119905119860213119905
minus 1198601211990511986023119905119879sum120591=119905
119876120591(39)
Corollary 5 Suppose that 120579119905 has a symmetric distributionfunction 119866(120579119905) then 119889119866(120579119905) = 119889119866(minus120579119905) Assume the firmexport products to the foreignmarket at the terminal timeThatis 119876119905 = 0 (119905 = 1 2 119879 minus 1) 119876119879 = 119876 In the case ofmultistage the optimal positions of exchange rate futures andoptions are
119883lowast119905 = [119875119905minus1 + 120573119905119878119905minus1 + 120573119905 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905 ]119876
119884lowast119905 = 2120573119905119876 120593119905 minus 11988111990512059021199051205902119905 2 minus 21198812119905(40)
China has become a major producer and consumer of silveralso a big importer and exporter According to the accessibledata we assume there is a firm in China exports silver tothe US dollar region countries The firm intends to buy theexchange rate futures and options for hedging Data resourcesare fromWind database Figure 1 shows the daily yield of thesilver exported price and the exchange rate (RMBUS dollar)
Mathematical Problems in Engineering 7
Table 1 Basic statistical characteristics of returns
Objective Maximum Minimum Mean Median Standard Deviation Skewness KurtosisSilver price 01736 -01556 -00005 -00006 00199 -04517 146842CNYUSD 00287 -00176 00000 0 00021 18943 406029
Table 2 Normality test of returns
Objective P-value J-B statistics Critical valueSilver price 10000e-03 81429e+03 59490CNYUSD 10000e-03 84688e+04 59490Note 119886119890plusmn c denote that a times119890plusmn119888 the same after
From Figure 1 we can observe that the returns ofthe silver price and the exchange rate show cluster effect(that is large fluctuations are often accompanied by largefluctuations and small fluctuations are often accompaniedby small fluctuations) We further test the ARCH effectswherein the statistics of the returns are presented in Table 1
Table 1 gives the statistical description of ARCH effectstest results with regard to the silver price and the exchangerate We find that the returns of the silver price and theexchange rate CNYUSD have nonzero skewness of minus04517and 18943 respectively The returns also have peak thicktails Based on LM statistic the ARCH effects correspondingto the first 20 lags of the two exchange rate returns are furtherconfirmed (In order to save space the specific numericalvalue is omitted here) The main objective of this paper isto examine the hedging roles of the currency futures andoptions We further to test the nonnormal of the returns
Table 2 shows that the J-B statistic is larger than the criti-cal value at the 5 significance level and the P-value is smallerthan the significant level That is the hypothesis of normaldistribution under J-B test is rejected Six typical modelsof GARCH-n GARCH-t GIR-n GIR-t EGARCH-n andEGARCH-t are tested to depict the marginal distributions ofthe silver price and the exchange rate of CNYUSD Threefitting criteria measured by LLF (optimized log-likelihoodobjective function value) AIC (Akaike information criteria)and BIC (Bayesian information criteria) are used to test howthe outlined models fit the in-sample data The values of LLFAIC and BIC for different models are in Tables 3 and 4
Since the bigger the LLF the better is and the smaller theAIC and BIC the better is from Tables 3 and 4 we find thatmodels of GARCH-t model capture the fatness tails of theunconditional distributions both silver price and CNYUSDbetter Then we use GARCH-t model to fit the return seriesfor the returns of silver price and the exchange rateThe detaildescription of GARCH-t model can be referred to Huang etal [33] if the yield sequence 119903119905 has the forms
where 119905(119889) obeys Student-t distribution with degree offreedom 119889 K is the conditional variance constant GARCHmeans the coefficients related to lagged conditional variancesARCH is the coefficients related to lagged innovations (resid-uals) and C denotes the conditional mean constant Themethod to estimate for the parameters could be artificialalgorithm In this paper we use MLE method to estimatethe parameters in GARCH model We let the informationset Ωtminus1 = 1198860 1198861 119886119905minus1 The joint density function canthen be written as 119891(1198860 1198861 119886119905) = 119891(119886119905 | Ω119905minus1)119891(119886119905minus1 |Ω119905minus2) sdot sdot sdot 119891(1198861 | Ω0)119891(1198860) Given data 1198861 119886119905 the log-likelihood is as follows
This can be evaluated using the model volatility equationfor any assumed distribution for the error term Here119871119871119865 can be maximized numerically to obtain 119872119871119864 Theestimation results can be seen in Table 5
Table 5 shows the maximum likelihood estimation ofparameters the validity of ARCH and Ljungbox tests forGARCH-t models and the fitted values of AIC (Akaikeinformation criterion) and BIC (Bayesian information cri-terion) It can be seen from Table 5 that the Ljung-Boxtest for GARCH-t model residuals at 5 confidence levelcannot reject the original hypothesis of order 1 3 5 and 7autocorrelation and the original hypothesis of ARCH effectin orders 4 6 8 and 10 cannot be rejected The values ofAIC and BIC are smaller but the LLF is larger According tothe P-value of parameter estimation it shows that GARCH-t model can fit the residual sequence of silver price and theexchange rate well We then give the expression of GARCH-t distributions for the silver price and the exchange rate ofCNYUSD as follows
119903119905 = minus52123119890 minus 04 minus 92936119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 32615119890 minus 06 + 961261205902119905minus1 + 27763119890 minus 021205762119905minus1119911119905 sim 119905 (49082)
(43)
119903119905 = minus37917119890 minus 05 + 18163119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 20000119890 minus 07 + 73552119890 minus 011205902119905minus1 + 26448119890minus 01119890 minus 021205762119905minus1
119911119905 sim 119905 (34282)
(44)
where (43) and (44) are the expressions for the silver priceand the exchange rate respectively
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 7
Table 1 Basic statistical characteristics of returns
Objective Maximum Minimum Mean Median Standard Deviation Skewness KurtosisSilver price 01736 -01556 -00005 -00006 00199 -04517 146842CNYUSD 00287 -00176 00000 0 00021 18943 406029
Table 2 Normality test of returns
Objective P-value J-B statistics Critical valueSilver price 10000e-03 81429e+03 59490CNYUSD 10000e-03 84688e+04 59490Note 119886119890plusmn c denote that a times119890plusmn119888 the same after
From Figure 1 we can observe that the returns ofthe silver price and the exchange rate show cluster effect(that is large fluctuations are often accompanied by largefluctuations and small fluctuations are often accompaniedby small fluctuations) We further test the ARCH effectswherein the statistics of the returns are presented in Table 1
Table 1 gives the statistical description of ARCH effectstest results with regard to the silver price and the exchangerate We find that the returns of the silver price and theexchange rate CNYUSD have nonzero skewness of minus04517and 18943 respectively The returns also have peak thicktails Based on LM statistic the ARCH effects correspondingto the first 20 lags of the two exchange rate returns are furtherconfirmed (In order to save space the specific numericalvalue is omitted here) The main objective of this paper isto examine the hedging roles of the currency futures andoptions We further to test the nonnormal of the returns
Table 2 shows that the J-B statistic is larger than the criti-cal value at the 5 significance level and the P-value is smallerthan the significant level That is the hypothesis of normaldistribution under J-B test is rejected Six typical modelsof GARCH-n GARCH-t GIR-n GIR-t EGARCH-n andEGARCH-t are tested to depict the marginal distributions ofthe silver price and the exchange rate of CNYUSD Threefitting criteria measured by LLF (optimized log-likelihoodobjective function value) AIC (Akaike information criteria)and BIC (Bayesian information criteria) are used to test howthe outlined models fit the in-sample data The values of LLFAIC and BIC for different models are in Tables 3 and 4
Since the bigger the LLF the better is and the smaller theAIC and BIC the better is from Tables 3 and 4 we find thatmodels of GARCH-t model capture the fatness tails of theunconditional distributions both silver price and CNYUSDbetter Then we use GARCH-t model to fit the return seriesfor the returns of silver price and the exchange rateThe detaildescription of GARCH-t model can be referred to Huang etal [33] if the yield sequence 119903119905 has the forms
where 119905(119889) obeys Student-t distribution with degree offreedom 119889 K is the conditional variance constant GARCHmeans the coefficients related to lagged conditional variancesARCH is the coefficients related to lagged innovations (resid-uals) and C denotes the conditional mean constant Themethod to estimate for the parameters could be artificialalgorithm In this paper we use MLE method to estimatethe parameters in GARCH model We let the informationset Ωtminus1 = 1198860 1198861 119886119905minus1 The joint density function canthen be written as 119891(1198860 1198861 119886119905) = 119891(119886119905 | Ω119905minus1)119891(119886119905minus1 |Ω119905minus2) sdot sdot sdot 119891(1198861 | Ω0)119891(1198860) Given data 1198861 119886119905 the log-likelihood is as follows
This can be evaluated using the model volatility equationfor any assumed distribution for the error term Here119871119871119865 can be maximized numerically to obtain 119872119871119864 Theestimation results can be seen in Table 5
Table 5 shows the maximum likelihood estimation ofparameters the validity of ARCH and Ljungbox tests forGARCH-t models and the fitted values of AIC (Akaikeinformation criterion) and BIC (Bayesian information cri-terion) It can be seen from Table 5 that the Ljung-Boxtest for GARCH-t model residuals at 5 confidence levelcannot reject the original hypothesis of order 1 3 5 and 7autocorrelation and the original hypothesis of ARCH effectin orders 4 6 8 and 10 cannot be rejected The values ofAIC and BIC are smaller but the LLF is larger According tothe P-value of parameter estimation it shows that GARCH-t model can fit the residual sequence of silver price and theexchange rate well We then give the expression of GARCH-t distributions for the silver price and the exchange rate ofCNYUSD as follows
119903119905 = minus52123119890 minus 04 minus 92936119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 32615119890 minus 06 + 961261205902119905minus1 + 27763119890 minus 021205762119905minus1119911119905 sim 119905 (49082)
(43)
119903119905 = minus37917119890 minus 05 + 18163119890 minus 02119903119905minus1 + 120576119905120576119905 = 120590119905119911119905
1205902119905 = 20000119890 minus 07 + 73552119890 minus 011205902119905minus1 + 26448119890minus 01119890 minus 021205762119905minus1
119911119905 sim 119905 (34282)
(44)
where (43) and (44) are the expressions for the silver priceand the exchange rate respectively
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Mathematical Problems in Engineering
Table 3 Fitting criteria of silver price
LLF 36748e+03 37755 e +03 36805 e +03 37747 e +03 36813 e +03 37749 e +03AIC -73415 e +03 -75390 e +03 -73490 e +03 -75374 e +03 -73506 e +03 -75378 e +03BIC -73223 e +03 -75102 e +03 -73201 e +03 -74990 e +03 -73218e+03 -75042 e +03
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
14
142
144
146
148
15
152
154
Retu
rns
Figure 2 Gains under three circumstances
The current time is June 1 2016 We suppose that a yearlater ie at June 1 2017The enterprise exports 10000 ouncesof silver to the country settled in US dollars By solvingthe proposed model we obtain the returns in three cases ofoptions and futures hedging only futures hedging and nohedging shown in Figure 2
In order to avoid the silver export risk the enterprise faceswith the return of futures and options joint hedging is greaterthan that of only futures hedging The result shows thatthe combination of options and futures is better than usingfutures hedging alone Without any hedging the profit isminimal which reflects the advantage of using derivatives forhedging In fact when the enterprise uses futures or optionsfor hedging futures and options themselves are regardedas financial assets to participate in the investment That isto say in the process of hedging the enterprise can obtainadditional income by trading derivatives At the same timelinear instrument futures and nonlinear instrument optionshave their own advantages in hedging against linear andnonlinear risks The combination of futures and options canbring different hedging experiences to hedgers and obtain
Hedging with futures and optionsHedging only with futuresNo hedging
50 100 150 200 2500Simulation times
0445
045
0455
046
0465
047
0475
048
Risk
of t
he re
turn
Figure 3 Volatility in the process of wealth accumulation
better hedging effect The original intention of investors touse derivatives hedging is to keep assets to preserve andincrease appreciation We say that hedging is the primarygoal followed by appreciation Therefore for hedgers inpursuit of appropriate returns they will also pay attentionto the volatility in the process of wealth accumulation Evenif the return is high if the hedger is exposed to excessivefluctuation process the psychological cost of the hedgeris relatively high and the hedger may have to bear thepsychological impact of ups and downsThis paper comparesand analyzes the stability of wealth accumulation processwhen using derivatives hedging We compare the standarddeviation of the simulated wealth in three cases as shown inFigure 3
From Figure 3 it is not difficult to find that in theprehedging stage the volatility of wealth accumulation underthe joint hedging of futures and options is relatively smallfollowed by only using futures hedging and no hedging Inthe latter stage the opposite is true but in general there is notmuch difference in the volatility of wealth paths Moreover
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Source SS df MS Chi-sq ProbgtChi-sqColumns 248687e+07 2 1243432816 57486 147798e-125Error 623528e+06 717 869635Total 311039e+07 719
Table 7 Comparison of three effective boundary values
Group name Confidence interval1 2 15865 20315 247641 3 40988 45438 498882 3 20673 25123 29573
the empirical part of this paper assumes that the utility func-tion of silver export enterprises is a quadratic utility functionwe can not only study profits or volatilityTherefore we studythe effective boundary (revenuestandard deviation) in threecases In order to avoid using ANOVAmultiple comparisonswhen comparing multiple groups when the groups do notsatisfy the normal distribution Kruskal-Wallis test is used tocompare the effective boundaries under different strategiesThe results of statistical analysis of Kruskal-Wallis test areshown in Table 6
As can be seen fromTable 6 because themean of the threegroups are unequal the effective boundaries are different inthe three cases of using options and futures using only futuresand no hedging Table 7 further compares the three effectiveboundaries
From the comparison matrix in Table 7 we can see thethree effective boundaries from large to small correspondingto options and futures hedging futures hedging and nohedging Generally speaking if the exporters do not adoptany hedge their effective boundaries are small From theconfidence intervals of themean between different groups wecan see that the effective boundary increases by nearly 45438on average which is larger than the average value-added of25123 of futures hedgingThe results show that hedging withfutures or options can effectively avoid the risk faced by theexport enterprise
6 Conclusion
Since the inception of floating exchange rates firms engagedin international operations have been highly interested indeveloping ways and means to protect themselves againstexchange rate risk Price or exchange rate uncertainty resultsin a reduction of production and exports Therefore themajor role of financial markets enabling firms to reduce priceor currency risks is their impact on production and exportlevels
This paper studies the problem of exchange rate futuresand options hedging for a risk averse export enterpriseFirstly under unbiased markets of exchange rate futures andoptions we prove that the prudent exporter is profitable tobuy exchange rate options for hedging The result validatesthe necessity of options hedging when the risk exposureis nonlinear Considering that in actual hedging practiceinvestors will adjust their investment strategies dynamicallyaccording to themarket conditions this paper generalizes the
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Mathematical Problems in Engineering
existing static futures and options hedging to the dynamic sit-uationWe establish the dynamic hedging model of exchangerate futures and options and present the explicit positionsunder the quadratic utility function by using the dynamicprogramming method Finally through empirical analysisit is found that the export enterprisersquos profit and effectiveboundary are the largest by using both exchange rate futuresand option hedging followed by only using exchange ratefutures hedging and then without any hedging strategy Wedemonstrate the role of exchange rate futures and optionshedging in the exchange rate risk management of the exportenterprise Therefore it is suggested that exporter shouldadopt hedging strategies of exchange rate futures and options
This paper focuses on the application of exchange ratefutures and options in hedging Therefore the transactioncost is neglected in this study and the firm is assumed to havesufficient futuresmargin In the future study the daily peeringrisk of futures hedging and the transaction cost of futures andoptions can be considered
Data Availability
The data are the export price of silver and the exchangerate of CNYUSD The data can be downloaded fromWind database whose website is httpwwwwindcomcnnewsitedatahtml However downloading data is payableFortunately our school (CCNU) bought the database So wehave a barrier-free access to download the data
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
Thispaper is financially supported by the raising initial capitalfor High-Level Talents of Central China Normal University(30101190001) Fundamental Research Funds for the CentralUniversities (CCNU19A06043 CCNU19TD006 and CCNU19TS062) Humanities and Social Science Planning FundfromMinistry of Education (Grant no 16YJAZH078)
References
[1] G Allayannis and J P Weston ldquoThe use of foreign currencyderivatives and firm market valuerdquo Review of Financial Studiesvol 14 no 1 pp 243ndash276 2001
[2] E Clarka and S Mefteh ldquoForeign currency derivatives usefirm value and the effect of theexposure profile evidence fromfrancerdquo International Journal of Business vol 15 no 2 pp 183ndash196 2010
[3] K P Wong ldquoOperational and financial hedging for exportingfirmsrdquo International Review of Economics amp Finance vol 16 no4 pp 459ndash470 2007
[4] K P Wong ldquoInternational trade and hedging under jointprice and exchange rate uncertaintyrdquo International Review ofEconomics amp Finance vol 27 no 2 pp 160ndash170 2013
[5] U Broll andK PWong ldquoTrade and cross hedging exchange rateriskrdquo International Economics and Economic Policy vol 12 no4 pp 509ndash520 2015
[6] S H Lence ldquoOn the optimal hedge under unbiased futurespricesrdquo Economics Letters vol 47 no 3-4 pp 385ndash388 1995
[7] D Lien and K P Wong ldquoMultinationals and futures hedgingunder liquidity constraintsrdquo Global Finance Journal vol 16 no2 pp 210ndash220 2005
[8] U Broll P Welzel and K P Wong ldquoExport and strategiccurrency hedgingrdquo Open Economies Review vol 20 no 5 pp717ndash732 2009
[9] K P Wong ldquoProduction and hedging in futures markets withmultiple delivery specificationsrdquo Decisions in Economics andFinance vol 37 no 2 pp 413ndash421 2014
[10] Y Machnes ldquoFurther results on comparative statics underuncertaintyrdquo European Journal of Political Economy vol 9 no1 pp 141ndash146 1993
[11] K P Wong ldquoProduction decisions in the presence of options anoterdquo International Review of Economics amp Finance vol 11 no1 pp 17ndash25 2002
[12] D Lien and K P Wong ldquoDelivery risk and the hedging role ofoptionsrdquo Journal of Futures Markets vol 22 no 4 pp 339ndash3542002
[13] H L Battermann and U Broll ldquoInflation risk hedging andexportsrdquo Review of Development Economics vol 5 no 3 pp355ndash362 2001
[14] E Bajo M Barbi and S Romagnoli ldquoOptimal corporatehedging using options with basis and production riskrdquo TheNorth American Journal of Economics and Finance vol 30 pp56ndash71 2014
[15] E Bajo M Barbi and S Romagnoli ldquoA generalized approachto optimal hedging with option contractsrdquo European Journal ofFinance vol 21 no 9 pp 714ndash733 2015
[16] K P Wong ldquoCross-hedging ambiguous exchange rate riskrdquoJournal of Futures Markets vol 37 no 2 pp 132ndash147 2017
[17] H Lapan G Moschini and S Hanson ldquoProduction hedgingand speculative decisions with options and futures marketsrdquoAmerican Journal of Agricultural Economics vol 75 no 3 pp748ndash750 1993
[18] D L Frechette ldquoThe demand for hedging with futures andoptionsrdquo Journal of Futures Markets vol 21 no 8 pp 693ndash7122001
[19] K P Wong ldquoCurrency hedging with options and futuresrdquoEuropean Economic Review vol 47 no 5 pp 833ndash839 2003
[20] K P Wong ldquoProduction and hedging under state-dependentpreferencesrdquo Journal of Futures Markets vol 32 no 10 pp 945ndash963 2012
[21] Y SakongD J Hayes andAHallam ldquoHedging production riskwith optionsrdquo American Journal of Agricultural Economics vol75 no 2 p 408 1993
[22] G Moschini and H Lapan ldquoThe Hedging Role of Optionsand Futures Under Joint Price Basis and Production RiskrdquoInternational Economic Review vol 36 no 4 p 1025 1995
[23] J Kotkatvuori-Ornberg ldquoDynamic conditional copula correla-tion and optimal hedge ratios with currency futuresrdquo Interna-tional Review of Financial Analysis vol 47 pp 60ndash69 2016
[24] W G Zhang X Yu and Y J Liu ldquoTrade and currencyoptions hedging modelrdquo Journal of Computational and AppliedMathematics vol 343 pp 328ndash340 2018
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 11
[25] E C Chang and K P Wong ldquoCross-hedging with currencyoptions and futuresrdquo Journal of Financial and QuantitativeAnalysis vol 38 no 3 pp 555ndash574 2003
[26] T G Chi P F YU and Y G Wang ldquoResearch on multi-period futures dynamic hedging modelrdquo Chinese Journal ofManagement Science vol 18 no 3 pp 17ndash24 2010
[27] Q Li Y Zhou X Zhao and X Ge ldquoDynamic hedging basedon fractional order stochastic model with memory effectrdquoMathematical Problems in Engineering vol 2016 Article ID6817483 8 pages 2016
[28] M S Kimball ldquoStandard risk aversionrdquo Econometrica vol 61no 3 pp 589ndash611 1993
[29] P Jorion Value at Risk The New Benchmark for ManagingFinancial Risk McGraw-Hill New York NY USA 2nd edition2001
[30] B Steil ldquoCurrency options and the optimal hedging of contin-gent foreign exchange exposurerdquo Economica vol 60 no 240pp 413ndash431 1993
[31] D Lien ldquoThe effect of liquidity constraints on futures hedgingrdquoJournal of Futures Markets vol 23 no 6 pp 603ndash613 2003
[32] T Bodnar N Parolya and W Schmid ldquoA closed-form solutionof the multi-period portfolio choice problem for a quadraticutility functionrdquo Annals of Operations Research vol 229 no 1pp 121ndash158 2015
[33] J-J Huang K-J Lee H Liang andW-F Lin ldquoEstimating valueat risk of portfolio by conditional copula-GARCH methodrdquoInsurance Mathematics amp Economics vol 45 no 3 pp 315ndash3242009
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
![Page 1: ResearchArticle Dynamic Currency Futures and Options Hedging …downloads.hindawi.com/journals/mpe/2019/8074384.pdf · and study the eectiveness of currency futures hedging. Zhangetal.[]employedGARCH-Copulamodelstoexam-ine](https://reader035.documents.pub/reader035/viewer/2022062605/5fda4d8c89fe4775df196d01/html5/thumbnails/1.jpg)
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