Longevity Risks Modelling and Financial Engineering

8/14/2019 Longevity Risks Modelling and Financial Engineering

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Universitt Ulm Fakultt fr Mathematik und Wirtschaftswissenschaften

Longevity Risks:Modelling and

Financial Engineering

Dissertationzur Erlangung des Doktorgrades

Dr. rer. nat.der Fakutt fr Mathematik und Wirtschaftswissenschaften

der Universitt Ulm

vorgelegt vonDipl.-Math. oec. Shaohui Wang

ausBaixiang, China

Ulm, im Mai 2008


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Amtierender Dekan Prof. Dr. Frank Stehling

Gutachter1. Gutachter: Prof. Dr. Rdiger Kiesel2. Gutachter: Prof. Dr. Hans- Joachim Zwisler

Tag der Promotion 07. Juli 2008


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Acknowledgements

This thesis would not have been possible without the nancial support by GermanScience Foundation Research Training Group: modelling, analysis and simulation ineconomy mathematics.

I take the opportunity to thank my academic advisor Professor Dr.Rdiger Kiesel who opened me the door to the inter-discipline research of longevity risks from nan-cial mathematical perspectives and who supported my studies and stays in every phase.I highly appreciate his strategical advises to my work and his encouragement, whichbuilded up an ideal research environment beyond my expectation.

I would also like to thank Professor Dr. Hans- Joachim Zwiesler who introducedthe actuarial science to me and was friendly available as second supervisor to my thesis.

Further, I thank my parents for their nancial and emotional supports for my stud-ies and stays in Germany. I thank my young brother and his wife to take care of ourparents since I was not at home when they needed us.

Last but not least, my thanks go to my wife Jiao Xueli for her caring support andencouragement during the writing of this thesis.


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Contents

Contents ii

List of Figures v

List of Tables vi

1 Introduction 1

1.1 Moti vation to Systematic Mortality Risks . . . . . . . . . . . . . . . . . 1

1.2 Objective of the Thesis and Contribution . . . . . . . . . . . . . . . . . 2

1.3 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Mortality Ri sks 9

2.1 Life Contingencies in Classical Actuarial Perspectives . . . . . . . . . . 9

2.1.1 Deterministic Mortality Rate . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Evaluation of Insurance Contracts: Principle of Equivalence . 12

2.1.3 Net Reserves and Thieles Diff erential Equation . . . . . . . . . 16

2.2 Life Contingencies on Systematic Mortality Risks . . . . . . . . . . . . 18

2.2.1 Stochastic Mortality Rate . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Systematic Mortality Risks . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3 Analysis of JPMorgan LifeMetrics . . . . . . . . . . . . . . . . . 26

2.2.4 Concrete Models and Fittings with JPMorgan LifeMetrics . . 31

ii


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CONTENTS iii

3 Alternative Approach to No -Arbitrage Hypothesis 42

3.1 Motivation: non-tradable Bank Account . . . . . . . . . . . . . . . . . . 42

3.2 Alternative Denition of No- Arbitrage Opportunity . . . . . . . . . . . 433.3 Hedg ing of Contingent Claims . . . . . . . . . . . . . . . . . . . . .. . . 47

4 Financial Market involving Survivor Bonds 54

4.1 Arbitrage-free Financial Market: Lvy Finance . . . . . . . . . . . . . . 54

4.1.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.2 Default-free Zero-coupon Bonds Market . . . . .. . . . . . . . 55

4.1.3 Arbitrage-free Risky Securities Market . . . . . . . .. . . . . . 60

4.2 Calibration of The Risk-neutral Measure with Spot Prices . . . . . . . . 64

4.2.1 Calibration of the Term Structure Model . . . . . . . .. . . . . 65

4.2.2 Calibration of Equity Model . . . . . . . . . . . . . . .. . . . . . 68

4.3 Arbitrag e-free Survivor Bonds Market . . . . . . . . . . . . . . . . . . . .69

4.3.1 Term Structure of Survivor Bonds . . . . . . . . . . . .. . . . . 70

4.3.2 Evalua tion of Contingent Claims related toLongevity Risks . . 79

4.4 Application to Unit-linked Insurance Contracts . . .. . . . . . . . . . . 88

4.4.1 Risk-neutral Evaluation of Unit-linked Contract . . . . . . . . . 89

4.4.2 Market Reserve and Thieles Diff erential Equation for Unit-linked Pure Endowment Contract . . . . . . . . . . . .. . . . . 91

5 Longevity Derivatives 97

5.1 Longevity Risk Mangement . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.1.1 Analysis of the Maximum of Future Lifetimes . . . . . . . . . . 98

5.1.2 Pure Longevity Derivatives . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Risk-neutral Valuation and Hedging of Longevity Derivatives . . . . . . 100

5.2.1 Risk-neutral Valuation of Longevity Derivatives . . . . . . .. . 100

5.2.2 Hedging Strategies for Longevity Derivati ves . . . . . . . . . . . 104

5.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106


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CONTENTS iv

A Technical Backgrounds 114

A.1 Calculus with Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . 114

A.2 Calculus with Lvy Processes . . . . . . . . . . . . . . . . . . . . . . . . . 124

Bibliography 128


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List of Figures

2.1 QQnorm plot of standardized residuals . . . . . . . . . . . . . . . . . . . 28

2.2 Empirical density of standardized residuals. . . . . . . . . . . . . . . . . 302.3 Model W1 with M2: generalized Hyperbolic distribution . . . . . . . . 36

2.4 Model W1 with M2: Hyperbolic distribution . . . .. . . . . . . . . . . 37

2.5 Model W1 with M2: Normal-inverse Gaussian distribution . . . . . . . 38

2.6 Model W1 with M2: Variance Gamma distribution . . . . . . . . . . . .39

v


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List of Tables

2.1 Table of skewness and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Table of normality tests I . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Table of normality tests II . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Fitted pa rameters of model W1 with M2 . .. . . . . . . . . . . . . . . . 40

2.5 Fitted pa rameters of model W2 with M2 . . . . . . . . . . . . . . . . . . 40

2.6 Fitted pa rameters of model W3 with M2. . . . . . . . . . . . . . . . . . 40

2.7 Fitted pa rameters of model W4 with M2 . . . . . . . .. . . . . . . . . . 41

vi


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Chapter 1

Introduction

1.1 Motivation to Systematic Mortality Risks

Let us observe a given population that satises some homogeneity conditions, for ex -ample, the people are of the same age, sex, and similar health situations. How many

will survive to a specic date in the future, say one year? If we know the one- year survival probability of this population, we can estimate the number of survivors in one year by the expected number of this population, that is, just multiplying the survivalprobability and the number of initial livings. For a sufficient large population, such a prediction works pretty well. The life insurance industry ( or the classical actuarial sci-ence ) lays itself principally on this idea to calculate the mortality risks in the insurancecontracts. In eff ect, by selling sufficient many policies, an insurance company bearstheoretically no mortality risks in its business.

But if we are not sure of the survival probability, how are the things going to be?Suppose there are two possible values of the one year survival probability,we will then get two estimators by calculating the expected number of survivors for the population. Therefore no matter how many policies an insurance company sells,the mortality riskscan not be diversied. The insurance companies can not sit on their own chairs to runthe business, they need to call on outside help, for example, the help coming from thecapital market to share the risks.

Such non-diversiable mortality risks are called systematic mortality risks. In par-ticular, they are called longevity risks when survival risks are concerned.

1


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CHAPTER 1. INTRODUCTION 2

1.2 Objective of the Thesis and Contribution

Chapter 2 : Mortality Risks

Life Contingencies in Classical Actuarial Perspectives Under the assump-tion of a deterministic mortality rate,we review some main concepts in classical actu-arial science.

We investigate the survival probabilities and the deterministic force of mortalityun-der the framework of a single jump processassociated with each underlying individual.Under the assumption that all individuals have independent future lifetimes, we canapproximate the number of future livings by the number of expected future livings.

Furthermore, the independent assumption of future lifetimes allows us to evaluatethe insurance contracts by the principle of equivalence . We explain this principle underthe framework of the standard-deviation principle. In particular, we investigate theindividual safety loading . In the current setting the individual safety loading converges tozero, which shows that the principle of equivalence is suitable to evaluate the mortality risks involved in the insurance contracts.

Finally, we state the classical time-continuous version of Thieles di ff erential equationfor the net reserves of pure endowment contract.

Life Contingencies on Systematic Mortality Risks Empirical research suchas Currie et al.[2004] shows that the number of future livings can not be precisely estimated by the expected future livings where the calculation uses a deterministicmortality rate.

We generalize the concept of conditional independence of future lifetimes( see Nor-berg [1989]) into a dynamic setting. Now under the framework of a double stochastic single jump process, we dene the stochastic survival processesfor the underlying individ-

uals. The concept of a stochastic mortality rat e is then generalized in a consistent man-ner compared to the deterministic mortality rate. Our approach can also be viewedas a dynamicalgeneraliza tion of the work of Blake and Burrows[2001] and Milevsky et al. [2006]. In particular, we show that some popular modeling approaches are notconsistent to the initial motivation to the stochastic mortality rate, for example, the work of Cairns et al.[2006],Dahl[2004], Biffis [2005], Ballotta and Haberman[2006],Milevsky and Promislow [2001] andMilevsky et al.[2005].

Under theassumption of a stochastic mortality rate, we show that the future life-

times are not independent any more. The number of future livings can only be es-timated by theconditi onal expecte d futur e livin g s. In a ddition, the individual safety


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loading does not converge to zero but to a positive number, which indicates that theprinciple of equivalence is not capable to evaluate the mortality risks involved in in-surance contracts. Thus we have introduced the systematic mortality risks.

Having done the theoretical analysis of the stochastic mortality rate, we also per-form an empirical analysis with respect to the JPMorgan LifeMetrics( seewww.Jpmorgan.com/lifemetrics ). The experts of JPMorgan start their modeling of mortality ratesunder the independence assumption of future lifetimes. Viewed from our approach,the models in LifeMetrics do not yield the stochastic mortality rate but a determinis-tic mortality rate. In addition, we test the normality assumption of the standardized residualsof their models. All of the tests reject this assumption. This result, however,motivates us to model the mortality rate under our stochastic approach.

Then we suggest some concrete models capturing both the deterministic and sto-chastic evolution of mortality rates. We model the random evolutions in our models by the class of generalized HyperbolicLvy processes. Using the LifeMetrics models as thedeterministic components and the data set of LifeMetrics,we estimate the parametersof our models. The tting results show that the generalized Hyperbolic distributionscapture the empirical data quite well.

Although the systematic mortality risk can not be diversied by just issuing moreinsurance policies, it can be transfered to the nancial market by trading on somenancial instruments on mortality risk. Before we build up the arbitrage-free marketmodel for systematic longevity risks, we rst review the classical no- arbitrage theoryabout nancial markets.

Chapter 3 : Alternative Approach of No -Arbitrage Hypothesis

Bank Account is Not Tradable In the standard literature, for example Harri-son and Pliska [1981], Karatzas and Shreve[1998], Delbaen and Schachermayer[2006],the no- arbitrage hypothesisis derived using thebank account as risk-free asset. Althoughone can generate a bank account by hypothetically trading ( with a self -nancing strat-egy ) in risk-free zero-coupon bonds( see Bjrk et al.[1997a ], Dberlein and Schweizer[2001]), we ca n show that the bank account is practicall y not a tradable asset to be involved in the hedging strategies for contingent claims. Actually, in their classical workneither Black and Scholes[1973] nor Merton [1973] use the bank account to derivethe hedging strategies for contingent claims. Black a nd Scholes[1973] use the risk-free short rateto model the risk-less return of the replicating portfolio( although it is

not self -nancing, see Musiela and Rutkowski[2005] Proposition 3.1.6 or Binghamand Kiesel[2004] Exercise 6.1 ). And Merton[1973] uses the risk-free zero-coupon
http://www.jpmorgan.com/lifemetricshttp://www.jpmorgan.com/lifemetricshttp://www.jpmorgan.com/lifemetrics


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T -bond to build up the replicating portfolio for a contingent claim and derives theevaluation formula for it.

Alternative Denition of a No -Arbitrage Opportunity We deneloca & y the no- arbitrage opportunity with respect to tradable( just maturing ) default-free zero-coupon T -bonds, and modelgloba & y the arbitrage-free assets dynamics with respectto the bank account. We investigate time-discrete and time-continuous trading re-spectively.

For time-discrete trading the one step risk-less return is given by the risk-less bondmaturing at the end of the step, which is predictable at the beginning of the trading step. The no-arbitrage opportunity means that the expected return of each risky assetover that trading step should not be greater than the risk-less return over that step.

For time-continuous trading we use the result of Bjrk et al.[1997a ] ( see also Bjrk[2004] section 20.2.3 ) to dene theinstantaneousrisk-less return yielded by risk-freebonds, which is equal to the spot short rate and assumed to be predictable. The in-stantaneous return of each risky asset is dened with respect to its diff erential form. Then similar to the time-discrete model, the no-arbitrage condition means( heuristi-cally ) that the e xpected instantaneous return of each risky asset should be equal to theinstantaneous risk-less return.

The no- arbitrage hypothesisis then stated as usual:the nancial market is said tocontain no- arbitrage opportunity( or be arbitrage- ee ) , if and only if there exists a proba-bility measure such that divided by the bank account,each such normalized asset is a( local - ) martingale.We prove this( partially with heuristic for time-continuous model ), how -ever, under our alternative denition of no-arbitrage opportunity.

Hedging of Contingent Claims Our ultimate aim is to derive the evaluationformula and in particular the hedging strategies for the contingent claims. As the bankaccount is not a tradable asset, we need to revise some of the classical results.

We rst generalize the numeraire invariance theore m in its more general versionin Protter[2001], whereProtter[2001] uses the bank account as a tradable asset. Basedon the generalized numeraire invariance theorem, we derive the classical risk- neutral v aluation formula( see, for exampleBingham and Kiesel[2004]) for contingent claims.Finally, using thechange of numerair e technique( for its technical background see Jacodand Shiryaev [2003] section II I. 3b ) w e express the risk-neutal valuation formula using various tradable assets such as the risk-free T -bond and the risky equity as numeraires.

As an application, we evaluate the European ca & option on a risky equit y andderive the hedging strategiesinvolving only the risk-free T -bond and the equity.


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We should remark that not every hedging strategy is self - nancing . The Kunita-Watanabe decomposition ( see, for example Schweizer[2001]) allows us to determine whether the derived hedging strategies are self -nancing or not.

Chapter 4 : Financial Markets involving Survivor Bonds

Arbitrage - free Bonds and Equity Markets We use a multi-dimensional in-dependent Lvy process as our risk driving factor( see Sato[1999], Cont and Tankov [2003], Kallsen and Tankov [2006], Eberlein[2007]).

The no-arbitrage condition, also called HJM - dri , condition , for the default-freeT -bond market is characterized with respect to thecumulant function of the Lvy process( see Raible[2000], zkan [2002]). Based on Its formula as in Ikeda andWatanabe [1989], Applebaum[2004], wederivethe diff erential dy namics of T -bondsand the term structure equation in a technically less involved way than the proofsin Raible[2000] or zkan[2002].

We then transfer the H JM-drif t condition fordefault-free T -bond into the risky equity market. The arbitrage-free condition for the equity markets is also character-ized with respect to the cumulant function of a Lvy process.

Since we work directly under the risk-neutral measure,we also outline a theoreticalcalibration procedure of the martingale measure with respect to the spotT -bondprices and equity prices. Here we generalized the idea of Eberlein and Kluge[2007]into a multi-dimensional Lvy process model forT -bonds and equity markets.

Arbitrage - free Survivor Bonds Market Motivated by the work of Blake andBurrows[2001], we build up an arbitrage-free survivor bondsmarket which serves asan underlying market to trade the systematic longevity risks on the nancial markets.

That is, for each individual of an initial homogeneous population there exists a survivor bonw hich pays one unit ( say Euro ) if the individual survives at the maturity.We dene the forward rat e through the sur vived survivor bonds and the related spot rate which isactually the sum of the risk-free short rate and the( risk-neutral ) stochastic force of mortality rate.

With respect to a generic representation of the individuals future lifetimes, wederive the arbitrage-free term structure for the survivor bonds market. This is donein two steps. In the rst step, we dene a generalized normalizing factor by using the

spot rate of survived survivor bonds. Normalized by this factor, the survived survivorbond is a martingale if a drift-condition, called rst HJM - dri , condition for survivor


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bonds, is satised, which is also characterized with respect to the cumulant function of the driving Lvy process. In the second step, we dene the risk- neutral stochastic force of mortalityas a diff erence of the spot rate of survivor bonds and the risk-free shortrate. Then we show that if the single jump process associated to the future lifetimecan becanonica & y constructed with the risk-neutral stochastic force of mortality, thenthe survivor bonds market is arbitrage-free. That is, normalized by the bank account,each survivor bond price is a martingale. We refer to Bielecki and Rutkowski[2004]for general technical background about the double stochastic single jump process.

Compared to the derivation of arbitrage-free defaultable bonds models( see Bi-elecki and Rutkowski[2004], Eberlein et al.[2006]), our two step approach is moreintuitive and technically less involved.

Evaluation of Longevity Risks related Contingent Claims Within our framework we evaluate the longevity risk related contingent claims. Two approaches are investigated. The risk- neutral valuation approach is just an application of the risk-neutral valuation principle and its extension with the jump process( see Bingham and Kiesel[2004], Bielecki and Rutkowski[2004]). We also apply the work of Blanchet-Scallietand Jeanblanc[2004] to derive a special form of hedging strategies for some survivalrelated contingent claims.

The PDE approach of evaluation is discussed in detail. We rst derived the partial -integro di ff erential equation ( PIDE for short ) for the survived value of the contingentclaim, which is assumed to be a sufficiently smooth function of default-free bond,equity, and survived survivor bond. Then by applying a suitable version of Its productlaw ( see Jacod and Shiryaev [2003]) we derive the PIDEfor the usual risk-neutral priceof the survival related contingent claim.

Application to U nit - linked Contr a ct a nd Thieles Di f erential Equation The unit-linked pure endowment contract is essentially an European call option onsome equity ( or funds ) related to the survival situation of the policyholder at the matu-rity. Based on the risk-neutral valuation, we apply the change of numeraire techniqueto derive the hedging strategies involving the equity and the survivor bonds as nu-meraires respectively. This can be viewed as an extension of the Black-Scholes formula for unit-linked pure endowment contract, where instead of the default-free T -bond we use the survivor bond in the short position of the hedging strategies.

The market consistent valuation of the reserves of the unit-linked pure endowmentcontract is discussed under various assumptions. In particular, the generalization of


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the classicalThieles di ff erential equation is concerned. Aase and Persson[1994], Stef -fensen[2000] generalize the Thieles DE for unit-linked contract with equity risks,Persson [1998] generalizes the Thieles DE for stochastic interest rate. Both of the works are performed under deterministic mortality rate assumption.

Within our frame w ork, we study the Thieles DE for unit-linked pure endowmentcontracts withrespect to the equity, default-free T -bond and survivor bond. Thus we generalize the Thieles DE under equity, stochastic interest rate and systematiclongevity risks. Both of the survived value and the usual risk-neutral price are investigated.

Chapter 5 : Longevity Derivatives

Longevity Risk Management To manage the longevity risks, we investigatethe longevity derivatives. We start with the analysis of the maximum of the futur e lifetimesof insured policyholders. Using the conditional independent property of thefuture lifetimes, we also derive the corresponding stochastic survival processes and

hazard rate of the maximum of future lifetimes.

For a given annuity portfolio we suggest two types of longevity derivativesto hedgethe involved longevity risks. One is dened onthe number of future living peopl e andthe other on the number of conditional expected future living peopl e . Both of them areassociated with the survival situation of the maximum of future lifetimes, since thepayoff exists only if at least one individual of the portfolio survives at maturity.

Using these two types of longevity derivatives,we evaluate the longevity derivatives

by applying the risk-neutral valuation technique and derive the corresponding hedging strategies involving the underlying survivor bonds and the default-free zero couponbonds. We study with care the role of diff erent ltrations involved by evaluation andcompare the diff erent pricing results.

Applications Two examples are investigated, the longevity securitization sug - gested by Lin and Cox [2005] and the guaranteed annuity options( see Ballotta and

Haberman[2006]). We also compare our results to those obtained so far in the liter-atures.


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1.3 Conclusion and Outlook

In this dissertation, we investigate various aspects of the systematic mortality risks,in particular, the longevity risks. To describe the precise meaning of random survivalprobability, we model the systematic mortality risks by generalizing the idea of conditional independence into a dynamical settings. The eff ects of systematic mortality risks on actuarial science are also analyzed. To evaluate and hedge the longevity riskson capital markets, we build up an arbitrage-free nancial market model including survivor bonds. Within an arbitrage-free nancial market containing default-free zero-coupon bonds and equities, we study the risk-neutral valuation and hedging strategiesfor the contingent claims involving longevity risks. The classical Thieles diff erential

equation is also generalized under our framework. Furthermore, we study the risk management of longevity risks. Two types of pure longevity derivatives are dened andevaluated with respect to various ltrations. We apply our results on some examplessuch as longevity securitization and the guaranteed annuity options. We also compareour results to those obtained so far in the literatures.

We may extend our research in the future with respect to the following topics. Wecan investigate more plausible stochastic process in the concrete mortality rate mod-els, such as subordinator processes or subordinator driving OU-processes. Further

empirical analysis for those models can be performed with respect to the JPMorganLifeMetrics. We can also apply our arbitrage-free survivor bonds market model onthe practical transaction and perform some empirical research. In particular, we cancalibrate the term structure of the survivor bonds using the outlined procedure fordefault-free T -bond and equity models. We can study the risk management of insur-ance portfolio under some risk measures with respect to the generalized Thieles DEfor the market reserves of the unit-linked pure endowment contract. A real theoreti-cal challenge is how to state and proof precisely the no-arbitrage hypothesis under ouralternative observation that the bank account is not a tradable asset. Some advancedmathematical techniques need to be studied or even to be invented.


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Chapter 2

Mortality Risks

2.1 Life Contingencies in Classical Actuarial Per -

spectives

2.1.1 Deterministic Mortality Rate

Let us observe a population consisting of x individuals. Each individual isx yearsold at time t = 0 . The future lifetime of each individual is denoted by ix with 0 ix T , i = 1 , , x . In this section we assumeAssumption 2.1.1. There exists a stochastic basis( , F , F = ( F t )0 t T , P ) such thatthe future lifetimes ix , i = 1 , , x , are identica & y distributed non- negative rando mv ariables.With the single jump processJ ix (t) := 1 { ix t} w e denote the survival situationof the i -th x aged at time t . Let H x := ( H xt )0 t T denote its natural ltration withthe product - algebraH xt := H 1t H

xt ,w her e H it = (J ix (s) : 0 s t) , i =

1, , x . In this section the whole information is just the jump information about ix , i =

1, , x , i.e.F t = H xt = H 1t H xt .

Remark 2.1.1. For the analysis of survival probability we dont require the independent property of the future lifetimes.

Let x have the same distribution as ix . We perform the analysis of the futur e lifetimes ix by studying their representation x if no confusion is caused.We dene the singl e

jump processJ x w ith J x (t) = 1 { x t} and the ltration H = ( H t )0 t T

generated by the single jump processJ x .

9


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CHAPTER 2. MORTALITY RISKS 10

We assume that the future lifetime x has the probability distribution

t qx := F x (t) := P ( x t). ( 2.1 ) The survival probability of thex -aged at timet is dened as

t px := Gx (t) := P ( x > t ) = 1 t qx . ( 2.2 ) The conditional probability that thex -aged will survive toT years, after having at-tained the agex + t , will be denoted as

T t px+ t := P ( x > T | x > t ). ( 2.3 )

In actuarial science, two numbers about life contingencies of the initial populationare important:

the number of living people Lx+ t at future timet

Lx + t := x

i=1 1 { ix >t }, ( 2.4 ) which is a random variable. And

the expected number of living people x+ t at future timet

x+ t = E P [Lx+ t ] = x t px , ( 2.5 )

which is a deterministic number.

We dene now the deterministic force of mortality.

Denition 2.1.1. We assume that F x (t) is absolutely continuous and denote the densityby f x . The deterministic force of mortality x (t) of an x - aged is dened as

x (t) = f x (t)1 F x (t)= ddt ln t px . ( 2.6 )

The force of mortality can be equivalent interpreted as

x (t) := lim 0

1

P [t < x t + | x > t ]= lim

0

1

P [t < x t + ]P [ x > t ]

. ( 2.7 )

And the survival probability is equal to

t px = e s t

0x (s ) ds , ( 2.8 )


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because t px = 1 F x (t) by ( 2.2 ) and 0 px = 1 . The conditional survival probability T t px+ t can be represented as

T t px+ t = e s T

t x (s ) ds ( 2.9 )

The deterministic force of mortality x (t) ca n be interpreted as the instantaneousdeath rate at timet , i.e.

x (t)dt P [t < x t + dt | x > t ]. ( 2.10 )

We derive some general expressions about the conditional survival probability of x under the natural ltration.

Lemma 2.1.1. let 0 < t < T T .P ( x > T |H t ) = 1 { x >t }P [ x > T | x > t ]

= 1 { x >t }e s

T

tx (s ) ds , ( 2.11 )

P (t < x T |H t ) = P [ x T |H t ]P [ x t |H t ]= P [ x > t |H t ]P [ x > T |H t ]= 1 { x >t } 31 e s

T

tx (s ) ds4. ( 2.12 )

Proof.We follow the line of the corollary 4.1.1 in Bielecki and Rutkowski[2004]. Letus deneY = 1 { x >T } .

P ( x > T ) = E [Y |H t ]

= 1 { x t}E [Y | x ] + 1 { x >t }E [1 { x >t }Y ]

P [ x > t ]

= 1 { x >t }E [1 { x >T }]P [ x > t ]

= 1 { x >t }P [ x > T | x > t ]

= 1 { x >t }e s

T

tx (s ) ds .

P (t < x T |H t ) = P ( x T |H t ) P ( x t |H t )= 1 P ( x > T |H t ) (1 P ( x > t |H t ))= 1 { x >t }

31

e

s T

tx (s ) ds

4.


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For introduction to life contingencies in classical actuarial science, one can referto part 1 of the book Jordan[1982], chapter 2 of Gerber[1997], chapter 3 of Gerberet al. [1997], chapter 3 of Norberg [2002], and chapter3 of Milevsky [2006]. Forthe background about single jump process, see chapter4 of Bielecki and Rutkowski[2004], chapter 1 of Jeanblanc[2007a ] and chapter2 of Jeanblanc[2007b].

2.1.2 Evaluation of Insurance Contracts: Principle of Equiv -

alen ce

In this section the independence property of the future lifetimes is needed.

Assumption 2.1.2. The future lifetimes ix , i = 1 , , x are independent and

identica & y distributed ( i.i.d. ).

This is the central assumption about future lifetime from the classical actuarialperspective, under which we can apply the principle of equivalence to evaluate theinsurance contracts. That is, the expected discounted future benet( premium to in-surance company ) is equal to the expected discounted future liability ( benet to thepolicyholder ). For more details about the principle of equivalence, see chapter5.1 of Gerber [1997] or chapter 5 of Schmidt[2006].

Let us observe theunit pure endowment contract issued on the populationx which pays each individual one unit if the insured individual survives up to timeT . The future liability of this insurance portfolio at timeT is V L (T ) := Lx + T . Letr (t) be the positive deterministic risk-free short rate ande s

T

tr (s ) ds the discount

factor. Let T E x denote the net single premium that is paid at the starting dateof the insurance. Thus the future benet at timeT of this insurance portfolio isV B (T ) := es

T 0

r (s ) ds x T E x .

We apply the equivalence principle to calculate the net single premiumT E x asfollows.

E P [e s T

0r (s ) ds V B (T )] = E P [e s

T

0r (s ) ds V L (T )]. ( 2.13 )

The lefthand side of above equation( 2.13 ) is equal to

x T E x .

And the righthand side of equation( 2.13 ) is equal to

E P [ x

i=1 1 { ix >T }e s T

0r (s ) ds ] = x E P 51 { ix >T }e s

T 0

r (s ) ds6,


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since ix , i = 1 , , x are i.i.d. .

This yields

T E x = T px e s T

0r (t ) dt = e s

T

ox (t )+ r (t ) dt . ( 2.14 )

The determination of premiumT E x can be motivated by the standard - deviation principl e .

Denition 2.1.2 ( Standard-deviation Principle ). Let X denote the liability of an insurance portfolio.For R + let

LH := ;X L2(R + )|P 5X > E [X ] + Var [X ]6> 0


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Markovs inequality we have for allx(c, )P [X

E [X ] + c] = P [Y

c]

= P [Y + x c + x]P [|Y + x | c + x]

E [(Y + x)2]

(c + x)2

=E [Y 2] + x2

(c + x)2

=Var [Y ] + x2

(c + x)2

= Var [X 2

] + x2

(c + x)2

The last expression is twice diff erentiable inx (c, ) and takes its minimum atx = Var [X ]/ c . Thus we have

P [X E [X ] + c] Var [X ] + (Var [X ]/ c)2

(c + Var [X ]/ c)2

=(Var [X ]/ c) (c + Var [X ]/ c)

(c + Var [X ]/ c)2

= Var [X ]/ cc + Var [X ]/ c

=Var [X ]

c2 + Var [X ].

Thus we get an upper bound of the probability for the event{X E [X ] + c} . This can be further characterized by the safety loading to the liability X .

Lemma 2.1.3. Assume X = qni=1 Z i w ith Z iL2(R + ) . Let (0, 1) . Then for a & c(0, ) w ith c2 1 Var [X ] w e have

P [X E [X ] + c] .

Proof. This lemma is also stated in Schmidt[2006] Folgerung6.1.3 . We give anexplicit proof based on the Cantellis inequality.


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Since c2 1 Var [X ] > 0 () , we haveP [X

E [X ] + c]

Var [X ]

c2

+ Var [X ]Cantellis inequality

Var [X ]

1 Var [X ] + Var [X ]

( * )

=1

1 + 1

=

Denition 2.1.3 ( Safety Loading ). We ca & S w ith

sX := 1 Var [X ] ( 2.17 )safety loading to X .

If Z i , i = 1 , , n are identica & y distributed,then we ca & sZ w ith

sZ :=1n 1 Var [X ] ( 2.18 )

individual safety loading to eachZ i .

We can now explain the rationality of equivalence principle by observing the asymp-totical property of the individual safety loading to the insurance liability.

Lemma 2.1.4. Let Z i L2(R + ), i = 1 , , n be independent identica & y distributed

random variables.Then the individual safety loading to eachZ i is

sZ = 1 n Var [Z i]. ( 2.19 ) And sZ goes to0 asn goes to

Proof. This is stated in Schmidt[2006] Folgerung6.1.6 . We also give a proof.Since Z iL

2(R + ), i = 1 , , n are i.i.d , we have

Var [n

i=1 Z i ] =n

i=1 Var [Z i ] = nVar [Z i]Substitute it into the equation of individual safety loading ( 2.18 ), we have

sZ =1

n 1

nVar [Z i ] =

1

n Var [Z i]


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According to this lemma, we see that the mortality risk in pure endowment con-tract can be well-diversied. The standard-deviation premiumH [V L (T )] of the unitpure endowment contract is

H [V L (T )] = E [V L (T )] + 1 Var [V L (T )]= x es

T

0r (s ) ds

T E x + 1 Var [Lx+ T ].Under the independent assumption of future lifetimes, the variance of the liability Var [Lx+ T ] becomes

Var [Lx+ T ] = x Var [1 { x >T }].

Thus the individual safety loading sZ

for each policyholder is

sZ = 1 x Var [1 { x >T }], which goes to0 as policyholdersx goes to.

More details about equivalence principle can be found in chapter5 of Gerber[1997], chapter 6 of Gerber et al.[1997], chapter 4 of Norberg [2002], and chapter5of Schmidt[2006].

2.1.3 Net Reserves and Thieles Di f erential Equation

We observe now the time continuous model. Letp(t) represent the deterministicperiod premium. We can apply the equivalence principle to determinep(t) by

E P C T 0 e s s

0r (u ) du 1 { x >s } p(s) dsD= E P 51 { x >T }e s

T

0r (s ) ds6. ( 2.20 )

By using change of integration and the general conditional survival probabilityn( 2.11 ),

we have

T 0 e s s

0x (u )+ r (u ) du p(s) ds = e s

T 0

x (s )+ r (s ) ds . ( 2.21 )

The individual based net( prospective ) reserve for the pure endowment insurancecontract is the diff erence of future liability and future premium income of the insurerat time t

E P 51 { x >T }e s T

tr (s ) ds |H t6E P C T t e s

s

tr (u ) du 1 { x >s } p(s) ds |H tD

= 1 { x >t }e s T

t x (s )+ r (s ) ds 1 { x >t } T

te s

st x (u )+ r (u ) du p(s) ds. ( 2.22 )


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Let V (t) denoted the reserve on the event{ x > t } ,

V (t) = e

s T

tx (s )+ r (s ) ds

T

te

s s

tx (u )+ r (u ) du p(s) ds. ( 2.23 )

Diff erentiation of V (t) with respect to timet yields the classical Thieles diff er-ential equation

Proposition 2.1.1 ( Thieles Diff erential Equation ).dV (t)

dt = ( r (t) + x (t)) V (t) + p(t).( 2.24 )

The period premiu m p(t) can be decomposed as

p(t) = AdV (t)

dt r (t)V (t)B+ ( x (t)V (t)) . ( 2.25 )

Proof.We apply the a ffine integral equation( see chapter0 of Kallsen[2006]) to derivethe Thieles diff erential equation.

We dene

X (t) := t0 x (t) + r (s) ds, Y (t) := t0 p(s) ds. Thus the prospective reserve reads as

V (t) = eX (t ) X (T ) T t eX (t ) X (s ) dY (s). ( 2.26 )Let us writeW (t) := eX (t ) X (T ) . The diff erential form of W (t) is

dW (t) = W (t)dX (t). ( 2.27 )

Let us write

Z (t) :=

T

t

eX (t ) X (s ) dY (s)

= T t eX (t ) X (T )+ X (T ) X (s ) dY (s)= A T t eX (T ) X (s ) dY (s)BeX (t ) X (T ) .

Thus the diff erential form of Z (t) is

dZ (t) = eX (t ) X (T ) 1eX (T ) X (t ) dY (t)2+ A

T

teX (T ) X (s ) dY (s)BeX (t ) X (T ) dX (t) ( 2.28 )

= Z (t)dX (t) dY (t). ( 2.29 )


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Combine the equations( 2.27 ) and ( 2.29 ) we have the diff erential form forV (t) .

dV (t) = dW (t) dZ (t)= ( W (t) Z (t)) dX (t) + dY (t)= V (t)dX (t) + dY (t), ( 2.30 )

That isdV (t)

dt = ( x (t) + r (t)) V (t) + p(t).

For more details about Thieles diff erential equation, one can refer to chapter6of Gerber[1997], chapter 5 of Koller[2000], and chapter1 of Kiesel[2006]. Theinterested reader can also see Norberg [2004a,b] for more background on Thiele.

2.2 Life C ontinge ncies on S yst emat ic Mortality Ris k s

2.2.1 Stochastic Mortality Rate

The empirical research, for example Currie et al.[2004], indicates that the realizednumber of future livings can not be precisely predicted by the deterministic mortality law. This motivates us to model the life contingencies by some stochastic process.

From now on, let us assume

Assumption 2.2.1. On the probability basis( , F , F = ( F t )0 t T , P ) there is a non-empty auxiliary ltration G := ( G t )0 t T . The future lifetimes of each individual i nthe initial population are now assumed to be conditiona & y independent identica & y distributed w ith respect toG . That is, for a & t1, , t x

[0, T ] ,

P 1x t1, , xx t x |G T = P 1x t1|G T P xx t x |G T , ( 2.31 ) and

P [ ix t |G T ] = P [ jx t |G T ] ( 2.32 ) for a & ix , jx , i = j, i, j = 1 , , x .

As in the last section, the survival information is dened by the ltration H x :=(H xt )0 t T w ith H xt := H 1t

, ,

H xt , H it := (1 { ix

s} : 0

s

t) .

The ltration F is now dened byF t = G tHxt .


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Furthermore,w e assume that for a & ix , i = 1 , , x

P [ ix s |G t ] = P [ ix s |G T ], ( 2.33 ) for a & s t T .

Similar to last section, let x represent the ix s . We dene the single jump processJ x and the ltrationH generated by J x . If no confusion is caused, we will performthe analysis of the ix s by studying x .

Remark 2.2.1. The last assumption 2.33 is essentia & y equivalent to the next assumption.We state it explicitly because we use it dene the survival processes in the fo & owing.See Bielecki and Rutkowski [ 2004 ] section 6.1 for a general formulation.

We make the following assumption for convenienceas we calculate.

Assumption 2.2.2. The - algebraG T and H t are assumed to be conditiona & y independent given G t ,i.e.

E [ |G t ] = E [ |G t ]E [|G t ] ,

w her e is a bounded G T measurable random variable, bounded H t measurable.

This assumption is also calledH hypothesis in reduced form modeling for credit

risk. It is the crucial assumption for the so-calledhazard process models . See Jean-blanc and Cam[2007a,b] for more details.

Remark 2.2.2. The above assumption and the assumption 2.33w i & be satised when xis constructed through the canonical method ( see Bielecki and Rutkowski [ 2004 ] section 6.5 and 8.2.1 ). In the last part of this section we wi & also construct x under our current settings.

We dene some processes about the conditional distributions of x .

Denition 2.2.1 ( Survival Processes ). The survival processes of x :t px := P [ x > t |G T ] = P [ x > t |G t ] = Gx (t) ( 2.34 )

t qx := P [ x t |G T ] = P [ x t |G t ] = F x (t) ( 2.35 ) The conditional survival processes of x :

T t px+ t :=P [ x > T |G T ]P [ x > t |G T ]

=P [ x > T |G T ]

P [ x > t |G t ].

( 2.36 )


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We also dene theG - hazard process x (t) as x (t) = ln t px = lnGx (t). ( 2.37 )

The following lemma is quite useful.

Lemma 2.2.1. For everyF - measurable random variabl e Y and every sub- algebraG of F w e have

E P 1 { x >t }Y |H tG = 1 { x >t }E P 1 { x >t }Y |G P [ x > t |G ] . ( 2.38 )

In particular, for everyt s w e have P [

x> s |H

t

G ] = 1{ x >t }

P [ x > s |G ]P [ x > t |G ]

. ( 2.39 )

Proof.See lemma 5.1.4 of Bielecki and Rutkowski[2004].

Proposition 2.2.1 ( Conditional Survival Probability ). The conditional survival probability of x is

P [ x > T |H tG T ] = 1 { x >t } T t px+ t . ( 2.40 )

Proof.Combine the lemma 2.2.1 and the assumption( 2.33 ), we have

P [ x > T |H tG T ]

(2 .39)= 1 { x >t }P [ x > T |G T ]P [ x > t |G T ]

(2 .33)= 1 { x >t }P [ x > T |G T ]P [ x > t |G t ]

(2 .34)= 1 { x >t } T t px+ t .

The expected realization of future number of livingsLx+ t = q xi=1 1 { ix >t } is now conditionally approximated by theconditional expected number of livings.Denition 2.2.2. The conditional expected number of livings at time t is cal -culated as:

x + t = E [Lx+ t |G T ] = E [ x

i=1 1 { ix >t } |G T ]= x E [1 { x >t } |G T ]

= x E [1

{ x >t } |G t ]= x t px . ( 2.41 )


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Now conditioned on the -algebra G t , the expected number of livings is a random variable.

We dene the stochastic force of mortalityas followsDenition 2.2.3 ( Stochastic Force of Mortality ). We assume that the survival processF x (t) in denition ( 2.2.1 ) is absolutely continuous w.r.t t and denote its density asf x (t) .The stochastic force of mortality of the initial x - aged at time t (w ith ag e x + t )is dened as

x (t) =f x (t)

1 F x (t)=

ddt ln t px .

( 2.42 )

The stochastic force of mortality x (t) can also equivalently interpreted as

x (t) := lim 0

1

P [t < x t + | x > t, G t ]= lim

0

1

P [t < x t + |G t ]P [ x > t |G t ]

. ( 2.43 )

Remark 2.2.3. x (t) is neither equal to

x (t) := lim 0

1

P [t < x t + |G tH t ] nor equal to

f x (t) := lim 0

1

P [t < x t + |G t ]. Indeed,

x (t) = 1 { x >t }x (t) see theore m 1 of Guo and Zeng [ 2007 ]

and

f x (t) = x (t) t px =dF x (t)

dt .

One may compare our denition of the hazard rate of x w ith that by Guo and Zeng [ 2007 ] , Guo et al.[ 2007 ] , Jeanblanc and Ca m[ 2007a,b ].

Furthermore, we assume thatx is a G -progressive measurable non-negative( in-deed, increasing ) process. TheG - hazard process x (t) can be dened as

x (t) = ln(t px ) = t

0x (s) ds. ( 2.44 )


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At time t , the observed individual is of agex+ t and x (t) represents the( stochas-tic ) instantaneous death rate. The stochastic survival process at timet can be repre-sented as

t px = P [ x > t |G t ] = exp( x (t)) = exp3 t0 x (s) ds4. ( 2.45 )For a xed timeT , the conditional survival process can now be represented as

T t px+ t = e x (t ) x (T ) = expA T t x (s) dsB, ( 2.46 )

And the conditional survival probability can be represented as

P [ x > T |H tG T ] =1 { x >t } P [ x > T |G T ]P [ x > t |G t ]

= 1 { x >t } T t px+ t . ( 2.47 )

Remark 2.2.4. The( conditional ) survival processes dened in( 2.2.1 ) are di ff erent thanthe usual denitions in literatures,w here the stochastic mortality rate is modeled para & el to the zero-coupon risk- ee or defaultabl e T -bonds models. We denote the( conditional ) survival probabilities aspx (t, T ) for this approach,w hich are calculated under current information G t .

px (t, T ) = E P

5e

s T

tx (s ) ds |G t

6( 2.48 )

= E P 5e s T

0x (s ) ds es

t

0x (s ) ds |G t6

= B x (t)E P C1B x (T ) |G tD, ( 2.49 )w ith B x (t) = es

t

0x (s ) ds . If we observe a zero-coupon default - ee bond market model

w ith short interest rat e x (t) , px (t, T ) is just the T bond price at time t , assuming the risk- neutral measure isP . Under this approach,the survival probability at time t = 0is

px (0, T ) = E P C1B x (T ) |G 0D, ( 2.50 )w hich is a determinist value.Such an approach is not consistent with the initial motivationto random survival probability and with introducing of stochastic mortality rate.

We dene the conditional survival process,however, on the whole information up toT .The survival probabilityT px is a stochastic process for a & T at the initial time t = 0 , and the conditional survival probabilityT t px+ t is a stochastic process for a & t, T, 0 t T .The interpretation of the force of mortality rat e x (t) under our approach is consistent withits deterministic modeling approach.


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The conditional survival probabilitypx (t, T ) is equal toE P [T t px+ t |G t ] . We don t have any actuarial interpretation of px (t, T ) .

The works of Cairns et al.[ 2006 ] , Dahl [ 2004 ] , Bi ffi s[ 2005 ] , Ba & otta and Haberman[ 2006 ] , Milevsky and Promislow[ 2001 ] , Milevsky et al.[ 2005 ] fa & into the px (t, T ) approach. The work of Blake and Burrows[ 2001 ] , Milevsky et al.[ 2006 ] can be viewed astime discrete i & ustration and motivation of our current approach.

So far to my knowledge, Norberg Norberg [ 1989 ] pioneered the denition of systematic mortality risksthrou gh the conditional indepe n dent ap proach as weh ave done in this section. Compared with our dynamical modeling, the work of Norberg [ 1989 ] models the systematic mortality risks only om the starting time 0 . A summary of Norbergs work can

be found i n M & er and Ste ff ensen[ 2007 ] section 5.5 .

Given a hazard process x , we can construct the future lifetime canonically suchthat all the previous assumptions are well-satised.

Canonical Construction of x We assume that we are given some probability space( , G , P ) . Let x (t) = s t0 x (s) ds with x being a non-negative,G -progressively measurable process. Thus x is a G -adapted, continuous, increasing process on( , G , P )such that

x (0) = 0 and

x () = + . We assume that is a random variable onsome probability space( , G , P ) , with the uniform probability law on[0, 1]. We take

the product space( = , F = G G ) with P = P P as the enlargedprobability space. Now let us dene x : ( , F , P ) R + by setting

x = inf {t 0 : exp( x (t)) } = inf {t 0 : x (t) ln}.We set now F t = G t H t for all t , where H t is the natural ltration of the singlejump processJ x (t) := 1 { x t} . And one can show that all the previous assumptions of this section are well-satised( see Bielecki and Rutkowski[2004] section 6.5 and inparticular Eberlein et al.[2006] for the case of Lvy models up to some maximal timeT ).

This canonical construction is also calledCox- processmodel. See, for example chap-ter 6 of Bielecki and Rutkowski[2004] and the paper of Blanchet-Scalliet and Jean-blanc[2004], Jeanblancand Cam[2007a,b].

For more technical background about hazard processes of ( conditionally indepen-dent ) single jump processes, see chapter5 and 9 of Bielecki and Rutkowski[2004],

the paper of Rutkowski and Yousiph[2006], Jeanbla nc and Cam[2007a,b], and thelecture notes of Jeanblanc[2007a,b].


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2.2.2 Systematic Mortality Risks

The systematic mortality risk is not diversiable. Conditioned onG t , i.e. for each

xed G t , the average rate of survival peopleL x + t x converges to t pt () ( see also

the work of Norberg [1989] and section5.5 of Mller and Steff ensen[2007]). But now t px is a random variable. The conditional expected number of living people at timet ,x+ t , is also a random variable. If the insurance company issues unit pure endowmentcontracts on the initial populationx , the payoff of this insurance portfolio is not a well-controlled random number a s the deterministic morta lity case. Since now theinsurance company bears the systematic mortality risk, no matter how largex is.

The probabilities of future lifetime of each individual are now equal to

t qx = P [ ix t] = E E [1 { ix t }|G t ]= E [ t qx ] = 1 E 5exp3 t0 x (s) ds46, ( 2.51 )t px = P [ ix > t ] = 1 t qx

= E 5exp3 t0 x (s) ds46. ( 2.52 )Proposition 2.2.2. Assume the future lifetimes ix , i = 1 , , x are conditiona & yindependent,i.e. they possess the stochastic force of mortalityx . Then the future lifetimes i

x, i = 1 , , x , are not independent under P .

Proof.Let ix , jx be future lifetimes of two arbitrary individuals, we have following assertion about the joint survival distribution of them,

P [ ix > t, jx > t ]

= E P [ ix > t, jx > t |G t ]= E P [ ix > t |G t ]P [ jx > t |G t ]= E [ t px 2]

= E 5e 2s

t0 x (s ) ds6

> 3E 5e s t

0x (s ) ds642

= P [ ix > t ]P [ jx > t ]. ( 2.53 )

The last inequality is due to the Jensens inequality for convex function. So the futurelifetimes are not independent underP .

In the previous proposition, the equality is true only when thex (t) is determin-

istic. In that case, the ltrationG is trivial, thus theG -conditional independence of ix coincides with the absolute independence underP .


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If the future lifetimes are independent, the variance of average livings

Lx+ t

x=

q xi=1 1 { ix >t }

x

converges to0 as x goes to innity, since

Var 3Lx+ tx 4= 1x Var (1 { x >t })=

1x

t px (1 t px ). ( 2.54 )

But if the future lifetimes have the stochastic mortality rate, things are changed.

Lemma 2.2.2. If the future lifetimes have the stochastic mortality rate,the variance of average livings converges to a positive value.

Proof.Following Shiryaev [1996] page 214, we can calculate the variance of Lx+ t as

Var [Lx+ t ]

= E [Var [Lx+ t |G t ]] + Var [E [Lx+ t |G t ]] ( 2.55 )= E [ x t px (1 t px )] + 2x

1E [ t p2x ]E 2[ t px ]

2, ( 2.56 )

since Lx+ t |Gt is Binomial distributed with parameters(x , t px ) .

Thus the limit of the variance of average livings at timet is

Var 5Lx+ tx 6 x E [ t p2x ]E 2[ t px ] = Var [ t px ], ( 2.57 ) which is non-negative due to proposition 2.2.2. And the variance of the average livingsconverges to zero only when the mortality rate is deterministic,sinceVar [ t px ] = 0 inthis case.

An important consequence is that the stochastic mortality risk is not diversiable.

Proposition 2.2.3. If the future lifetimes have the stochastic mortality rate, then the individual safety loading does not converges to zero.

Proof.Let Z i := 1 { ix >T }, i = 1 , , x . Following ( 2.18 ) in lemma ( 2.1.3 ), the individ-ual safety loading for each ix is

sZ = 1x 1 Var [Lx+ T ].


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Due to the last lemma ( 2.2.2 ), we have

sZ =1

x 1

E [ x t px (1

t px )] + 2x (E [ t p2x ]

E 2[ t px ]).

As x goes to innity, the individual safety loading goes to

sZ 1 E [ t p2x ]E 2[ t px ], which is a positive number.

Thus the mortality risk is not diversiable.

An actuarial implication is that the equivalence principle is not applicable for in-surance contracts. If the premium is calculated according to the equivalence principle,the insurance company will bear the systematic risk, no matter how large the portfo-lio is. On contrast, the larger the portfolio becomes, the more liability losses will theinsurance company pay to the policyholders.

For an elementary but quite illustrative discussion,one can consult Milevsky et al.[2006]. Blake and Burrows[2001] also give a time-discrete analysis to motivate thenecessity of survivor bonds market.

2.2.3 Analysis of JPMorgan LifeMetrics

Recently, JPMorgan provides a toolkit called LifeMetricsto capture the stochasticevolution of the mortality rate, seehttp://www.jpmorgan.com/lifemetrics . Themortality models investigated in JPMorgan LifeMetrics are not stochastic mortality models as our current approach. Under the independence assumption for the individuals future lifetimes, the experts of JPMorgan LifeMetrics review the popular ap-

proaches to project the deterministic mortality ra te. Although they use stochasticparameters or processes to t the model, they are actually model the deterministicmortality rate.

Theoretically, as we show in proposition 2.2.2, if we assume that the mortality rateis stochastic, then the individuals future lifetimes are not independent. Since theindependence of all individuals future lifetimes is assumed in JPMorgan LifeMetrics( see Cairns et al.[2007]), its modeling approach can not yield stochastic mortality rate.

Empirically, we test the i.i.d. standard Normal assumption of the standardized

residuals( see equation(1) in 6.5.1 of Cairns et al.[2007]). All tests reject the stan-dard Normal assumption. Thus the model assumptions of JPMorgan LifeMetrics are
http://www.jpmorgan.com/lifemetricshttp://www.jpmorgan.com/lifemetrics


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actually not true. More important, this suggests us that the mortality rate should bemodeled with stochastic process.

Using the software released by JPMorgan LifeMetrics( see Cairns[2007]), we testthe normality of standardized residuals for each model, M1, M2, M3, M5, M6, M7,and M8, in JPMorgan LifeMetrics. The results indicate that the normal assumptionshould be rejected.

We show the QQplot of the standard residuals against the normality for eachmodel.


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The tted empirical density looks like followings. The red line represents the ttednormal distribution, while the blue one the empirical density


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! 2 0

! 1 0

0

5

1 5

0 . 0 0 0 . 0 4 0 . 0 8 E m p

i r i c a

l v s

N o r m a

l , M 1

s t a

n d

a r d

i s e

d r

e s

i d u a

l s

! 8 0

! 4 0

0

4 0

0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 E m p

i r i c a

l v s

N o r m a

l , M 2

s t a

n d

a r d

i s e d

r e s

i d u a

l s

! 4 0

! 2 0

0

1 0

0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 E m p

i r i c a

l v s N o r m a

l , M 3

s t a

n d

a r d

i s e

d r e s

i d u a

l s

! 2 0

0

1

0

2 0

0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 E m p

i r i c a

l v s

N o

r m a

l , M 5

s t a

n d

a r d

i s e

d r e

s i d

u a

l s

! 1 0

0

1 0

2 0

0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 E m p

i r i c a

l v s

N o r m a

l , M 6

s t a

n d

a r d

i s e

d r

e s

i d u a

l s

! 2 0

0

1 0

2 0

0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 E m p

i r i c a

l v s

N o r m a

l , M 7

s t a

n d

a r d

i s e d

r e s

i d u a

l s

! 5 0

0

5 0

1 0 0

0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 E m p

i r i c a

l v s N o r m a

l , M 8

s t a

n d

a r d

i s e

d r e s

i d u a

l s

Figure 2.2: Empirical density of standardized residuals.


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Some empirical results against standard normal assumption of the standardizedresiduals are listed as following.

We use the package fBasics from CRAN website( see Wuertz and many others[2007]) to calculate the skewness and kurtosis of the standardized residuals.

M1 M2 M3 M5 M6 M7 M8Skewness -0.3870489 -2.2695739 -2.6015951-0.2444897 0.6240143-1.0348435 2.0567398Kurtosis 0.5835406 30.6700704 9.0720633 0.7241267 3.6418226 5.1625196 29

Table 2.1: Table of skewness and kurtosis The p-valuesof the test against normality of standardized residuals are

Test M1 M2 M3 M5 M6 M7 M8Kolmogorov -Smirnov 0.07762 < 2.2e-16 < 2.2e-16 0.04104 0.0007808 2.537e-07 2.2e-16Shapiro-Wilk 1.730e-06 < 2.2e-16 < 2.2e-16 1.258e-06 < 2.2e-16 < 2.2e-16 < 2.2e-16 Jarque-Bera 5.304e-10 < 2.2e-16 < 2.2e-16 2.723e-08 < 2.2e-16 < 2.2e-16 < 2.2e-16

Table 2.2: Table of normality tests I

Test M1 M2 M3 M5 M6 M7 M8Cramer- von Mises 2.807e-05 Inf Inf 9.404e-06 3.816e-10 1.754e-06 Inf Lilliefors 0.000616 < 2.2e-16 < 2.2e-16 9.823e-05 6.849e-10 < 2.2e-16 < 2.2e-16Pearson 0.1270 < 2.2e-16 < 2.2e-16 3.957e-05 1.786e-08 1.768e-12 < 2.2e-16

Table 2.3: Table of normality tests II

For more details about JPMorgan Lifemetrics, see Cairns et al.[2007], Coughlanet al.[2007].

2.2.4 Concrete Models and Fittings with JPMorgan LifeMet -

rics

Concrete Models Motivated by the plausible characters of mortality evolution andthe statistical adaption of the model,we use the Lvy processes to capture the stochas-

tic mortality evolution. For anxaged population of some cohort, we model thestochastic mortality ratex (t) in the following four forms


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Model W1 :

x (t) = gx (t) + L t . ( 2.58 )

Model W2 :

x (t) = gx (t) + eL t . ( 2.59 )

Model W3 :

x (t) = gx (t)L t . ( 2.60 )

Model W4

x (t) =

gx (

t)eL t . (

2.61 )

Where gx (t) is a deterministic function which can represent the best estimation aboutfuture mortality rate, andL t is some stochastic process to be specied.

Remark 2.2.5. The model W 1 is essentia & y equivalent to the Ho- Lee model of interest rate theory.

The model W 1 can also be viewed as a time-continuous generalization of the model used by Blake and Burrows[ 2001 ]. In their paper,Blake and Burrows model the( time- discrete ) stochastic death process by adjusting data om a deterministic death probability by multiply-ing an exponential random variable. Now we fo & ow their idea to model the stochastic survival process by a product of a deterministic survival probability and an exponenti random variable. If we express this model with mortality rate, the corresponding stochas-tic mortality rate wi & become a sum of a deterministic mortality rate and a random vari - able. This is just what model W 1 says.

We investigate the Lvy processes of the class generalized Hyperbolic distribu-tions and its subclass. Principally we follow the chapter3.2 of McNeil et al.[2005].In particular, we follow the work of Breymann and Lthi[2007].

Denition 2.2.4. Arandom variabl e W is said to have a Generalized Inverse Gaussian( GIG ) distribution with parameters , , and , for shor t W GIG ( , , ) , if it has a density given by

f GIG (w) = A B

2 w 1

2K ( ) exp;12 3w + w4 0, ( 2.62 )w her e

K (x) =1

2

0

w 1 exp

;1

2x(w + w 1)

0,

w ith parameters satisfying


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> 0, 0, < 0 > 0, > 0, = 0

0, > 0, > 0.

The class of multivariate generalized hyperbolic( GH ) distribution can be rep-resented as a normal mean- variance mixture, where the mixing variable is GIG dis-tributed.

Denition 2.2.5. The random vector X = ( X 1, , X d)is said to have a multivariat e generalized hyperbolic( GH ) distribution with parameters , , , , and , for shor tX

GH d( , , , , , ) ,if

X := + W + WAZ, ( 2.63 )w her e

1. Z N k (0, I k ) , i.e.Z fo & ows ak - dimensional standard normal distribution,

2. A

R d k is a matrix and = AA ,

3. , Rd are deterministic vector,and

4. W 0 is a scalar -valued random variable which is independent of Z and has aGeneralized Inverse Gauss distribution,W

GIG ( , , ) .

Since the conditional distribution of X givenW is gaussian with mean + W and varianceW , we derive the GH density as follows.

f X (x) = 0 f X |W (x |w)f W (w) dw=

0

e(x ) 1

(2 )d2 | |

12 w

d2

exp

IQ(x)

2w 1

2/ w Jf W (w) dw

=( / ) ( + 1 ) d2

(2 )d2 | |

12 K ( )

K d2 ( ( + Q(x))( + 1 ))e(s ) 1 ( ( Q(x))( + 1 )) d2 ( 2.64 )

where Q(x) = ( x ) 1(x ) .Some subclasses of Generalized Hyperbolic distributions are

Hyperbolicdistribution, where =12 (d + 1) . For = 1 and thusd = 1 we havea univariate Hyperbolic distribution.


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Normal Inverse Gaussian distribution, where = 12 . Variance Gammadistribution, where > 0 and which is a limiting case as

goes to 0 .

The ( , , , , , ) -parameterization has an identication problem. That is, thedistribution GH d( , , , , , ) and GH d( , / k, k , ,k , k ) are identical forany k > 0 .

To overcome this drawback, we can constrain the expected value of the mixing variableW to be equal to1 . We set

E [W ] =

K +1 ( )K ( )

= 1 , ( 2.65

)

and

= . ( 2.66 ) Thus we have

= K +1 ( )K ( )

and = 2

=

K ( )K +1 ( )

. ( 2.67 )

We call this( , , , , ) - parameterization . We will use this parameterizationof the generalized hyperbolic distribution to t the stochastic mortality models.

Empirical Investigation Assuming that we have the data sets as described in JP-Morgan Lifemetrics,( see Cairns[2007], Coughlan et al.[2007]). The age x rangesover (x1, . . . , x na ) , the calendar year ranges over(t1, . . . , t ny ) . The functiongx (t) is a pre-specied function, which is tted by according to the JPMorgan LifeMetrics withrespect to the modelM # , with # = 1 , 2, 3, 5, 6, 7, 8 in Cairns[2007]. The randomevolution part is captured by the stochastic process L t .

Applying the approximation that

1 px+ t exp(x (t)) , ( 2.68 ) we investigate the cohort modeling approach.

We observe a xed age60 with x j = 60 , j = 1 , . . . , n y 1 of all cohort i.e. thepopulation is aged60 at each time t j . We assume that the available historical data

set is just a realization of the stochastic survival process1 px+ t . We get the sampleincrement of stochastic mortality rate


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For modelW 1 with x (t) = gx (t) + L t

ICR1 j (i) := x j (t i ) x j (t i 1) (gx j (t i ) gx j (t i 1))= L t i L t i 1 ( 2.69 )

where j = 1 , . . . , n y 1 and i = 2 , . . . , n a . For modelW 2 with x (t) = gx (t) + eL t

ICR 2 j (i) := log Ax j (t i) (gx j (t i)x j (t i 1) gx j (t i 1)B= L t i

L t i 1 ( 2.70 )

where j = 1 , . . . , n y 1 and i = 2 , . . . , n a . For modelW 3 with x (t) = gx (t)L t

ICR 3 j (i) :=x j (t i )gx j (t i)

x j (t i 1)gx j (t i 1)

= L t i L t i 1 ( 2.71 )

where j = 1 , . . . , n y

1 and i = 2 , . . . , n a .

For modelW 4 with x (t) = gx (t)eL t

ICR 4 j (i) := log Ax j (t i )gx j (t i 1)x j (t i 1)gx j (t i)B= L t i L t i 1 ( 2.72 )

where j = 1 , . . . , n y 1 and i = 2 , . . . , n a .

We assume that t i

t i1

= 1 , for alli = 2 , . . . , n a . Thus the increments for eachmodel have identical distribution asL1 ,

ICR k j (i)d= L1, ( 2.73 )

for allk = 1 , 2, 3, 4 , all j = 1 , . . . , n y 1 , and alli = 2 , . . . n a .For (L t ) being Lvy process,we apply its stationary independent increments prop-

erty to estimate the parameters according to the data set.

We list some graphs of the tted results for modelW 1 w.r.t. the model M2.

For the generalized Hyperbolic distribution of L t we have


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G h y p

f o r

W 1 w i t h m

2

d a

t a

D e n s i t y

! 0

. 0 0 4

0 . 0

0 0

0 . 0

0 4

0 . 0

0 8

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

A s y m m e

t r i c G e n e r a

l i z e

d H y p e r b o

l i c

G a u s s i a n

!!!

!!

!!

!!

!!

!!

!!

!!!

!

!!!!

! 0 . 0

0 4

0 . 0

0 0

0 . 0

0 4

0 . 0

0 8

1 2 3 4 5 6

G h y p

f o r

W 1 w i t h m

2

g h y p . d

a t a

l o g ( D e n s i t y )

!!

!!

!

!

! 0

. 0 0 2

0 . 0

0 2

0 . 0 0

6

! 0 . 0 0 4 ! 0 . 0 0 2 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8

G h y p

f o r

W 1 w i t h m

2

T h e o r e

t i c a

l Q u a n

t i l e s

S a m p l e Q u a n t i l e s

!

A s y m m e

t r i c G e n e r a

l i z e

d H y p e r b o

l i c

G a u s s i a n

Figure 2.3: Model W1 with M2: generalized Hyperbolic distribution


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For the Hyperbolic distribution of L t we have

h y p

f o r

W 1 w i t h m

2

d a

t a

D e n s i t y

! 0

. 0 0 4

0 . 0

0 0

0 . 0

0 4

0 . 0

0 8

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

A s y m m

e t r i c H y p e r b o

l i c

G a u s s

i a n

!!!

!!

!!

!

!!

!!

!!

!!

!!

!

!!!!

! 0

. 0 0 4

0 . 0 0 0

0 . 0

0 4

0 . 0

0 8

1 2 3 4 5 6

h y p

f o r

W 1 w i t h m

2

g h y p . d

a t a


!!

!!

!

! 0

. 0 0 2

0 . 0

0 0

0 . 0

0 2

0 . 0

0 4

! 0 . 0 0 4 ! 0 . 0 0 2 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8

h y p

f o r

W 1 w i t h m

2

T h e o r e

t i c a

l Q u a n

t i l e s


!

A s y m m e

t r i c H y p e r b o

l i c

G a u s s i a n

Figure 2.4: Model W1 with M2: Hyperbolic distribution


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For the Normal-inverse Gaussian distribution of L t we have

N I G f o r

W 1 w i t h m

2

d a

t a

D e n s i t y

! 0

. 0 0 4

0 . 0

0 0

0 . 0

0 4

0 . 0

0 8

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

A s y m m e

t r i c N o r m a l I n v e r s e

G a u s s

i a n

G a u s s

i a n

!!!

!!

!!

!

!!

!!

!!

!!

!!

!

!!!!

! 0

. 0 0 4

0 . 0 0 0

0 . 0

0 4

0 . 0

0 8

1 2 3 4 5 6

N I G f o r

W 1 w i t h m

2

g h y p . d

a t a


!!

!

!

!

!

! 0

. 0 0 2

0 . 0

0 2

0 . 0

0 6

! 0 . 0 0 4 ! 0 . 0 0 2 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8

N I G f o r

W 1 w i t h m

2

T h e o r e

t i c a

l Q u a n

t i l e s


!

A s y m m e

t r i c N o r m a

l I n v e r s e

G a u s s

i a n

G a u s s i a n

Figure 2.5: Model W1 with M2: Normal-inverse Gaussian distribution


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For the Variance Gamma distribution of L t we have

V G f o r

W 1 w i t h m

2

d a

t a

D e n s i t y

! 0

. 0 0 4

0 . 0

0 0

0 . 0

0 4

0 . 0

0 8

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

A s y m m e

t r i c V a r i a n c e

G a m m a

G a u s s

i a n

!!!

!!

!!

!

!!

!!

!!

!!

!!

!

!!!!

! 0

. 0 0 4

0 . 0 0 0

0 . 0

0 4

0 . 0

0 8

1 2 3 4 5 6

V G f o r

W 1 w i t h m

2

g h y p . d

a t a


!

!

!!

!

! 0

. 0 0 2

0 . 0

0 2

0 . 0

0 6

! 0 . 0 0 4 ! 0 . 0 0 2 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8

V G f o r

W 1 w i t h m

2

T h e o r e

t i c a

l Q u a n

t i l e s


!

A s y m m e

t r i c V a r i a n c e

G a m m a

G a u s s i a n

Figure 2.6: Model W1 with M2: Variance Gamma distribution


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For modelW 4 : x (t) = gx (t)eL t we have

Distr. log -likelihood gHyp 24.13267075 32.29683139-0.02554900 0.01587546 0.02759323 1637.12Hyp 1 47.71805766-0.03019872 0.01577556 0.03223673 1637.138NIG 12 75.51132814-0.05014181 0.01534125 0.05218837 1637.147VG 102.85618656 -0.07450063 0.01464559 0.07654978 1637.134

Table 2.7: Fitted parameters of model W4 with M2

Although the systematic mortality risk can not be diversied by just issuing moreinsurance policies, it can be transfered to the nancial market by trading on some -nancial instruments on mortality risk. Such a transaction is calledSecuritization andis a popular solution on the nancial market. During the last years, the Swiss Re is-sues the mortality bond concerning on the systematic increasing death rate, and BNPParibas tries the longevity bonds allowing the transformation of systematic increasing survival rate.

In chapter 4 we will suggest an approach to model these transaction of systematicmortality risks, in particular the longevity risks in the nancial market. We will inves-tigate the longevity risks from nancial market viewpoint. The mathematical work will be performed under the risk-neutral probability measures.

Before we start it, let us rst review the no- arbitrage theoryabout nancial markets.


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Chapter 3

Alternative Approach to

No -Arbitrage Hypothesis

3.1 Motivation: non - tradable Bank Account

Assumption 3.1.1 ( Model Setup ). As usual,w e assume that there exists a stochasticbasis( , F , F = ( F t )0 t T , Q ) w hich satises the usual conditions, see Protter [ 2005 ] rst denition in section I. 1 , or Jacod and Shiryaev[ 2003 ] denitionsI. 1.2 and I. 1.3 .Two types of securities are supported by( , F , F , Q ) : the default - free zero - coupon bonds and the risky securities.

The price of the default - ee zero-coupon bond p(t, T ) w hich pays one unit at matu- rityT , T -bond for short,is given by

p(t, T ) = expA T t f (t, s ) dsB, w ith p(T, T ) = 1 , for a & 0 t T T , ( 3.1 )w her e (f (t, T ))0 t T is the instantaneous forward rate of the T -bond. We also dene the bank account as

B (t) = exp( t0 r (s) ds), ( 3.2 )w ith short rat e r (t) := f (t, t ) .

Any tradable security other than default - ee T - Bonds is considered as risky security. Examples are stocks( or funds ) , defaultable xed income bonds, survivor bonds( whichw i & be dened in chapter 4 ) , and contingent claims on underlyings of any traded securi -ties.

Bank account is not a tradable asset, although one can hypothetically generate itby self -nancing strategies withT -bonds.

42


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CHAPTER 3. ALTERNATIVE APPROACHTONO- ARBITRAGEHYPOTHESIS43

At time T > 0 , if one needs a position of a deterministic value on bank account,say oneB (T ) , one can just position one unit money at time0 and roll over instan-taneously by trading thejust maturing t -bonds up to timeT , and gets oneB (T )at time T , see Bjrk et al.[1997a ], Dberlein and Schweizer[2001]. However, if oneneeds a position of a random value on bank account at timeT , for example, a shortposition of one stock much on bank account, one can not perform this trading strat-egy by selling one stock bank account at time0 , since at the beginning one does notknow the exact value of the stock at timeT .

As the bank account is not tradable, the conventional d + 1 assetsapproachinvolving d risky assets and the one riskless asset of bank account( seeHarrison andPliska [1981], Karatzas and Shreve[1998], Delbaen and Schachermayer[2006], ) is notcapable to dene theno arbitrage opportunity . Since to generate a bank accountB (N ) on a N -steps time discrete trading nancial bonds market, there are neededat least N zero-coupon bonds with maturities up toN . To generate a bank accountB (T ) on the time continuous trading bonds ma rket, there are needed uncountableinnitely many bonds up to timeT .

In fact, neither Black and Scholes[1973] nor Merton[1973] use the bank accountas tradable asset in their seminal discussions about arbitrage-free evaluation of con-tingent claims.

3.2 Alternative Denition of No -Arbitrage Oppor -

tunity

We will deneloca & y the no - arbitrage opportunity with respect to tradable( just

maturing )

default-free T

-bonds, and modelgloba & y the arbitrage

-free assets dynam

-

ics with respect to the bank account.

Denition 3.2.1 ( No- Arbitrage Opportunity ). The nancial market is said to be of no -arbitrage opportunity (or arbitrage - free ) , if over every time step, the expected return of any tradable asset ( including default - ee T -bonds ) , conditioned on the infor - mation at the beginning of the step,is equal to the riskless return of the defualt - ee bond that matures at the end of the step.

Let us make this descriptive denition precisely.


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Time -discrete model Without loss of generality, let us assume that the time step = 1 . We dene the one step return of a tradable assetp(t) over (t 1, t ) as

R t = p(t) p(t 1) p(t 1), ( 3.3 )

and the one step riskless return of thet -bond p(t 1, t ) over (t 1, t ) asR (t )t =

1 p(t 1, t ) p(t 1, t )

. ( 3.4 )

The no arbitrage opportunity means that

E Q [R t |F t 1] = R(t )t . ( 3.5 )

Time -continuous model Since the trading occurs instantaneously, we look at thediff erential form of the asset returns. Let us dene the return of a tradable assetp(t)over the instantaneous time step dt as

R t dt =d p(t) p(t )

, ( 3.6 )

and the instantaneous riskless return of thet -bond p(., t ) as( see Bjrk[2004] section20.2.3 for an heuristic interpretation )

r (t) dt ( 3.7 )

Now the no-arbitrage opportunity means that

E Q [R t dt |F t ] = r (t) dt, ( 3.8 )

in which we assumer (t) is predictable,i.e. r (t) is F t measurable

The following hypothesis is usually assumed to be true to exclude arbitrage possi-

bility from the nancial market model.

Theorem 3.2.1 ( No Arbitrage Hypothesis ). The nancial market is said of nor - arbitrag e opportunity( or arbitrage- ee ) if and only if the normalized tradable asset prices( i.e. divided by the non-tradable bank account ) are martingales under Q .

Proof.We discuss time discrete and time continuous models respectively.

1. Time discrete model: Without loss of generality,we assume = 1 . The bank

account B (t) at time t is generated by the risk-free bonds up tot with B (t) =

rti=1 1 p(i 1,i ) . Let the one step riskless return beR(t )t =

1 p(t 1,t ) p(t 1,t )

( see ( 3.4 )).


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The arbitrage - free T - bonds market.ONLY IF part: Let the one step return of an arbitrary T -bond withT

t be R (T )t := p(t,T ) p(t 1,T ) p(t 1,T ) ( cf. ( 3.3 )). Since the normalizedT -bond

1 p(t,T )B (t ) 2is martingale, we have0 = E Q C p(t, T )B (t) p(t 1, T )B (t 1) |F t 1D

E Q Cp(t, T ) p(t 1, T ) |F t 1D=B (t)

B (t 1)

E Q Cp(t, T ) p(t 1, T ) 1|F t 1D=

B (t)B (t 1)

1

E

Q

C p(t, T )

p(t

1, T )

p(t 1, T ) |F t 1D=1

p(t

1, t )

p(t 1, t )

E Q [R (T )t |F t 1] = R(t )t .

Thus the T -bonds market is arbitrage-free ( see denition 3.2.1 ).IF part: For the time discrete model, it is sufficient to show that

0 = E Q C p(t, T )B (t) p(t 1, T )B (t 1) |F t 1D. This can be done reversely as above. Thus

1 p(t,T )

B (t )

2is a martingale under

Q . The arbitrage - free risky assets market

ONLY IF part: Let the one step return of an arbitrary risky assetX (t)be R t := X (t ) X (t 1)X (t 1) ( cf. ( 3.3 )). Since the normalized asset price1X (t )B (t )2ismartingale, we have

0 = E Q CX (t)B (t) X (t 1)B (t 1) |F t 1D

E Q

CX (t)

X (t 1)|F t 1

D=

B (t)

B (t 1)

E Q CX (t)X (t 1) 1|F t 1D=B (t)

B (t 1) 1

E Q CX (t) X (t 1)X (t 1) |F t 1D=

1 p(t 1, t ) p(t 1, t )

E Q [R t |F t 1] = R

(t )t .

Thus the T -bonds market is arbitrage-free ( see denition 3.2.1 ).IF part: For the time discrete model, it is enough to show that

0 = E Q CX (t)B (t) X (t 1)B (t 1) |F t 1D.


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This can be done reversely as above. Thus1X (t )B (t )2is a Q -martingale.For technical background on time discrete martingale,see part Bof Williams[1991] or chapter 24

26 of Jacod and Protter[2004].

2. Time continuous trading model:

Now the bank account at timet is B (t) = es t

0r (s ) ds , where r (t) = f (t, t ) is the

short rate. The instantaneous riskless return becomes r (t)dt ( see ( 3.7 )).

The arbitrage - free T - bonds marketONLY IF part: Let the instantaneous return of an arbitrary T -bond bed p(t,T )

p(t ,T )( cf. ( 3.6 )). Since

1 p(t,T )

B (t )

2is a martingale, the conditioned expected

instantaneous return is equal to zero. That is

0 = E Q CdA p(t, T )B (t) B|F t DIt sformula

0 =E Q Cd p(t, T )B (t) r (t) p(t , T )B (t) dt |F t D

E Q Cd p(t, T ) p(t , T ) |F t D= r (t)dt.

Thus the T -bond market is arbitrage-free ( see denition 3.2.1 ).IF part: Similar to the case of time-discrete model, it is sufficient to show that

E Q CdA p(t, T )B (t) B|F t D= 0 . This can be done recursively as above. That is

E Q Cd p(t, T ) p(t , T ) |F t D= r (t)dt

E Q Cd p(t, T )B (t) |F t D= r (t) p(t , T )B (t) dt0 =

E Q Cd p(t, T )B (t) r (t) p(t , T )B (t) dt |F t DIt sformula

0 = E Q CdA p

Date post:	30-May-2018
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Longevity Risks Modelling and Financial Engineering

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