Arun ChockalingamSchool of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907, [email protected]
Kumar MuthuramanMcCombs School of Business, The University of Texas at Austin, Austin, Texas 78712, [email protected]
The problem of pricing an American option written on an underlying asset with constant price volatility has been studiedextensively in literature. Real-world data, however, demonstrate that volatility is not constant, and stochastic volatilitymodels are used to account for dynamic volatility changes. Option pricing methods that have been developed in literature forpricing under stochastic volatility focus mostly on European options. We consider the problem of pricing American optionsunder stochastic volatility, which has had relatively much less attention from literature. First, we develop a transformationprocedure to compute the optimal-exercise policy and option price and provide theoretical guarantees for convergence.Second, using this computational tool, we explore a variety of questions that seek insights into the dependence of optionprices, exercise policies, and implied volatilities on the market price of volatility risk and correlation between the asset andstochastic volatility. The speed and accuracy of the procedure are compared against existing methods as well.
Subject classifications : American option; stochastic volatility; free boundary.Area of review : Financial Engineering.History : Received August 2008; revisions received January 2010, July 2010, August 2010; accepted October 2010.
1. IntroductionOptions are contracts that give the holder the right to sell(put) or buy (call) an underlying asset at a predeterminedstrike price. A European option allows the holder to exer-cise the option only on a predetermined expiration date,while an American option allows the holder to exercise theoption at any point in time until the expiration date. Optionpricing has always played a prominent role in financial the-ory as well as real derivative markets. In a celebrated paper,Black and Scholes (1973) derive a closed-form solution forthe price of a European option by characterizing the priceas the expected payoff under a risk-neutral measure. Theirmodel assumes that the underlying asset price follows ageometric Brownian motion with constant volatility.
Even under the constant volatility assumption, that is,the classical Black-Scholes setting, closed-form solutionsdo not exist for American options. Due to the possibilityof early exercise, the American option price depends onthe optimal-exercise policy, which can be represented byan exercise boundary (also known as the free boundary) onthe price-time space. The exercise boundary partitions theprice-time space into hold and exercise regions.
Most of option pricing literature consider the con-stant volatility model. Rubinstein (1994), however, providesempirical evidence, using implied volatilities obtained fromindex options on the S&P 500, that suggests that the con-stant volatility assumption does not hold. Using data forthe OEX contract, Broadie et al. (2000) find that dividendsalone are not accountable for all aspects of option pric-ing and exercise decisions, and they suggest that stochastic
volatility needs to be included as well. Furthermore, Scott(1987) and the references therein provide ample evidence ofvolatility changing over time. This can also be readily seenfrom the implied volatilities calculated from market prices.Implied volatilities are the volatilities that, when used inthe Black-Scholes formula, provide European option pricesconsistent with option prices observed in the market. Whenimplied volatilities are plotted against strike prices, the plotsexhibit a “smile” effect, which refers to the resulting U-shaped curve, as opposed to a straight line that one wouldexpect if asset prices had constant volatility. Implied volatil-ities for in- and out-of-the-money options are observed tobe higher than at-the-money options. Assuming constantvolatility therefore leads to considerable mispricing. Hence,models that allow the volatility of the underlying assetprice to be stochastic are needed to better capture marketbehavior.
Several models have been proposed to better model theevolution of volatility. One approach has been to use ARCHmodels and their variants (Bollerslev et al. 1992). Usinganother diffusion process to model volatility, however, hasbecome a more popular choice. Hull and White (1987),Scott (1987), Stein and Stein (1991), and Heston (1993)each propose different diffusion processes to represent thedynamics of asset price volatility.
1.1. Relevant Literature
The pricing of American options under constant volatil-ity and European options under stochastic volatility haveboth received considerable attention in literature. Numer-ical methods available to compute the price and/or
exercise boundary for American options under constantvolatility include simulation-based techniques (Broadie andGlasserman 1997, Longstaff and Schwartz 2001), bino-mial lattices (Cox et al. 1979), partial differential equa-tion (PDE) solution methods (Brennan and Schwartz1977, Muthuraman 2008), front-fixing methods (Wu andKwok 1997, Nielsen et al. 2002), front-tracking meth-ods (Pantazopoulos et al. 1998), and integral methods(Kim 1990, Jacka 1991, Carr et al. 1992). Myneni (1992),Karatzas and Shreve (1998), and Broadie and Detemple(2004) provide an overview of American option pricingunder the Black-Scholes setting.
The pricing of European options under stochastic volatil-ity has been looked at for various choices of diffusiondynamics to represent the volatility process. Hull and White(1987) use a lognormal process, while Scott (1987) andStein and Stein (1991) use a mean-reverting Ornstein-Uhlenbeck (OU) process. Closed-form solutions have beenobtained in Heston (1993) when volatility is modeled asa square-root process. Ball and Roma (1994) also use thesame square-root process.
Relatively little attention has been paid to the problemof pricing an American option under stochastic volatility.Literature on American options under stochastic volatil-ity can be classified into PDE-based and non-PDE-basedapproaches. The PDE methods solve the free-boundaryproblem arising from the use of classical dynamic pro-gramming arguments and provide the entire price functionand the optimal-exercise policy explicitly. Non-PDE-basedapproaches compute the price for any given time, assetprice and underlying volatility by computing the condi-tional expectation under a suitable martingale measure.
A popular approach to solve the related free-boundaryPDE problem is to reformulate it as a linear complemen-tarity problem (LCP). The projected successive over relax-ation (PSOR) method proposed by Cryer (1971) is widelyused to solve these LCPs. Clarke and Parrott (1999) usea stretching transformation and an adaptive-upwind finitedifference approximation to discretize the LCP, resultingin the need to solve many discrete complementarity prob-lems. A multigrid iteration method is developed to solvethese problems. Oosterlee (2003) states that the projectedline Gauss-Seidel smoother used in Clarke and Parrott(1999) is too involved, and studies alternate smoothersthat can be used in conjunction with the multigrid itera-tion method, finding that an alternating line Gauss-Seidelsmoother is a better choice, bringing about better conver-gence. Oosterlee (2003) also improves upon Clarke andParrott (1999) using a recombination of iterants. Ikonen andToivanen (2004) solve each of the discrete complementar-ity problems obtained after time and space discretizationusing operator splitting methods. The method divides eachtime step into two fractional steps, integrates the PDE withan auxiliary variable over the time step, then updates thesolution to satisfy the linear complementarity conditions
due to the early-exercise constraint. Componentwise split-ting methods are utilized in Ikonen and Toivanen (2007a) tosolve the discrete complementarity problems. The compo-nentwise splitting method splits each discretized LCP intothree LCPs, and the use of Strang symmetrization furtherdecomposes the three LCPs into five LCPs. Each LCP con-sists of tridiagonal matrices and is solved using the Brennanand Schwartz (1977) algorithm. Another technique, knownas the penalty method, replaces the unknown free-boundarywith a nonlinear penalty term and solves the resulting non-linear fixed-boundary problem. Zvan et al. (1998) use astandard Galerkin finite element method to discretize thearising PDE and use penalties to force the discrete prob-lems to satisfy the early-exercise constraint. The authorshighlight the equivalence of the penalty method and thelinear complementarity formulation. Ikonen and Toivanen(2007b) compare the five methods for pricing Americanoptions under stochastic volatility and find that, while theerror in prices computed by any of these methods is com-parable, the componentwise splitting method is consider-ably faster. They do, however, acknowledge that the PSORmethod is the easiest to implement, while the component-wise splitting method is the most difficult to implement.
As for non-PDE approaches, a nonparametric approachis utilized in Broadie et al. (2000). Several simulation meth-ods such as the least-squares Monte Carlo approach inLongstaff and Schwartz (2001), the primal-dual simulationalgorithm of Andersen and Broadie (2004), and the stochas-tic mesh method of Broadie and Glasserman (2004) thathave all been developed for American options under con-stant volatility can also be adapted to compute Americanoption prices under stochastic volatility.
1.2. Contribution and Outline
In this paper, we show that the problem of pricingAmerican options under stochastic volatility can, in fact, betransformed into a sequence of European-type option pric-ing problems. More specifically, the free-boundary problemarising in the pricing of American options can be trans-formed into a sequence of fixed-boundary problems. Bya European-type option, we mean that the exercise pol-icy is set a priori in the option contract. It is importantto note that solving fixed-boundary problems is not verydifficult in three dimensions if one leverages on existingPDE solver packages or libraries that are available to solvefixed-boundary problems. Software packages such as Com-sol even allow the reuse of contours from one solution asboundaries for another problem—making the implementa-tion of our method very easy. We also prove the conver-gence of the sequence of fixed-boundary problems. Ourresults show the methodology proposed here is faster thanother available methods while being as accurate as thePSOR method. Moreover, all it involves is solving linearPDEs in domains with fixed boundaries.
The approach we present differs in several ways fromthe other papers on American option pricing under stochas-tic volatility. First, the model we consider is far moregeneral. The numerical methods listed above are tailoredfor the Heston (1993) stochastic volatility model. Themethodology presented here works for the Hull and White(1987), Scott (1987), and Stein and Stein (1991) modelsas well. An assumption that is common to most existingpapers is that the market price of volatility risk is zero.As will be discussed in §2.2, the price of an Americanoption is, in general, not unique and depends on the mar-ket price of volatility risk. Most of these papers can beextended to handle nonzero volatility risk premiums. How-ever, because they do not consider nonzero market price ofvolatility risk, they could not study the effects of volatil-ity risk premium on the exercise policy and the price.The approach described in this paper readily accommodatesnonzero volatility risk premiums, and we study its effects.Several of the existing papers also use a wrong boundarycondition, which we correct.
In terms of underlying methodology, the solution tech-nique proposed in this paper is very different from any thatis available for pricing American options under a stochasticvolatility setting. Methods, like the penalty method, covertthe linear PDE with a free boundary to a nonlinear PDEwith a fixed boundary. On the other hand, the transforma-tion we propose retains the linear PDE but solves it withinfixed boundaries in each iteration. Because the complexityof dealing with a single nonlinear PDE is much harder thandealing with a sequence of linear PDEs, it is not surprisingthat runtimes are much better in our case. Methods like thecomponentwise splitting method introduce new approxima-tions other than the ones due to discretization, while oursdoes not. The accuracy depends only on the method usedto solve the fixed-boundary problem.
The method proposed in this paper extends the class ofmethods that are being called moving-boundary methods.Such methods were developed initially for singular con-trol problems (Muthuraman and Kumar 2006, Kumar andMuthuraman 2004) and were demonstrated to work numer-ically for higher-dimensional cases. For American optionpricing in the classical setting, Muthuraman (2008) extendsthis method and provides theoretical guarantees. This paperextends the method to optimal stopping problems in ahigher setting. Most importantly, due to the challengesin dealing with higher dimensions theoretically, there hadbeen absolutely no theoretical guarantees for the methodin any problem with more than one space dimensionality.This paper provides the first set of such guarantees.
The layout of the paper is as follows. Section 2 presentsthe model formulation. The transformation procedure ispresented in §3. A computational illustration of the pro-cedure is provided in §4, together with insights intohow stochastic volatility affects option pricing. Speed andaccuracy comparisons are also presented in §4. We con-clude in §5. All proofs are collected in Appendix A,
and a detailed discussion of the finite difference schemeused to solve the fixed-boundary problem is presented inAppendix B.
2. Model FormulationWe start this section with a discussion on the use of asecond stochastic process to model the evolution of volatil-ity. We then formulate the free-boundary problem thatthe American option price and the optimal-exercise policyjointly solve.
2.1. Stochastic Volatility Models
In stochastic volatility models, asset price (Xt) evolution isrepresented by
dXt =�Xt dt +�tXt dWt1 (1)
where � is the constant mean rate of return and Wt is astandard Brownian motion (a Wiener process). The instan-taneous volatility at time t is represented by �t . The volatil-ity �t = f 4Yt5, where Yt is another stochastic process andf 4 · 5 is a nonstochastic function. The evolution of Yt isrepresented by
dYt =�Y 4Yt5dt +�Y 4Yt5dZt1 (2)
where �Y and �Y are nonstochastic functions. In Equa-tion (2), Zt is another standard Brownian motion correlatedwith Wt . We assume a constant correlation � ∈ 6−1117,i.e., dWt dZt = �dt. Hence, Zt can be written as a lin-ear combination of Wt and an independent Wiener processZt such that Zt = �Wt + √
1 −�2Zt . Here, we have twosources of randomness, namely Wt and Zt , but only onetradeable asset, leading to market incompleteness. We referreaders to Björk (2004) and Fouque et al. (2000) for fur-ther discussions on stochastic volatility models and marketincompleteness.
Much of the literature on stochastic volatility focuses ona few specific models (Table 1). Financial data show that�< 0 (Fouque et al. 2000). Of the four models, the Hestonmodel is the most popular and the only one that allowsfor a nonzero correlation. Also, Dragulescu and Yakovenko(2002) provide evidence that the Heston model is in agree-ment with real-world data, leading to wider adoption of themodel by researchers and practitioners. Thus, for exposi-tional ease, we will restrict much of our attention to theHeston model and provide additional comments relevant tothe other models when necessary. The transformation pro-cedure developed in this paper works for all models listedin Table 1.
Consider an American put option written on an underly-ing asset with price Xt at time t given by Equation (1).The instantaneous volatility �t is represented by the Hes-ton model. The put option with strike price K and matu-rity time T written on this asset pays max 8K − Xt109 ≡4K −Xt5
+ at any time t ∈ 601 T 7. The price of an option,p4�1 x1 y5, is represented as a function of the time to expiry,� ≡ T − t, the underlying asset price x and the value, y,of the process Yt . As noted earlier, the market consideredis an incomplete one as a result of volatility not being atradable asset.
Assuming that the market selects a unique equivalentmartingale measure P∗4å5, derivatives should be priced withrespect to this measure to disallow arbitrage giving
p4�1x1y5
= sup�∈401�5
E∗4å5{
e−r4�−�54K−XT−�5+ ∣∣X� =x1Y� =y
}
1 (3)
where r > 0 is the risk-free rate of interest. As mentionedearlier, when the volatility of the asset price is constant,the optimal-exercise policy can be represented by a contin-uous, nonincreasing exercise boundary (e.g., Karatzas andShreve 1998). Analogous to the constant volatility case, theoptimal-exercise policy under stochastic volatility can berepresented by a surface (Fouque et al. 2000). The exercisesurface is a continuous, nonincreasing surface x = b4�1 y5,b2 �+ ×�→�+. This boundary partitions the state spaceand dictates the optimal-exercise policy. The region whereit is optimal to hold, known as the continuation region, isdefined as C = 84�1 x1 y5 ∈ �2
+ ×� � x > b4�1 y59, and theregion where it is optimal to exercise, known as the stop-ping region, is defined as S = 84�1 x1 y5 ∈ �2
+ × � � x ¶b4�1 y59.
In the continuation region, the price of the Americanoption satisfies the PDE
12�2t x
2 ¡2p
¡x2+�v�2
t x¡2p
¡x¡y+ 1
2v2�2
t
¡2p
¡y2+ rx
¡p
¡x
+ 4�4m′ − y5− v�tå5¡p
¡y− rp− ¡p
¡�= 0 (4)
for all 4�1 x1 y5 ∈C.In Equation (4), å denotes the market price of volatil-
ity risk. Because volatility cannot be traded, we have anincomplete market under the stochastic volatility setting.Although we lose the uniqueness of derivative pricing, forany given å there is a unique price på. An infinitesimalincrease in the volatility risk ��t increases the infinitesi-mal rate of return on the option by å. The market priceof volatility risk and its effect on pricing is discussedin §4.3. For a detailed discussion on the interpretationof å and its relation to equivalent martingale measureswe refer the reader to Bakshi and Kapadia (2003) andHenderson (2005).
In the exercise region, the price is the payoff
p4�1 x1 y5= 4K − x5+ (5)
for all 4�1 x1 y5 ∈S. For notational convenience, we definethe differential operator L such that the LHS of Equa-tion (4) is denoted by Lp.
To solve Equation (4) in the region C, the followingboundary conditions are needed:
p401 x1 y5= 4K − x5+1 (6)
p4�1 b4�1 y51 y5= 4K − b4�1 y55+1 (7)
limx→�
¡p
¡x= 01 (8)
limy→�
¡p
¡y= 01 and (9)
Ap ≡ rx¡p
¡x+�m′ ¡p
¡y− rp− ¡p
¡�= 0 at y = 00 (10)
Figure 1 illustrates the state space and boundaryconditions.
Equations (6) and (7) prescribe the value at expiry andat exercise. As the underlying asset price increases, theprobability that the asset price falls below K, before orat expiry, decreases. This increasingly guarantees a zeropayoff. Equation (8) reflects this behavior of the option.Equation (9) captures the argument that when volatility isextremely large, a marginal increase in volatility has littleeffect on the price.
The boundary condition at y = 0 is directly derivedby taking y = 0 in Equation (4). Several papers (includ-ing Clarke and Parrott 1999, Oosterlee 2003, Ikonen andToivanen 2004, Ikonen and Toivanen 2007a) that developinnovative numerical methods to price American optionsunder the Heston model argue that when volatility is zero,since there is no randomness, the payoff as well as theprice are deterministic, leading to p4�1 x105 = 4K − x5+.However, this is not the case because Yt is a mean-revertingprocess in the Heston model, meaning that if Yt = 0, then
Figure 1. The state space and boundary conditions.
almost surely dYt is positive, and hence Yt+ is greaterthan zero, making the asset price process nondeterministic.Therefore, the use of p4�1 x105= 4K − x5+, although eas-ier to handle, is incorrect and needs to be replaced by thePDE in Equation (10).
It is important to note that Equations (4)–(10) can besolved for any sufficiently smooth surface, i.e., exercisepolicy b. The optimal-exercise policy, however, is the onlypolicy for which p is smooth across the boundary b. Thiscondition, commonly known as the smooth pasting con-dition, implies that p as well as ¡p/¡x are continuousacross b and gives rise to Fouque et al. (2000):
limx↓b
¡p
¡x= −1 (11)
for optimality. A solution p4�1 x1 y51 b4�1 y5 to Equa-tions (4)–(11) also has to satisfy the condition
p4�1 x1 y5¾ 4K − x5+ ∀ 4�1 x1 y5 ∈S∪C (12)
in order to be the true price function and optimal-exercisepolicy.
2.3. Other Stochastic Volatility Models
The structure of the pricing problem described by Equa-tions (4)–(11) remains the same for the other modelsdiscussed in §2.1. However, the specific PDE and theboundary condition given in Equation (10) are model spe-cific. This is due to the differences in the evolution of Yt ,its domain, and the function f 4 · 5 that determines thevolatility.
Table 2 summarizes the PDE that replaces Equation (4)and the boundary condition that replaces Equation (10) foreach model.
3. Transforming the Free-BoundaryProblem
In this section, we first present the transformation pro-cedure that solves the free-boundary problem described
Table 2. Boundary conditions.
Model PDE boundary condition, boundary
Stein and Stein (1991)12�2t x
2 ¡2p
¡x2+���tx
¡2p
¡x¡y+ 1
2�2 ¡
2p
¡y2+ rx
¡p
¡x+ 4�4m− y5−�å5
¡p
¡y− rp− ¡p
¡�= 0
¡p
¡y= 01 y → −�
Scott (1987)12�2t x
2 ¡2p
¡x2+���tx
¡2p
¡x¡y+ 1
2�2 ¡
2p
¡y2+ rx
¡p
¡x+ 4�4m− y5−�å5
¡p
¡y− rp− ¡p
¡�= 0
12�2 ¡
2p
¡y2+ rx
¡p
¡x+ 4�m−�å5
¡p
¡y− rp− ¡p
¡�= 01 y → −�
Hull and White (1987)12�2t x
2 ¡2p
¡x2+�c2�
3t x
¡2p
¡x¡y+ 1
2c2
2�4t
¡2p
¡y2+ rx
¡p
¡x+ 4c1�
2t − c2�
2t å5
¡p
¡y− rp− ¡p
¡�= 0
p = 4K − x5+1 y = 0
in §2.2. We then demonstrate the mechanics of the trans-formation procedure with a numerical illustration.
The American option pricing problem is defined byEquations (4)–(11) and is satisfied by the price functionp4�1 x1 y5 and the optimal-exercise policy b4�1 y5. Nowconsider an arbitrary policy defined by a surface b04�1 y5.When such an arbitrary exercise policy is used by an optionholder, the value to the holder will be referred to as theassociated value p04�1 x1 y5. Clearly “the price” is the asso-ciated value of using the optimal-exercise boundary, thatis, p04�1 x1 y5 = p4�1 x1 y5 when b0 = b and p04�1 x1 y5¶p4�1 x1 y5 for any b0.
For an arbitrary exercise policy b0, one can find theassociated price p0 by solving Equations (4)–(10), whichcan be done using standard PDE techniques such as finitedifference and finite element schemes. A finite differencescheme for computing the associated values is detailedin Appendix B. For a given b0, the uniqueness of theassociated value function is established by the followingproposition.
Proposition 3.1. If pn satisfies Equation (4) with theboundary conditions given by Equations (6)–(10) for agiven bn4�1 y5, then pn is unique.
All proofs are collected in Appendix A.Our aim is to construct a transformation procedure that
will converge and provide the price of the option onconvergence. Starting from a guess b0, if we can con-struct a sequence of policies b01 b11 0 0 0 that are monotonicincreasing, i.e., bn4�1 y5 < bn+14�1 y5 for all (�1 y) and isbounded above, then convergence is inevitable. Exercise-policy improvement would also imply that pn4�1 x1 y5 <pn+14�1 x1 y5 for all (�1 x1 y). We construct and demonstratesuch a transformation below.
We require that the initial guess policy b0 be such that
¡p0
¡x
∣
∣
∣
∣
4�1 b04�1 y5+1 y5
<−10 (13)
Proposition 3.2 guarantees us that such a b0 can be obtainedwhen the chosen b0 lies below the optimal policy b. This is
necessary because we are interested in constructing a trans-formation procedure that monotonically converges to theoptimal b (Theorem 3.1). It has to be noted that although bis unknown, it is not difficult to choose a b0 < b. A wrongchoice would imply that Condition (13) is violated and arestart with a lower choice b0 would eventually work.
Proposition 3.2. If b04�1 y5 < b4�1 y5, then ¡p0/¡x�4�1 b04�1 y5+1 y5 <−1.
Say we begin with an arbitrary guess b0 and solve Equa-tions (4)–(10) and obtain p0. Now, on the boundary b0,p0 = 4K − x5+ and immediately above b0, the derivativew.r.t. x is less than −1. These together imply that p0 <4K − x5+ in a region immediately above b0. In this region,the policy b0 can easily be bettered because an immediateexercise will yield improvement. Hence, we could chooseany b1 in the region where p0 < 4K − x5+ and be guar-anteed of improving the exercise policy. Further guidingour choice of a new exercise policy is our preference tochoose a b1 that is also guaranteed to lie below b (that isb14�1 x1 y5 < b4�1 x1 y5) so that we can iterate. Such achoice is indeed possible and is given by
bn+14�1 y5={
sup4bn4�1 y51�5
x∣
∣
¡pn
¡x
∣
∣
∣
∣
4�1 x1 y5
<−1}
(14)
for all � and y. The theoretical guarantee that such abn+1 exists implies exercise policy improvement and is stillbelow b is provided by Theorem 3.1.
Theorem 3.1. Ifpn ∈C81129 solves Equations (4), (6)–(10),with ¡pn/¡x�4�1 bn4�1 y5+1 y5 < −1, then bn+1 as defined byEquation (14) exists. Furthermore, the price function pn+1
obtained using bn+1 is such that pn+1 > pn and ¡pn+1/¡x�4�1 bn+14�1 y5+1 y5 <−1.
The convergence of the improvement proceduredescribed by Equation (14) to ¡p/¡x = −1 at the exerciseboundary implies the satisfaction of Equation (11), yieldingoptimality.
The mechanism described above can viewed as decom-posing the American option pricing problem into asequence of European-style option pricing problems. ByEuropean style, we mean that the exercise policy is knowna priori, unlike for American options. This reveals thatpricing American options is only as difficult as pricing asequence of European-style options.
3.1. An Illustration
We now illustrate the mechanics of the transformation pro-cedure using a computational example. Consider the Hestonmodel with parameters used in Clarke and Parrott (1999):strike price K = 10, time to expiry T = 3 months, risk-freerate of interest r = 10% per annum, mean rate of reversion� = 5, long-term mean variance level m′ = 0016, volatilityof the Yt process � = 009, correlation � = 001, and market
Figure 2. Initial and final exercise policies.
01
23 0
0.20.4
0.60.8
1.00
2
4
6
8
10
Variance, yTime to expiry, �
Stoc
k pr
ice,
x
b
b0
price of volatility risk å = 0. We set the initial exercise-policy guess b04�1 y5 = 1 for all � and y. The transforma-tion procedure for this set of parameters converges to theoptimal exercise policy shown in Figure 2 after four itera-tions. Figure 2 also shows b0.
Figure 3 plots the exercise policies in each iteration forthe cut taken at y = 1. As can be recalled, for each exercisepolicy, the associated value function is first computed. Thenthe improved exercise policy is set as the contour of theassociated value function derivative w.r.t. x equalling −1.On each iteration, the maximum difference between theprevious and new associated value functions is measuredfor convergence to within a tolerance �. Alternatively, themaximum difference between the old exercise policy andthe new exercise policy can be measured for convergence.
Figure 4 shows the changes in the first derivative withrespect to x through the iterations for the cut taken at� = 105 months and y = 1. In each figure, the first dotted
Figure 4. Policy improvement at � = 105 months and y = 1.
0 1 2 3 4 5 6 7 8
–1.10
–1.05
–1.00
–0.95
–0.90
–0.85
–0.80
–0.75
Stock price, x
Firs
t der
ivat
ive
w.r
.t. x
b0
4 5 6 7 8
Stock price, x
b1
4 5 6 7 8
Stock price, x
b2
0 1 2 3 4 5 6 7 8
–1.10
–1.05
–1.00
–0.95
–0.90
–0.85
–0.80
–0.75
Stock price, x
Firs
t der
ivat
ive
w.r
.t. x
b3 ≈ b4 ≡ b
line represents the location of the current boundary, andthe second represents the location of the largest x such that¡p/¡x ¶ −1. As is evident from Figure 4, the transfor-mation procedure progressively seeks to convert any non-optimal exercise strategies that are exposed by regions with¡pn/¡x < −1. The converged price function does have¡pn/¡x¾−1 for all x, satisfying Equation (11).
For further illustrative purposes, also consider the Hull-White model with the following parameters: c1 = 003,c2 = 006, and � = 0. Other parameters are kept the sameas in the previously considered Heston model. Figure 5plots b0 and the converged b that is obtained using thetransformation procedure. Although the completed optimal-exercise boundaries in Figure 2 (Heston model) and Fig-ure 5 (Hull-White model) are not directly comparable dueto the difference in volatility dynamics and noncomparableparameters, the structural differences at the y = 0 boundaryis evident. Unlike the Heston model, under the Hull-Whitemodel, it is strictly optimal to exercise for any x below Kwhen y = 0. This obviously stems from the fact that thelognormal process used in the Hull-White model becomescompletely deterministic when y = 0, while this is not thecase in the Heston model.
Figure 5. Optimal exercise policy for the Hull-Whitemodel.
In this section, we use the transformation proceduredescribed in §3 as a computational tool to study vari-ous phenomena. Using an exhaustive set of computationalexperiments, we seek insights into the price premium due tostochastic volatility, effects of correlation on option pricingand exercise policies, effects of market price of volatilityrisk, and the structure of implied volatilities observed instochastic volatility models and their dependence on corre-lation and market price of volatility risk. Then we provideruntime/error comparisons against various existing compu-tational methods discussed earlier in §1.
4.1. Stochastic Volatility
Assuming constant volatility, as in the Black-Scholesmodel, leads to the mispricing of options. In this section,we study these price differences and the effects of intro-ducing stochastic volatility into the Black-Scholes model.
In Figure 6, we plot the price differences betweenoptions priced in a constant volatility setting, and optionspriced in a Heston stochastic volatility model setting, forthree different volatility levels. We use the same modelparameters considered in §3.1 but assume that the volatil-ity process and the underlying price process are uncorre-lated, i.e., �= 0. Letting p�24�1 x5 denote the price of a putoption with constant volatility � , we plot p4�1 x100045 −p00224�1 x5 in Figure 6(a), p4�1 x100165−p00424�1 x5 in Fig-ure 6(b), and p4�1 x100365− p00624�1 x5 in Figure 6(c), forvarious values of � and x. The optimal-exercise boundariesfor the constant volatility case (dotted line) and stochasticvolatility case (solid line) are plotted on the ceiling/floor ofthe 3-d plot as well.
Figure 6(a) shows that the constant volatility modelsyields a lower price because it does not account for thevery high likelihood of volatility increasing (� = 002, y =�2 = 0004 and m′ = 0016), and higher volatilities yieldhigher option prices in the classical American and Euro-pean constant volatility model. Figure 6(c), however, showsthat the constant volatility model yields a higher price thanthe stochastic volatility model, as the stochastic volatility
Figure 6. Price difference (stochastic volatility—con-stant volatility) at different volatility levels.
00.05
0.100.15
0.200.25
05
1015
200
0.05
0.10
0.15
0.20
Time to expiry, �
Stock price, x
Dif
fere
nce
in o
ptio
n pr
ice
(a) � = 0.2
00.05
0.100.15
0.200.25
05
1015
20–0.020
–0.015
–0.010
–0.005
0
0.005
0.010
Time to
expiry
, �
Stock price, x
Dif
fere
nce
in o
ptio
npr
ice
(b) � = 0.4
00.05
0.100.15
0.200.25
05
1015
20–0.20
–0.15
–0.10
–0.05
0
0.05
Time t
o expir
y,�
Stock price, x
Dif
fere
nce
in o
ptio
npr
ice
(c) � = 0.6
model accounts for the likely drop in volatility. These pricedifferences can be explained by the mean-reverting natureof volatility in the model. Looking at Figure 6(b), the casewhen the spot variance is equal to its long-term average,we find that there is no constant overpricing or underpric-ing of the option. In this situation, volatility mean-reversionhas negligible effect on the option price. Instead, price dif-ferences occur because of the effect of implied volatilities,because implied volatilities for options at-the-money tendto be lower than implied volatilities for options in- andout-of-the-money, whereas the constant volatility model
assumes that volatilities are constant everywhere. We alsofind that the exercise boundaries for the constant volatilitycase and stochastic volatility case intersect in Figure 6(b),unlike in Figures 6(a) and 6(e), where the boundary for thestochastic volatility case is consistently above or below theconstant volatility exercise boundary.
It would be insightful to study the effects of stochasticvolatility on the two components of the American optionprice—the associated European option price and the earlyexercise premium. We plot European option prices andearly exercise premiums, at time T , obtained in the con-stant volatility case and the stochastic volatility case inFigures 7 and 8. The effects described in Figure 6 can bereadily observed in Figure 7 as well. The European com-ponent price reflects that for lower volatility levels, themean-reverting nature of volatility prices the option higher.However, in Figure 8, for lower volatility levels the premi-ums are not consistently larger. We find that as the volatil-ity level increases, early exercise rights for the stochasticvolatility setting become relatively more valuable. This isbecause early exercise premiums benefit significantly fromhigher randomness and as volatility levels increase, ran-domness in the stochastic volatility setting becomes sig-nificantly higher. One also has to keep in mind that whenin-the-money, the option holder prefers randomness muchless than when out-of-the-money.
To study solely the effect of stochastic volatility onoption pricing, we plot option prices obtained using adeterministic mean-reverting volatility model, the stochas-tic volatility model, and the constant volatility model attime T in Figure 9, for � = 002, 004, and 006. The mean-reverting effect that was observed in Figure 6 can beobserved in Figure 9 as well, as is to be expected. In Fig-ure 9(b), the plot of option prices for the constant volatilitymodel is not clearly observable because it is actually veryclose to the plots of option prices for the stochastic anddeterministic mean-reverting volatility models.
In all three cases, with respect to the deterministic model,the stochastic nature of volatility underprices options forlower stock prices and overprices for larger stock prices.For stock prices well below the option’s strike price, theholder would prefer less randomness in the system toincrease the probability of the option remaining, or expir-ing, in-the-money. A deterministic mean-reverting volatil-ity model would thus be preferable to a stochastic model.Therefore, option prices for the deterministic model arehigher, reflecting the holder’s preference for the determin-istic model. But for higher stock prices, the holder wouldprefer more randomness to ensure that the stock price willfall below the strike price of the option. One can also notethat as � increases, the stock price at which the two setsof option prices intersect increases as well. This impliesthat the deterministic model is preferred over the stochas-tic model by the option holder for a larger range of stockprices. Now, in the deterministic model, volatilities tendtoward the mean volatility level in a known fashion. As the
Figure 7. European option price differences at time T .
7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.00
0.5
1.0
1.5
2.0
Stock price, x
Eur
opea
n op
tion
pric
e, p
Stochasticvolatility
Constantvolatility
10.5 11.0 11.5 12.0 12.5 13.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Stock price, x
Eur
opea
n op
tion
pric
e, p
Stochasticvolatility
Constantvolatility
7 8 9 10 11 12 13 14 15 16
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Stock price, x
Eur
opea
n op
tion
pric
e, p
Stochasticvolatility
Constantvolatility
(a) � = 0.2
(b) � = 0.4
(c) � = 0.6
volatility level increases, the downward drift of the volatil-ity process increases as well. Thus, as volatility increases,the deterministic model is capable of bringing about dropsin larger stock prices and increasing the probability of theoption expiring in-the-money. As a result, we find that thethreshold over which the stochastic model becomes pre-ferred to the deterministic model increases as volatilityincreases.
Figure 8. Changes in early exercise premium at time T .
This section considers the effects of correlation betweenthe underlying price process and the volatility process, oth-erwise known as the “leverage” effect. The leverage effectis well studied in the classical setting. In the stochasticvolatility setting correlation is often assumed to be zeroalthough inconsistent with real-world data. Option pricesand exercise policies for correlation � set to −1, 0, and 1
are shown in Figure 10. Correlation affects the option pricedifferently when the stock price x is above and below thestrike price K. It is interesting to note that the option priceis unaffected by � when x =K. As Figure 10(a) illustrates,option prices increase with correlation when x < K. Theopposite is true when x >K.
An increase in correlation can be thought of as adecrease in overall uncertainty in the system. Whenout-of-the-money (x >K), it is only natural that the holder
Figure 10. Effects of changing correlation �.
8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.00
0.5
1.0
1.5
2.0
Stock price, x
Opt
ion
pric
e, p
� = –1� = 0� = 1
(a) Price functions (� = 1.5 months and �t = 0.4)
0 1 2 36.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Time to expiry, �
Stoc
k pr
ice,
x
� = 1.5 months
(b) Exercise policies (�t = 0.4)
� = –1� = 0� = 1
prefers more randomness to increase chances of a posi-tive payoff, hence explaining option price decreases withincreases in correlation when out-of-the-money. When in-the-money (x < K), the holder’s preference for less ran-domness is indicated by the increase in price with increasesin correlation.
Exercise boundaries decrease as correlation increases, asshown by Figure 10(b). This aligns with the notion thatunder optimality, higher prices imply later exercise, becauseat the exercise boundary the payoff is always 4K − x5+.
4.3. Impact of Market Price ofVolatility Risk on Option Pricing
Next, let’s consider the effect of å, the market price ofvolatility risk. An interpretation for å can be obtained byconsidering delta-hedged option portfolios that are con-structed by buying an option and hedging it with a fractionof the underlying asset, such that in the Black-Scholes set-ting, the rate of return matches the risk-free interest rate.
The gain on this portfolio is the difference between the risk-free rate and the earnings from the portfolio. Obviously, ina Black-Scholes setting, when the delta-hedged portfolio iscontinuously rebalanced, the delta-hedged gain is alwayszero. Bakshi and Kapadia (2003) study and relate the delta-hedged gains to the market price of volatility risk. Theyshow that when the market price of volatility risk å is pos-itive (negative), the expected delta-hedged gain is positive(negative). It also implies that when volatility is stochasticand volatility risk not priced, i.e., when å= 0, the expecteddelta-hedge gain is zero. Furthermore, using market data,they demonstrate empirically that å is negative. The find-ing that å is negative is consistent with the notion thatmarket volatility rises when market return drops.
Figure 11(a) shows that as å increases, p decreases.Henderson (2005) shows that for European put options, asthe market price of volatility risk increases, the price ofthe option decreases. The question is whether this trans-lates to American option pricing as well. Figure 11 provides
Figure 11. Effects of changing volatility riskpremium å.
8 9 10 11 12 13 140
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Stock price, x
Opt
ion
pric
e, p
Λ = –2Λ = 0Λ = 2
0 1 2 37.0
7.5
8.0
8.5
9.0
9.5
10.0
Time to expiry, �
Stoc
k pr
ice,
x � = 1.5 months
Λ = –2Λ = 0Λ = 2
(a) Price functions (� = 1.5 months and �t = 0.4)
(b) Exercise policies (�t = 0.4)
numerical evidence that it does hold true for American putoptions as well.
An insight into this behavior is revealed by consideringthe delta-hedged portfolio. When the market does not pricevolatility risk, the delta-hedged gain should be zero, andthe price of the option matches the price of constructing theportfolio. When the market does price volatility risk, how-ever, the delta-hedged gain is proportional to the volatilityrisk premium. Hence, as å increases, the expected gainon this portfolio also increases. In this case, the price ofthe option would be the price of constructing the portfoliominus the expected gain.
4.4. On Implied Volatilities
One of the primary arguments favoring stochastic volatil-ity models is that the implied volatilities computed fromobserved option prices are not constant and often exhibit a“smile” curve when plotted against strike prices. We exam-ine the nature of implied volatility curves and their depen-dence on both correlation and market price of volatility riskin Figure 12. Implied volatilities are computed for variousstrike prices from 8 to 12, at an underlying stock pricex = 11. We use a bisection search in conjunction with thebinomial-tree method to calculate the implied volatilities.
The smile curves are well captured in Figure 12(a). Theypivot at K = 11, which is the underlying stock price atwhich the implied volatilities are computed. When K < x,the implied volatility decreases as correlation increases.On the other hand, when K > x, a decrease in correla-tion causes a decrease in the implied volatility as well.From §4.2, we know that for K < x, as � increases, pdecreases. This translates directly to decreasing impliedvolatility. A drop in p therefore implies a drop in volatility.By a similar reasoning, because p increases as � increasesfor K > x, the monotonic relationship between option priceand volatility leads to the implied volatility increasing.
The smile curves in Figure 12(b) demonstrate that aså increases, the implied volatility decreases. Recall from§4.3 that as å increases, the option price p decreases.This decrease in option price leads to a lower impliedvolatility, because option prices increase monotonicallywith volatility.
4.5. Runtime and Accuracy Comparisons
Ikonen and Toivanen (2007b) exhaustively compare the fivemethods available in literature for pricing American optionsunder stochastic volatility. These are the PSOR, multigrid,operator splitting, penalty, and the componentwise split-ting (CS) method. Of these, they find that CS performsfastest, with comparable accuracy to the other methods.From the standpoint of implementation, the authors notethat the PSOR method is the easiest and CS is the hardest.The remaining three methods fall in between these two interms of speed/accuracy and ease of implementation, high-lighting a trade-off between ease of implementation and the
Figure 12. Effects of changing � and å on impliedvolatilities.
8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.00.20
0.25
0.30
0.35
0.40
0.45
Strike price, K
Impl
ied
vola
tility
� = –0.8
� = –0.5
� = 0
� = 0.5
� = 0.8
(a) Changing �
8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.00.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
Strike price, K
Impl
ied
vola
tility
Λ = 2
Λ = 1
Λ = 0
Λ = –1
Λ = –2
(b) Changing Λ
speed/accuracy of the method. In this section, our objec-tive is to place the method presented in this paper. To thisextent, we compare the speed and accuracy to that of onlythe CS method and the PSOR method, because the othermethods are known to fall between these two, both in termsof speed/accuracy and implementation ease. The compar-isons were carried out using a C++ implementation withthe GMM++ library on a 2.8 GHz Intel Xeon Mac Prowith 2 GB RAM and 1.6 GHz bus speed.
It is also important to keep in mind that implementa-tion of the proposed method is easy and straightforwardbecause it solves only a fixed-boundary PDE problem ineach iteration, and that the PDE is the classical convection-diffusion PDE with a second-order cross-derivative term.Several off-the-shelf packages can do this readily. We alsoprovide in Appendix B a simple finite difference schemethat solves the fixed-boundary problem that is used for theresults in this section. In the finite difference scheme, timestepping is done using an implicit Euler method. Note that
the method proposed here, in its current form, works forall the popular stochastic volatility models and allows forboth positive and negative volatility risk premiums.
As earlier, for our method, the first boundary guess isb04�1 y5= 1 for all � and y, and we take the parameter setthat has been used in the illustration. The relaxation param-eter and stopping criterion used for the PSOR method arethe same as those found in Ikonen and Toivanen (2007b).Tables 3 and 4 list the prices obtained using the respectivemethods for five initial asset prices and two volatilities forvarious grid sizes. For the true values, we use the valueslisted in Ikonen and Toivanen (2007b), which the authorsobtain from using the CS method in conjunction with avery fine grid.
Figure 13 plots the root-mean-square errors (RMSE) andruntimes for the three different methods for the variousgrid sizes listed in Tables 3 and 4. The performance of theCS method in relation to the PSOR method is comparableto the relation found in Ikonen and Toivanen (2007b). Asthe figure shows, for the same accuracy the transformationprocedure is on average 10 times faster than the PSORmethod and more than twice as fast as the componentwisesplitting method.
As the figure shows, the transformation procedure hasbetter accuracy than the CS method, particularly on coarsergrids. This is understandable because the CS method intro-duces additional errors when the splitting is performed. Atthe same time, the speed of our scheme is greater thanthat of the PSOR method and the CS method. As men-tioned before, the CS method is harder to implement thanthe PSOR method, but this leads to the better performanceof the CS method. It is interesting to note, however, thatimplementing our transformation procedure is no more dif-ficult than implementing the PSOR method because ourtransformation procedure is essentially a standard finite dif-ference implementation, modified to update the boundaryduring each iteration.
The speed and accuracy of option prices obtained usingthe transformation procedure are highly dependent on thechoice of the fixed-boundary problem solver. As Figure 13shows, however, even with simple finite differences on auniform grid, the scheme performs well, obtaining accu-rate option prices quickly. In our C++ implementation,we have used the generalized minimal residual method(GMRES) provided by the GMM++ library to solve theresulting system of linear equations. The use of special-ized solvers such as finite element methods would increasethe speed and accuracy of the scheme by a significant fac-tor; however, it would do so by complicating implementa-tion if off-the-shelf packages are not taken advantage of.This is illustrated in Muthuraman (2008) when comparingAmerican option pricing methodologies in a Black-Scholessetting.
5. Concluding RemarksMuch of the literature on option pricing under stochasticvolatility focuses on European options. We considered in
this paper the harder problem of pricing American optionsin a stochastic volatility setting. It was shown that com-puting the price of an American option under stochasticvolatility is only as difficult as computing the price of a
Figure 13. A comparison of RMSE and computingtime.
series of European-type options when the exercise policiesare predetermined. A computational procedure for calculat-ing the price as well as the optimal-exercise boundary wasdeveloped. Using this procedure, we have sought insightsinto the dependence of American option prices, exercisepolicies, and implied volatilities on factors such as the mar-ket price of volatility risk and the correlation between stockprice and the volatility process. The method was demon-strated to be as accurate as the PSOR method, while havingbetter speeds than other existing methods in our numeri-cal experiments. An avenue for future research would be toextend such a pricing methodology for other American-typederivatives. Specifically, pricing options on multiple assetswould be interesting because there are also inherently mul-tidimensional problems. The theoretical guarantees estab-lished for the proposed method critically depend on themaximum principle (Theorem A.1). This begs the ques-tion of whether the methodology can be extended to higherdimensions if one can establish the maximum principle. Fora problem in higher dimensions, if one can determine theboundary update conditions and also prove the maximum
principle for the difference in value between two iterations,then certainly the critical part of establishing convergenceis in place. The other details would be problem dependent.Empirical studies that explore questions of investor exercisebehaviors against those computed by the optimal-exercisepolicy would be interesting as well.
Appendix
A. ProofsThe proofs of theorems and propositions are collected here.We begin with the statement and proof of Theorem A.1,which establishes the maximum principle that is needed forsubsequent proofs.
Theorem A.1. For a given T ∈ 401�5 and a continuousg4�1 y5 > 0 for all 4�1 y5 ∈ 401 T 5×�+, let C= 84�1 x1 y5 ∈401 T 5×�2
+ � x > g4�1 y59. Also, let h be the solution to
Lh= 0 in C (A1)
(L is defined in Equation (4)), with boundary conditionsgiven by
h401 x1 y5= 01 (A2)
h4�1 g4�1 y51 y5= F 4�1 y51 (A3)
limx→�
¡h
¡x= 01 (A4)
limy→�
¡h
¡y= 0 and1 (A5)
rx¡h
¡x+�m′ ¡h
¡y− rh− ¡h
¡�= 0 at y = 00 (A6)
If r > 0 and F 4�1 y5 > 0 for all 4�1 y5 ∈ 401 T 5 × �+,then the maxima of h are attained only on the boundary4�1 g4�1 y51 y5, and the minima of h are attained on theboundary 401 x1 y5.
Proof. For notational convenience, let A, B, and C repre-sent ¡2h/¡x2, ¡2h/4¡x¡y5, and ¡2h/¡y2, respectively.
We show that the maxima of h is attained only onthe boundary 4�1 g4�1 y51 y5 by ruling out other possibili-ties. First, say an internal maxima exists and is attained atsome 4� ′1 x′1 y′5. Now, this maxima will be no less thanh4�1 x1 y5 for all � , x, and y, meaning that h4� ′1 x′1 y′5 ¾h4�1 g4�1 y51 y5. This implies that h4� ′1 x′1 y′5 > 0 becauseF 4�1 y5 > 0. Also, by the necessary conditions for an inter-nal maxima, we have that ¡h/¡x = ¡h/¡y = ¡h/¡� = 0.Substitution into Equation (A1) yields
rh= 12Ax
2y+B��xy+ 12C�
2y0 (A7)
The Hessian needs to be a negative semidefinite matrix.This implies that the determinant of the leading n×n prin-cipal minor of the Hessian is nonnegative (nonpositive)
when n is even (odd); hence at 4� ′1 x′1 y′5, AC − B2 ¾ 0,and A ¶ 0. Thus at this point if A 6= 0, the second-orderdifferential terms in Equation (A7) can be rearranged as
12Ax2y+B��yx+ 1
2C�2y
= 12Ay
(
x2 + 2B��A
x+ C�2
A
)
= 12Ay
[(
x+ B��
A
)2
+(
AC −B2�2
A2
)
�2
]
0 (A8)
Now, because �2 ¶ 1, we have AC −B2�2 >AC −B2 > 0.Hence, from Equation (A7) we have rh < 0, implyingh4� ′1 x′1 y′5 < 0, a contradiction. If A = 0, it follows thatB = 0, leading to a similar contradiction. Therefore, themaxima cannot be attained in the interior.
Now assume that the maximum is attained at some4T 1 x′1 y′5, i.e., on the boundary � = T . We must have thath4T 1 x′1 y′5¾ max�1 y F 4�1 y5 > 0, with ¡h/¡x = ¡h/¡y=0and ¡h/¡� ¾ 0. Substitution into Equation (A1) now yields
rh= 12Ax2y+B��xy+ 1
2C�2y− ¡h
¡�0 (A9)
Considering the function h along the cut taken at T , wemust again have A¶ 0 and AC − B2 ¾ 0. Rearranging (ifA 6= 0), we have
h= 1r
(
12Ay
[(
x+ B��
A
)2
+(
AC −B2�2
A2
)
�2
])
−1r
¡h
¡�0
(A10)
Using arguments similar to those used earlier, even whenA= 0 we have h4T 1 x′1 y′5 < 0, again a contradiction. Themaxima cannot be attained on the boundary � = T , either.
Next, say that the maximum is attained at some4� ′1 x′105, i.e., on the boundary y = 0. Again, at 4� ′1 x′105,by conditions of maxima, we have ¡h/¡x = ¡h/¡� = 0 and¡h/¡y ¶ 0. Substitution into Equation (A6) gives
h= 1r�m′ ¡h
¡y1 (A11)
leading to h4� ′1 x′105¶ 0, another contradiction.Finally, we need to rule out the possibility of a max-
ima being reached as y → �. Consider a boundary at aY > max64å�/2�521m′7. Now assume that the maxima isachieved at some 4� ′1 x′1 Y 5 on the boundary y = Y . At thispoint, the coefficient of hy is negative and because this isa maxima, the Hessian is negative semidefinite, implying anegative LHS, therefore resulting in a contradiction.
Now because the maxima clearly cannot be attained onthe boundary � = 0, this leaves us with the result. Argu-ments and reasoning as in the above will establish that theminima of h is achieved on the boundary 401 x1 y5.
Proposition A.1. If F 4�1 y5 = 0 for all 4�1 y5 ∈ 401 T 5 ×�+, then h4�1 x1 y5= 0 for all 4�1 x1 y5 ∈ C.
Proof. The proof follows directly from the proof ofTheorem A.1. If F 4�1 y5 = 0 for all 4�1 y5 ∈ 401 T 5 ×�+, the maximum of h is also 0, because the maximaof h is attained at the boundary 4�1 g4�1 y51 y5, whereh4�1 g4�1 y51 y5= F 4�1 y5. Because the minima of h is also0, this must mean that h= 0 for all 4�1 x1 y5 ∈ C. �Proof of Proposition 3.1. Assuming there exist two solu-tions h1 and h2, considering the equations solved by h =h1 −h2 and using Theorem A.1 directly gives the result.
Proof of Proposition 3.2. Let C0 = 84�1 x1 y5 ∈ 401 T 5×�2
+ � x > b04�1 y59 and Cb = 84�1 x1 y5 ∈ 401 T 5×�2+ � x >
b4�1 y59. In the region C0 − Cb, it is optimal to exercise,but the exercise policy dictated by b0 chooses to subop-timally hold. Hence in C0 − Cb, p0 < p = 4K − x5+. Onthe boundary b0, we have that p0 = 4K − x5+. Also, byTheorem A.1, the maxima of p0 is attained on b0. Becausep0 = K − x on b0 and p0 < K − x in C0 − Cb, we musthave that ¡p0/¡x�4�1 b04�1 y5+1 y5 <−1. �
For the rest of this section, subscripts denote derivatives.
Proof of Theorem 3.1. Because pnx4�1 bn4�1 y5+1 y5 <
−1 and limx→� pnx4�1 x1 y5= 0, bn+1 exists by the continu-
ity of pnx in Cn, where Cn = 84�1 x1 y5 ∈ 401 T 5×�2
+�x >bn4�1 y59. The definition of bn+1 also implies bn+1 > bn.
From the definition of bn+1, for all x ∈ 4bn4�1 y51bn+14�1 y55, � , and y, we have pn
x4�1 x1 y5 < −1. Thisimplies
pn4�1 bn+14�1 y51 y5−pn4�1 bn4�1 y51 y5
<−4bn+14�1 y5− bn4�1 y551
pn4�1 bn+14�1 y51 y5
< pn4�1 bn4�1 y51 y5+ bn4�1 y5− bn+14�1 y5
=K − bn+14�1 y5
= pn+14�1 bn+14�1 y51 y50
Thus, pn+1 >pn on bn+1. Now, the difference p = pn+1 −pn
solves
Lp = 0 in Cn+11
p401 x1 y5= 01
p4�1 bn+14�1 y51 y5 > 01
limx→�
px4�1 x1 y5= 01
limy→�
py4�1 x1 y5= 01 and
Ap = 00
By Theorem A.1, p attains its maxima on bn+1 and its min-ima of 0 on the boundary 401 x1 y5 for 4x1 y5 ∈ 4bn+11�5×�+. This implies that p > 0 in Cn+1, i.e., pn+1 >pn.
Finally, we show that pn+1x 4�1 bn+14�1 y5+1 y5 <−1.
Because pn+1x = pn
x + px and pnx4�1 b
n+14�1 y51 y5= −1by the definition of bn+1, it suffices to show that px4�1bn+14�1 y5+1 y5 < 0. Assume that px4�1 b
n+14�1 y5+1 y5¾ 0 instead. Now because limx→� px4�1 x1 y5 = 0,px4�1 b
n+14�1 y5+1 y5 ¾ 0 implies that the maxima of pis attained in Cn+1. But this contradicts Theorem A.1,which states that the maxima of p is attained on bn+1.Therefore, we must have that px4�1 b
B. Finite Difference ImplementationThe fixed-boundary problem defined by Equations (4)–(10)can be solved using the finite difference method. In this sec-tion, we discuss relevant implementation issues using thefinite difference scheme for the Heston stochastic volatilitymodel.
For the sake of numerical implementation, the time axis,asset-price axis, and variance axis are truncated to 601 T 7,601 X7, and 601 Y 7, respectively, for some large enoughT 1 X1 Y ∈ �+. The boundary conditions for x → � andy → � are applied at X and Y , respectively.
The time axis, asset-price axis, and variance axis arediscretized into l, m, and n pieces yielding grid steps�� = T /l, �x = X/m, and �y = Y /n, respectively. Fork = 01 0 0 0 1 l, i = 01 0 0 0 1m, and j = 01 0 0 0 1 n, the price ofthe option at node 4k1 i1 j5 is denoted by p4�k1 xi1 yj5 =p4��k1�xi1 �yj5 = pk
i1j . Using this notation, a finite dif-ference discretization based on central differences can beobtained as follows:
Given an exercise policy b, the remaining boundary condi-tions are represented as follows:
p0i1 j = 4K − �xi5
+ for i = 01 0 0 0 1m1 (B2)
pki1 j = 4K − �xi5
+ if xi ¶ b4�k1 yj51 (B3)
pki1 j −pk
i−11 j = 0 if i =m 4xi = X51 and (B4)
pki1 j −pk
i1 j−1 = 0 if j = n 4yj = Y 50 (B5)
Given the price for a time to expiry �k−1, i.e., pk−1i1 j ∀ i1 j ,
the price at time �k can be obtained by solving a systemof linear equations Dpk = pk−1, where pk is a mn-vectorthat represents the option prices for all asset prices andvolatilities at time step �k, and the mn × mn matrix D isassembled using Equations (B1)–(B5) at each time step.This set of equations is solved for each k = 11 0 0 0 1 l. Theresulting matrix p is then the value function associated withthe exercise policy b.
AcknowledgmentsWe thank the associate editor, Mark Broadie, Haolin Feng,Jose Figueroa-Lopez, Stanley Pliska, Bruce Schmeiser,Stathis Tompaidis, and two anonymous referees for theircomments, suggestions, and feedback.
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