Inspired by Finance

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Finance

Yuri Kabanov · Marek RutkowskiThaleia Zariphopoulou Editors


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Yuri Kabanov Marek Rutkowski

Thaleia ZariphopoulouEditors

Inspired by Finance

The Musiela Festschrift


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Editors

Yuri KabanovLaboratoire de mathématiquesUniversité de Franche-ComtéBesançon, France

International Laboratory of QuantitativeFinance

Higher School of EconomicsMoscow, Russia

Marek RutkowskiSchool of Mathematics & StatisticsUniversity of SydneySydney, New South Wales, Australia

Thaleia ZariphopoulouDepts. of Mathematics and IROMMcCombs School of BusinessThe University of Texas at AustinAustin, USA

ISBN 978-3-319-02068-6 ISBN 978-3-319-02069-3 (eBook)

DOI 10.1007/978-3-319-02069-3Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013952730

Mathematics Subject Classification: 91GXX, 91G10, 91G20, 91G30, 91G40, 91G80

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Introduction

The present volume contains 25 papers, contributed by 47 authors, and dealing withhot topics of modern mathematical finance. They cover a broad spectrum of areas,including: pricing and hedging of derivative securities, modeling of term structure of interest rates, optimal stopping problems and pricing of contingent claims of Amer-ican style, performance criteria and portfolio optimization problems, counterpartycredit risk and valuation of defaultable securities.

In the paper “Forward Start Foreign Exchange Options under Heston’s Volatilityand the CIR Interest Rates”, Rehez Ahlip and Marek Rutkowski examine the val-

uation of forward start foreign exchange options in the Heston stochastic volatilitymodel for the exchange rate combined with the Cox–Ingersoll–Ross dynamics forthe domestic and foreign interest rates. They derive semi-analytical formulae forsuch contracts.

In “Real Options with Competition and Incomplete Markets”, Alain Bensoussanand Sing Ru (Celine) Hoe consider a Stackelberg leader-follower game for exploit-ing an irreversible investment opportunity with payoffs of a continuous stochasticincome stream for a fixed cost.

In the article “Dynamic Hedging of Counterparty Exposure”, Tomasz Bieleckiand Stéphane Crépey study mathematical aspects of dynamic hedging of CreditValuation Adjustment in a portfolio of OTC financial derivatives. Their analysis

justifies rigorously some market practice, thus making precise the proper definitionof the Expected Positive Exposure (EPE) and the way the EPE should be used in thehedging strategy.

Luciano Campi in “A Note on Market Completeness with American Put Options”shows that any contingent claim on a possibly incomplete two-asset market, satisfy-ing some natural hypotheses, can be approximated by investing dynamically in theunderlying stock and statically in all American put options of every strike price kand with the same maturity T .

The paper “An f -Divergence Approach for Optimal Portfolios in ExponentialLévy Models” by Susanne Cawston and Ludmila Vostrikova develops a unified ap-proach to derivation of explicit formulae for utility maximizing strategies in expo-nential Lévy models. This approach is related to f -divergence minimal martingale

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vi Introduction

measures and is based on a new concept of preservation of the Lévy property by f -divergence minimal martingale measures. For a certain class of f -divergences func-tions, they give conditions for the existence of corresponding maximizing strategiesas well as explicit formulae.

Bénamar Chouaf and Serguei Pergamenchtchikov consider, in their paper “Opti-mal Investment with Bounded VaR for Power Utility Functions”, the classical Mer-ton problem with a constraint involving Value-at-Risk. They obtain explicit expres-sions for the Bellman function and the optimal control.

In “Three Essays on Exponential Hedging with Variable Exit Times”, TahirChoulli, Junfeng Ma and Marie-Amélie Morlais address three main problems re-lated to exponential hedging with variable exit times. The first problem is to explic-itly parameterize the exponential forward performances and describing the optimalsolution for the corresponding utility maximization problem. The second problem

deals with the horizon-unbiased exponential hedging. The authors are interested indescribing the dynamic payoffs for which there exists an admissible strategy thatminimizes the risk—in the exponential utility framework—whenever the investorexits the market at stopping times. Furthermore, they explicitly describe the optimalstrategy when it exists. The third contribution deals with the optimal selling prob-lem, where the investor is simultaneously looking for the optimal portfolio and theoptimal time to liquidate the assets.

In the paper “Mean Square Error and Limit Theorem for the Modified LelandHedging Strategy with a Constant Transaction Costs Coefficient”, Sébastien Darses

and Emmanuel Denis obtain delicate results on the rate of convergence for the ap-proximate hedging strategy. This strategy was recently suggested by the second au-thor and it turns out that it performs well—in contrast to the Leland strategy—without rescaling.

In his paper “Yield Curve Smoothing and Residual Variance of Fixed IncomePositions”, Raphaël Douady treats the yield curve as an object lying in an infinite-dimensional Hilbert space, the evolution of which is driven by a cylindrical Brown-ian motion. He proves that the principal component analysis (PCA) can be appliedand he provides the best approximation of the yield curve evolution by the GaussianHeath–Jarrow–Morton model with a predetermined number of factors.

In the paper “Maximally Acceptable Portfolios”, Ernst Eberlein and Dilip Madanconsider an optimization problem, in a non-Gaussian setting, which performancecriterion is the Cherny–Madan index of accessibility. Using back-testing on realdata, they show that the corresponding optimal portfolios outperform those basedon the maximal Sharpe ratio.

The paper “Conditional Default Probability and Density”, co-authored by NicoleEl Karoui, Monique Jeanblanc, Ying Jiao, and Benhaz Zargari, is dedicated to thestudy of some interesting mathematically and practically important questions arisingin the theory of defaultable securities.

In “Some Extensions of Norros Lemma in Models with Several Defaults”, PavelGapeev extends the result mentioned in the title to the case of credit risk modelsin which the reference filtration is not trivial. He shows that if the reference filtra-tion satisfies the so-called immersion property with respect to every filtration which


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is progressively enlarged by any particular default time, then the terminal values of the compensators of the associated default processes are independent of the observa-tions. The author also provides links between various kinds of immersion propertiesand (conditional) independence of the terminal values of the compensators (with

respect to the reference filtration).Pavel Gapeev and Neofytos Rodosthenous in their paper “On the Pricing of Perpetual American Compound Options” present, in the framework of the Black–Scholes model, explicit pricing formulae for financial contracts which give theirholders the right to buy or sell some other options at certain times in the future. Therational pricing problems for such contracts are embedded into two-step optimalstopping problems for the underlying asset price processes. Their method consistsof decomposing these two-step problems into ordinary one-step ones and, in turn,solve them sequentially.

Emmanuel Gobet and Ali Suleiman in “New Approximations in Local VolatilityModels” propose new approximation formulae for the price of call options, moreprecise and numerically efficient than the existing ones. They extend previous re-sults where stochastic expansions were combined with the Malliavin calculus toobtain approximations based on the local volatility at-the-money and they derivealternative expansions involving the local volatility at strike.

The paper “Low-Dimensional Partial Integro-Differential Equations for High-Dimensional Asian Options” by Peter Hepperger deals with problems of pricingAsian options with their payoffs depending on large numbers of securities (forexample, an option on a stock basket index) whose prices are modeled by jump-

diffusion processes.Constantinos Kardaras contributes the work titled “A Time Before Which Insid-

ers Would not Undertake Risk”. The numéraire portfolio is the unique strictly posi-tive wealth process that, when used as a benchmark to denominate all other wealth,makes all wealth processes local martingales. If the minimum of the numéraire port-folio is known then risk-averse insider traders would refrain from investing in therisky assets before that time. This and other results of the paper shed light on theimportance of the numéraire portfolio as an indicator of an overall market perfor-mance.

The authors of “Sensitivity with Respect to the Yield Curve: Duration in aStochastic Setting”, Paul Kettler, Frank Proske, and Mark Rubtsov, study an ex-tension of the concept of bond duration to stochastic setting. They define stochas-tic duration as a Malliavin derivative in the direction of a stochastic yield surfacemodeled by the Musiela equation. Using this concept, they propose a mathemati-cal framework for the construction of immunization strategies (or delta hedges) of portfolios of interest rate securities with respect to the evolution of the whole yieldsurface.

In the paper “On the First Passage Time Under Regime-Switching with Jumps”,Masaaki Kijima and Chi Chung Siu present the analytical solution for the Laplace

transform of the joint distribution of the first passage time and undershoot/overshootvalue under a regime-switching jump-diffusion model. Their methodology can beapplied to a variety of stopping time problems under a regime-switching model with

jump risks.


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viii Introduction

The article “Strong Consistency of the Bayesian Estimator for the Ornstein-Uhlenbeck Process” by Arturo Kohatsu-Higa, Nicolas Vayatis, and Kazuhiro Ya-suda deals with a theoretical basis of a computational intensive parameter estima-tion method for Markov models. This method can be considered as an approximate

Bayesian estimator method or a filtering problem approximated using particle meth-ods.

The question how to retrieve the probability distributions of the underlying assetfrom the corresponding derivatives quotes is the main subject of the paper “Multi-asset Derivatives and Joint Distributions of Asset Prices” by Ilya Molchanov andMichael Schmutz. Their work is related to a geometric interpretation of multi-assetderivatives as support functions of convex sets. Various symmetry properties for bas-ket, maximum and exchange options are discussed alongside with their geometricinterpretations.

The paper “A Class of Homothetic Forward Investment Performance Processeswith Non-zero Volatility” by Sergey Nadtochiy and Thaleia Zariphopoulou is a con-tribution to the new and promising theory of forward investment. This approachallows for dynamic update of the investor’s investment criterion and offers an alter-native to the classical maximal expected utility objective, which is defined only ata single instant. The underlying object is a stochastic process, the so-called forwardinvestment performance process, which is defined for all times.

Alexander Novikov, Timothy Ling, and Nino Kordzakhia contributed to thevolume by the paper “Pricing of Volume-Weighted Average Options: Analyti-

cal Approximations and Numerical Results”. The volume weighted average price(VWAP), over rolling number of days in the averaging period, is used as a bench-mark price by market participants and can be regarded as an estimate for the pricethat a passive trader will pay to purchase securities in a market. The VWAP iscommonly used in brokerage houses as a quantitative trading tool and also ap-pears in Australian taxation law to specify the price of share-buybacks of publicly-listed companies. The volume process is modeled via a shifted squared Ornstein-Uhlenbeck process and a geometric Brownian motion is used to model the assetprice. The authors derive analytical formulae for moments of VWAP and use themoment matching approach to approximate a distribution of VWAP. Numerical re-sults for moments of VWAP and call option prices are verified by Monte Carlosimulations.

In the paper “Solution of Optimal Stopping Problem Based on a Modificationof Payoff Function”, Ernst Presman compares the idea of the Sonin algorithm of space reduction and sequential modification of the Markov chain with the one of thealgorithm of modification of the payoff function without modification of the chain.He provides some examples showing that the second approach can be extended tothe continuous time models and that, in turn, it leads to a better understanding of solutions of optimal stopping problems.

The aim of the paper “A Stieltjes Approach to Static Hedges” by MichaelSchmutz and Thomas Zürcher is to extend the Carr–Madan approach to hedgingfairly general path-independent contingent claims by static positions in standardtraded assets like bonds, forwards, and plain vanilla call and put options.


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The paper “Optimal Stopping of Seasonal Observations and Projection of aMarkov Chain” by Isaac Sonin is dedicated to an application of the state eliminationalgorithm, which was proposed by the author in his earlier work, and a study of therelationship of the fundamental matrices of the initial chain and its modification in

the reduced state space.Yuri Kabanov

Marek RutkowskiThaleia Zariphopoulou

Besançon, FranceSydney, AustraliaOxford, UK


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Marek Musiela graduated with M.Sc. degree in Mathematics from the Universityof Wrocław in 1973 and was awarded the Ph.D. degree from the Polish Academyof Sciences in 1976. During the first period of his academic career, his researchinterests focussed on statistics of stochastic processes and functionals of diffusionprocesses ([1, 2]). After a period of employment 1976–1980 at the Polish Academyof Sciences, he moved to France where he spent five years at the Institute NationalPolytechnique de Grenoble. During this period, he was awarded the degree of Doc-teur d’Etat in 1984. During his stay in France and afterwards, he very actively col-

laborated with Alain Le Breton with whom he has published several papers on esti-mation problems for diffusion processes and general semimartingales ([3, 4]).In 1985 he took the position at the University of New South Wales, where he

stayed till 2000. Encouraged by Alan Brace, he started research on the theory of termstructure of interest rates, as well as practical implementations of various GaussianHeath-Jarrow-Morton type models. In the first stage, his academic contributionswere concerned with development and deepening of the HJM methodology ([5, 6]).In particular, he proposed and developed a novel way of analyzing an HJM-typemodel that hinges on introducing infinite-dimensional processes representing theyield curve and the study of the so-called Musiela’s SPDE governing the dynamicsof the yield curve. This highly innovative approach underpinned further studies of consistency problems for HJM models for the next decade.

The next exciting step in Marek’s research was the development of original ap-proaches to arbitrage-free modeling of market rates. His research in this area origi-nally started in collaboration with Dieter Sondermann from the University of Bonnand was subsequently continued by the group concentrated around Marek at UNSWin Sydney. Their joint efforts and parallel studies by a group of researchers lead bySondermann at the University of Bonn resulted in what is now well-known as theLIBOR Market Model. The ground-breaking papers ([7, 8, 9]), which were com-

pleted in 1995 and published in 1997, completely revised the traditional paradigmof term structure modeling with continuous compounding. Before 1995, virtuallyall continuous-time term structure models used in the valuation of derivatives wereinvariably based on either the concept of the short-term rate or the instantaneous

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forward rate. The influence of the new paradigm on further research was immense;it suffices to mention that each of these works was since then cited in hundreds of papers by other researchers. In retrospective, one can make an opinion that this wasthe last major development in the field of term structure modeling.

After a highly successful academic career at universities in France and Australia,Marek made in 2000 a bold decision to leave the academia and start a new excitingperiod in his life as the head quant with BNP Paribas in London. After several yearsof experience in consulting for investment banks in Australia and Europe, he wasvery well prepared to the new challenge of leading the Fixed Income Research andSupport Team.

Around this time, Marek began a collaboration with Thaleia Zariphopoulou onindifference valuation in incomplete markets and forward investment performancecriteria. This was also the time that he had started being interested in utility-based

pricing in incomplete markets ([12, 13]). Subsequently Marek and Thaleia focussedthe evolution of risk preferences and their connection with numeraire and risk pre-mia. The goal was to understand the structure of indifference prices and what theytell us about pricing and optimal investment choice. This in turn generated manyquestions on the interface of derivative valuation and portfolio management and,gradually, led them to the development of the concept of forward investment perfor-mance measurement ([16, 17]). At the same time, Marek studied with Pierre-LouisLions the fundamental properties of stochastic volatility models ([14, 15]).

All his colleagues were always struck by his constant drive for a better under-standing and his uncanny ability to raise interesting and pertinent mathematical is-

sues. They were very impressed and stimulated by Marek’s inquisitive mind. Hequestioned almost everything in the classical setting and challenged many ideas andstandardized formulations. We look forward to getting inspired by him for manymore years to come.

References

1. Musiela, M.: Divergence, convergence and moments of some integral functionals of diffusions.

Z. Wahrscheinlichkeitstheorie Verw. Geb. 70, 49–65 (1985)2. Musiela, M.: On Kac functionals of one-dimensional diffusions. Stoch. Process. Appl. 22,

79–88 (1986)3. Musiela, M., Le Breton, A.: Strong consistency of least squares estimates in linear regression

models driven by semimartingales. J. Multivar. Anal. 23, 77–92 (1987)4. Musiela, M., Le Breton, A.: Laws of large numbers for semimartingales with applications to

stochastic regression. Probab. Theory Relat. Fields 81, 275–290 (1989)5. Musiela, M.: A multifactor Gauss-Markov implementation of Heath, Jarrow and Morton.

Math. Finance 4(3), 259–283 (1994)6. Brace, A., Musiela, M.: Swap derivatives in a Gaussian HJM framework. In: Dempster,

M.A.H., Pliska, S.R. (eds.) Mathematics of Derivative Securities. Cambridge University Press

(1996)7. Brace, A., Ga̧tarek, D., Musiela, M.: The market model of interest rate dynamics. Math. Fi-

nance 7, 127–154 (1997)8. Miltersen, K., Sandmann, K., Sondermann, D.: Closed form solutions for term structure

derivatives with log-normal interest rates. J. Finance 52, 409–430 (1997)


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9. Musiela, M., Rutkowski, M.: Continuous-time term structure models: Forward measure ap-proach. Finance Stoch. 1, 261–291 (1997)

10. Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modeling. Springer, Berlin,New York, First edition, 1997; Second edition, 2005.

11. Goldys, B., Musiela, M., Sondermann, D.: Lognormality of rates and term structure models.

Stoch. Anal. Appl. 18(3), 375–396 (2000)12. Musiela, M., Zariphopoulou, T.: An example of indifference prices under exponential prefer-

ences. Finance Stoch. 8, 229–239 (2004)13. Musiela, M., Zariphopoulou, T.: A valuation algorithm for indifference prices in incomplete

markets. Finance Stoch. 8, 399–414 (2004)14. Musiela, M., Lions, P.L.: Some properties of diffusion processes with singular coefficients.

Commun. Appl. Anal. 1, 109–125 (2006)15. Musiela, M., Lions, P.L.: Correlations and bounds for stochastic volatility models. Ann. IHP,

Analyse Nonlinéaire 24(1), 1–16 (2007)16. Musiela, M., Zariphopoulou, T.: Portfolio choice under dynamic investment performance cri-

teria. Quant. Finance 9(2), 161–170 (2009)

17. Musiela, M., Zariphopoulou, T.: Portfolio choice under space-time monotone performancecriteria. SIAM J. Finance Math. 1, 326–365 (2010).


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Contents

Forward Start Foreign Exchange Options Under Heston’s Volatility

and the CIR Interest Rates . . . . . . . . . . . . . . . . . . . . . . . 1Rehez Ahlip and Marek Rutkowski1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Foreign Exchange Model . . . . . . . . . . . . . . . . . . . . . 33 Forward Start Foreign Exchange Options . . . . . . . . . . . . . 44 Bond Pricing and Forward Exchange Rate . . . . . . . . . . . . 45 Auxiliary Probability Measures . . . . . . . . . . . . . . . . . . 6

5.1 Bond Price Numéraire . . . . . . . . . . . . . . . . . . . 75.2 Savings Account Numéraire . . . . . . . . . . . . . . . . 10

6 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . 127 Valuation of Forward Start Foreign Exchange Options . . . . . . 14

7.1 Options Pricing Formula in the Bond Numéraire . . . . . 157.2 Options Pricing Formula in the Savings Account

Numéraire . . . . . . . . . . . . . . . . . . . . . . . . . 208 Put-Call Parity for Forward Start Foreign Exchange Options . . . 23

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Real Options with Competition and Incomplete Markets . . . . . . . . 29

Alain Bensoussan and SingRu (Celine) Hoe1 Investment Game Problems and General Model Assumptions . . 302 Follower’s Problem and Solution . . . . . . . . . . . . . . . . . 31

2.1 Postinvestment Utility Maximization . . . . . . . . . . . 322.2 Preinvestment Utility Maximization . . . . . . . . . . . . 342.3 Follower’s Optimal Stopping Rule . . . . . . . . . . . . 37

3 Leader’s Problem and Solution . . . . . . . . . . . . . . . . . . 38

3.1 Postinvestment Utility Maximization . . . . . . . . . . . 383.2 Leader’s Optimal Stopping Rule . . . . . . . . . . . . . 44

4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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Dynamic Hedging of Counterparty Exposure . . . . . . . . . . . . . . . 47Tomasz R. Bielecki and Stéphane Crépey1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.1 General Set-up . . . . . . . . . . . . . . . . . . . . . . . 48

2 Cashflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.1 Re-hypothecation Risk and Segregation . . . . . . . . . . 512.2 Cure Period . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Collateral Modeling . . . . . . . . . . . . . . . . . . . . 57

4 Common Shock Model of Counterparty Credit Risk . . . . . . . 594.1 Unilateral Counterparty Credit Risk . . . . . . . . . . . . 594.2 Model of Default Times . . . . . . . . . . . . . . . . . . 60

4.3 Credit Derivatives Prices and Price Dynamicsin the Common Shocks Model . . . . . . . . . . . . . . 635 Hedging Counterparty Credit Risk in the Common Shocks Model 64

5.1 Min-Variance Hedging by a Rolling CDSon the Counterparty . . . . . . . . . . . . . . . . . . . . 64

5.2 Multi-instruments Hedge . . . . . . . . . . . . . . . . . 69References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A Note on Market Completeness with American Put Options . . . . . . 73Luciano Campi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Hedging with American Put Options . . . . . . . . . . . . . . . 764 A Counterexample to Hedging with European Call Options . . . 80References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

An f -Divergence Approach for Optimal Portfolios in Exponential Lévy

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83S. Cawston and L. Vostrikova1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2 Utility Maximization in Exponential Lévy Models . . . . . . . . 853 A Decomposition for Lévy Preserving Equivalent Martingale

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 Utility Maximizing Strategies . . . . . . . . . . . . . . . . . . . 96References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Optimal Investment with Bounded VaR for Power Utility Functions . . 103Bénamar Chouaf and Serguei Pergamenchtchikov1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . 107

3.1 The Unconstrained Problem . . . . . . . . . . . . . . . . 1073.2 The Constrained Problem . . . . . . . . . . . . . . . . . 108

4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110


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4.1 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . 1104.2 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . 114

Appendix Properties of the Function (35) . . . . . . . . . . . . . . . 115References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Three Essays on Exponential Hedging with Variable Exit Times . . . . 117Tahir Choulli, Junfeng Ma, and Marie-Amélie Morlais1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172 Mathematical Model and Preliminaries . . . . . . . . . . . . . . 1193 Complete Parameterization of Exponential Forward Performances 1234 Horizon-Unbiased Exponential Hedging . . . . . . . . . . . . . 1365 Optimal Portfolio and Investment Timing for Semimartingales . . 140Appendix 1 Some Auxiliary Lemmas . . . . . . . . . . . . . . . . . 148Appendix 2 MEH σ -Martingale Density Under Change of Probability 154References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Mean Square Error and Limit Theorem for the Modified Leland

Hedging Strategy with a Constant Transaction Costs Coefficient . 159Sébastien Darses and Emmanuel Lépinette1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1592 Notations and Models . . . . . . . . . . . . . . . . . . . . . . . 161

2.1 Black–Scholes Model and Hedging Strategy . . . . . . . 1612.2 Reminder About Leland’s Strategy . . . . . . . . . . . . 162

2.3 A Possible Modification of Leland’s Strategy . . . . . . . 1632.4 Assumptions and Notational Conventions . . . . . . . . . 1643 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.1 Geometric Brownian Motion and Related Quantities . . . 1664.2 Basic Results Concerning the Revision Dates . . . . . . . 168

5 Proof of the Limit Theorem . . . . . . . . . . . . . . . . . . . . 1705.1 Step 1: Splitting of the Hedging Error . . . . . . . . . . . 1715.2 Step 2: The Mean Square Residue Tends to 0 with Rate

n

1

2 +2p

. . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.3 Step 3: Asymptotic Distribution . . . . . . . . . . . . . . 1845.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 190

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191A.1 Explicit Formulae . . . . . . . . . . . . . . . . . . . . . 191A.2 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 193A.3 Technical Lemmas . . . . . . . . . . . . . . . . . . . . . 198

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Conditional Default Probability and Density . . . . . . . . . . . . . . . 201

N. El Karoui, M. Jeanblanc, Y. Jiao, and B. Zargari1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2022 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2023 Examples of Martingale Survival Processes . . . . . . . . . . . . 203


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xviii Contents

3.1 A Dynamic Gaussian Copula Model . . . . . . . . . . . 2043.2 A Gamma Model . . . . . . . . . . . . . . . . . . . . . 2073.3 Markov Processes . . . . . . . . . . . . . . . . . . . . . 2073.4 Diffusion-Based Model with Initial Value . . . . . . . . . 208

4 Density Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.1 Structural and Reduced-Form Models . . . . . . . . . . . 2104.2 Generalized Threshold Models . . . . . . . . . . . . . . 2114.3 An Example with Same Survival Processes . . . . . . . . 212

5 Change of Probability Measure and Filtering . . . . . . . . . . . 2135.1 Change of Measure . . . . . . . . . . . . . . . . . . . . 2135.2 Filtering Theory . . . . . . . . . . . . . . . . . . . . . . 2145.3 Gaussian Filter . . . . . . . . . . . . . . . . . . . . . . . 217

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Yield Curve Smoothing and Residual Variance of Fixed Income Positions 221Raphaël Douady1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212 History, Tribute and Recent Bibliography . . . . . . . . . . . . . 2253 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . 225

3.1 Term Structure of Interest Rates . . . . . . . . . . . . . . 2263.2 Risk-Neutral Probability . . . . . . . . . . . . . . . . . . 2263.3 Diffusion of Discount Factors and Forward Rates . . . . 2273.4 Function Valued Random Processes . . . . . . . . . . . . 231

4 Market Data on the Term Structure . . . . . . . . . . . . . . . . 2334.1 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 2334.2 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.3 Cash and Future Short Rates . . . . . . . . . . . . . . . . 2344.4 STRIP, or the Decomposition of Bonds . . . . . . . . . . 2354.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 236

5 Brownian Motions in a Hilbert Space . . . . . . . . . . . . . . . 2366 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6.1 Almost Complete Market . . . . . . . . . . . . . . . . . 2376.2 Finite Variance . . . . . . . . . . . . . . . . . . . . . . . 2386.3 Gaussian Rates . . . . . . . . . . . . . . . . . . . . . . . 238

7 Principal Component Analysis . . . . . . . . . . . . . . . . . . 2387.1 The Volatility Operator . . . . . . . . . . . . . . . . . . 2387.2 Principal Component Analysis . . . . . . . . . . . . . . 2407.3 Infinite Dimensional H.J.M. Representation . . . . . . . 241

8 Optimal Representation with an N -Factor Model . . . . . . . . . 2429 Possible Choice in the Hilbert Space V . . . . . . . . . . . . . . 24610 Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 24711 Computation of Eigenmodes . . . . . . . . . . . . . . . . . . . 249

11.1 Reconstruction and Smoothing of the Yield Curve . . . . 24911.2 Eigenmode Computation from the Historical Series . . . 250

12 Dimension Reduction . . . . . . . . . . . . . . . . . . . . . . . 251


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Contents xix

12.1 The Drift Term and the Real Option Pricing . . . . . . . 25212.2 Practical Option Hedging . . . . . . . . . . . . . . . . . 25312.3 Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . 253

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Maximally Acceptable Portfolios . . . . . . . . . . . . . . . . . . . . . . 257Ernst Eberlein and Dilip B. Madan1 Acceptability Indices . . . . . . . . . . . . . . . . . . . . . . . 2592 Constructing Maximally Acceptable Portfolios . . . . . . . . . . 2633 Nonlinearity and Acceptability in Economies . . . . . . . . . . . 2654 In Sample Application to Portfolios Constructed for the Year 2008 2665 Backtesting Portfolio Rebalancing from 1997 to 2008 . . . . . . 2686 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Some Extensions of Norros’ Lemma in Models with Several Defaults . . 273Pavel V. Gapeev1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2732 Default Times and Filtration Immersions . . . . . . . . . . . . . 274

2.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . 2742.2 Immersion Properties . . . . . . . . . . . . . . . . . . . 275

3 Extensions of Norros’ Lemma . . . . . . . . . . . . . . . . . . . 2763.1 The Case of One Default Time . . . . . . . . . . . . . . 2763.2 The Case of Two Default Times . . . . . . . . . . . . . . 278

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

On the Pricing of Perpetual American Compound Options . . . . . . . 283Pavel V. Gapeev and Neofytos Rodosthenous1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2832 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

2.1 Formulation of the Problem . . . . . . . . . . . . . . . . 2852.2 The Structure of the Optimal Stopping Times . . . . . . . 2862.3 The Free-Boundary Problem . . . . . . . . . . . . . . . 288

3 Solutions of the Free-Boundary Problems . . . . . . . . . . . . . 2883.1 The Call-on-Call Option . . . . . . . . . . . . . . . . . . 2893.2 The Call-on-Put Option . . . . . . . . . . . . . . . . . . 2893.3 The Put-on-Call Option . . . . . . . . . . . . . . . . . . 2903.4 The Put-on-Put Option . . . . . . . . . . . . . . . . . . . 291

4 Main Results and Proofs . . . . . . . . . . . . . . . . . . . . . . 292

5 Chooser Options . . . . . . . . . . . . . . . . . . . . . . . . . . 2975.1 Formulation of the Problem . . . . . . . . . . . . . . . . 2975.2 Solution of the Free-Boundary Problem . . . . . . . . . . 298

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303


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xx Contents

New Approximations in Local Volatility Models . . . . . . . . . . . . . 305E. Gobet and A. Suleiman1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

1.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . 305

1.2 Literature Background . . . . . . . . . . . . . . . . . . . 3061.3 Standing Assumptions for the Approximations . . . . . . 3071.4 Definitions and Other Notations . . . . . . . . . . . . . . 308

2 Expansion Formulas . . . . . . . . . . . . . . . . . . . . . . . . 3092.1 A General Result . . . . . . . . . . . . . . . . . . . . . . 3092.2 Application to Expansion Formulas for Call Price . . . . 3122.3 Other Expansions Based on the Local Volatility at Strike . 3132.4 Expansion Formulas for Implied Volatility . . . . . . . . 3162.5 Applications to Time-Dependent CEV Model . . . . . . 317

3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 3184 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . 3255 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . 3286 Computations of Derivatives of the Black–Scholes Price

Function with Respect to S and K . . . . . . . . . . . . . . . . . 328References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Low-Dimensional Partial Integro-differential Equations for High-

Dimensional Asian Options . . . . . . . . . . . . . . . . . . . . . . 331

Peter Hepperger1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 Hilbert Space Valued Jump-Diffusion . . . . . . . . . . . . . . . 332

2.1 Driving Stochastic Process . . . . . . . . . . . . . . . . 3322.2 Value of an Asian Option . . . . . . . . . . . . . . . . . 334

3 Approximate Pricing with POD . . . . . . . . . . . . . . . . . . 3393.1 POD for the Driving Process . . . . . . . . . . . . . . . 3393.2 POD for the Average . . . . . . . . . . . . . . . . . . . . 3413.3 Approximate Pricing . . . . . . . . . . . . . . . . . . . . 345

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

A Time Before Which Insiders Would not Undertake Risk . . . . . . . . 349Constantinos Kardaras1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3492 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

2.1 The Set-up . . . . . . . . . . . . . . . . . . . . . . . . . 3512.2 The First Result . . . . . . . . . . . . . . . . . . . . . . 3532.3 A Partial Converse to Theorem 1 . . . . . . . . . . . . . 355

3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3563.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . 3563.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . 359

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362


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Contents xxi

Sensitivity with Respect to the Yield Curve: Duration in a Stochastic

Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363Paul C. Kettler, Frank Proske, and Mark Rubtsov1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

2 An Expanded Concept of Duration via Malliavin Calculus . . . . 3673 Estimation of Stochastic Duration and the Construction

of Immunization Strategies . . . . . . . . . . . . . . . . . . . . 375Appendix Macaulay Duration and Portfolio Immunization . . . . . . 381

A.1 Discrete Case . . . . . . . . . . . . . . . . . . . . . . . 381A.2 Continuous Case . . . . . . . . . . . . . . . . . . . . . . 382A.3 Portfolio Immunization . . . . . . . . . . . . . . . . . . 382

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

On the First Passage Time Under Regime-Switching with Jumps . . . . 387Masaaki Kijima and Chi Chung Siu1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3872 Regime-Switching Jump-Diffusion Process . . . . . . . . . . . . 390

2.1 A Special Case: Two Regimes . . . . . . . . . . . . . . . 3943 First Passage Time Under Regime-Switching

Double-Exponential Jump Model . . . . . . . . . . . . . . . . . 3963.1 Conditional Independence and Memoryless Properties . . 3973.2 The First-Passage-Time Problem . . . . . . . . . . . . . 399

4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 4035 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Strong Consistency of the Bayesian Estimator for the Ornstein–

Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 411Arturo Kohatsu-Higa, Nicolas Vayatis, and Kazuhiro Yasuda1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4122 Framework and General Theorem . . . . . . . . . . . . . . . . . 413

2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . 4132.2 General Theorem of Kohatsu-Higa et al. [9] . . . . . . . 4152.3 Parameter Tuning for Assumption (A) (6)-(a) . . . . . . . 416

3 The Ornstein–Uhlenbeck Process . . . . . . . . . . . . . . . . . 4203.1 The Euler–Maruyama Approximation of the OU Process . 4213.2 About Assumptions (A) (1)–(5) . . . . . . . . . . . . . . 4223.3 Assumption (A) (6) . . . . . . . . . . . . . . . . . . . . 427

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Multiasset Derivatives and Joint Distributions of Asset Prices . . . . . . 439Ilya Molchanov and Michael Schmutz1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4392 Basket Options and Options on the Maximum of Several Assets . 441


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3 Characterisation of the Distribution of the Underlying AssetPrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

4 Recovery of Asset Distributions from Option Prices . . . . . . . 4475 Symmetry Properties and Basket Options . . . . . . . . . . . . . 448

6 Symmetries of Exchange and Max-Options . . . . . . . . . . . . 4517 Joint Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 4528 Combinations, Lift Zonoids and General Univariate European

Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Pricing of Volume-Weighted Average Options: Analytical

Approximations and Numerical Results . . . . . . . . . . . . . . . 461Alexander A. Novikov, Timothy G. Ling, and Nino Kordzakhia1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

2 The VWAP Model and the Moment Matching Approach . . . . . 4633 Computing the VWAP Moments . . . . . . . . . . . . . . . . . 464

3.1 The VWAP First Moment . . . . . . . . . . . . . . . . . 4643.2 Computing the Second Moment . . . . . . . . . . . . . . 4683.3 Generalized Inverse Gaussian Distribution . . . . . . . . 469

4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 469Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

A Class of Homothetic Forward Investment Performance Processes

with Non-zero Volatility . . . . . . . . . . . . . . . . . . . . . . . . 475Sergey Nadtochiy and Thaleia Zariphopoulou1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4752 The Stochastic Factor Model and Investment Performance

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4772.1 Forward Investment Performance Process . . . . . . . . . 4782.2 The Forward Performance SPDE . . . . . . . . . . . . . 4792.3 The Zero Volatility Case . . . . . . . . . . . . . . . . . . 481

3 Homothetic Forward Investment Performance Processes . . . . . 483

3.1 The Zero-Volatility Homothetic Case . . . . . . . . . . . 4833.2 Non-zero Volatility Homothetic Case . . . . . . . . . . . 484

4 Non-negative Solutions to an Ill-Posed Heat Equationwith a Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 4854.1 The Backward Heat Equation . . . . . . . . . . . . . . . 494

5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4965.1 Mean Reverting Stochastic Volatility . . . . . . . . . . . 4965.2 Heston-Type Stochastic Volatility . . . . . . . . . . . . . 500

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

Solution of Optimal Stopping Problem Based on a Modification

of Payoff Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 505Ernst Presman1 Discrete Time Case . . . . . . . . . . . . . . . . . . . . . . . . 505


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2 Some Examples for One-Dimensional Diffusion . . . . . . . . . 509References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

A Stieltjes Approach to Static Hedges . . . . . . . . . . . . . . . . . . . 519Michael Schmutz and Thomas Zürcher1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5192 Static Hedging with the Lebesgue Measure . . . . . . . . . . . . 5203 Static Hedging with Lebesgue–Stieltjes Integrals . . . . . . . . . 523References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

Optimal Stopping of Seasonal Observations and Projection of a Markov

Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535Isaac M. Sonin1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

2 Optimal Stopping of MC . . . . . . . . . . . . . . . . . . . . . 5363 Recursive Calculation of Characteristics of MC and the StateReduction (SR) Approach . . . . . . . . . . . . . . . . . . . . . 538

4 State Elimination (SE) Algorithm . . . . . . . . . . . . . . . . . 5395 Projection of MC and Seasonal Observations . . . . . . . . . . . 5396 Open Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 542References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543


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Forward Start Foreign Exchange OptionsUnder Heston’s Volatility and the CIR InterestRates

Rehez Ahlip and Marek Rutkowski

Abstract We examine the valuation of forward start foreign exchange options inthe Heston (Rev. Financ. Stud. 6:327–343, 1993) stochastic volatility model for

the exchange rate combined with the CIR (see Cox et al. in Econometrica 53:385–408, 1985) dynamics for the domestic and foreign interest rates. The instantaneousvolatility is correlated with the dynamics of the exchange rate, whereas the domes-tic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate volatility. The main results are derived using the probabilistic ap-proach combined with the Fourier inversion technique developed in Carr and Madan(J. Comput. Finance 2:61–73, 1999). They furnish two alternative semi-analyticalformulae for the price of the forward start foreign exchange European call option.As was argued in Ahlip and Rutkowski (Quant. Finance 13:955–966, 2013), the

setup examined here is the only analytically tractable version of the foreign ex-change market model that combines the Heston stochastic volatility model for theexchange rate with the CIR dynamics for interest rates.

Keywords Option pricing · Heston stochastic volatility model · Forward startoptions · Interest rates

Mathematics Subject Classification (2010) 91G20 · 91G30

1 Introduction

Forward start options are financial derivatives belonging to the class of path-dependent contingent claims, in the sense that their pay-off depends not only on

R. AhlipSchool of Computing and Mathematics, University of Western Sydney, Penrith South, NSW1797, Australia

e-mail: [email protected]

M. Rutkowski (B)School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australiae-mail: [email protected]

Y. Kabanov et al. (eds.), Inspired by Finance, DOI 10.1007/978-3-319-02069-3_1,© Springer International Publishing Switzerland 2014

1

mailto:[email protected]:[email protected]://dx.doi.org/10.1007/978-3-319-02069-3_1http://dx.doi.org/10.1007/978-3-319-02069-3_1mailto:[email protected]:[email protected]


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2 R. Ahlip and M. Rutkowski

the final value of the underlying asset, but also on the asset price at an intermediatetime between the initiation date of a contract and its expiry date. Typically, a for-ward start contract gives the holder the right to enter into a call (or put) option witha strike level that will be a fixed percentage of the underlying asset price at a future

date, termed the strike determination date.Forward start options can be seen as building blocks to so-called cliquets or

ratchets. Cliquet options are equivalent to a series of forward start at-the-moneyoptions with a single premium determined upfront. These are often sold by invest-ment banks to institutional investors who seek to benefit from market oscillationsin the price of the underlying during the lifetime of the contract. Cliquets are usu-ally tailored to provide protection against downside risk, while retaining significantupside potential; see, for instance, Lipton [12] or Windcliff et al. [19]. However,in principle, it is also possible to design cliquet options to profit from bear mar-

kets.In the financial literature, the most widely popular model for stochastic volatility

is Heston’s [9] model. Valuation of forward start equity options under a stochas-tic volatility model was addressed by several authors. Kruse and Nögel [11] de-rived closed-form solutions for the forward start call option in Heston’s stochas-tic volatility model by integrating the call pricing formula with respect to theconditional density of the variance value at strike determination date. A numer-ical evaluation of their expression is rather complicated, however, since in or-der to obtain the desired distribution function, it introduces another level of in-

tegration to already complex integrals in Heston’s formula. Independently, Lucic[13] established an exact pricing formula for forward start options in Heston’sstochastic volatility model by representing the distribution functions in the formof a single integral. Amerio [2] provided a general framework for pricing for-ward start derivatives using Monte Carlo simulations and demonstrated the sen-sitivity with respect to future volatility. All of the above mentioned results havebeen obtained assuming a constant interest rate and for the case of equity call op-tions.

More recently, Van Haastrecht et al. [17] extended the stochastic volatility model

of Schöbel and Zhu [15] to equity/currency derivatives by including stochastic in-terest rates and assuming all driving model factors to be instantaneously correlated.It is notable that their model is based on Gaussian processes and thus it enjoys an-alytical tractability, even in the most general case of a full correlation structure. Bycontrast, when the squared volatility is driven by Heston’s model and the interestrate is driven either by the Vasicek’s [18] process or by the CIR process introducedby Cox et al. [4], a full correlation structure leads to intractability of equity op-tions even under a partial correlation of the driving factors. This feature has beendocumented, among others, by Van Haastrecht and Pelsser [16] and Grzelak and

Oosterlee [6] who examined, in particular, the Heston/Vasicek and Heston/CIR hy-brid models (see also Grzelak and Oosterlee [7] and Grzelak et al. [8], where theSchöbel-Zhu/Hull-White and Heston/Hull-White models for foreign-exchange andequity derivatives are studied).


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Forward Start Foreign Exchange Options 3

The goal of this work is to derive semi-analytical solutions for the price of theforward start foreign exchange option in a model in which the instantaneous volatil-ity of the exchange rate is specified by Heston’s model, whereas the short-terminterest rate processes for the domestic and foreign economies are assumed to fol-

low mutually independent CIR processes. It is worth noting that we extend here thepricing formula for the plain-vanilla foreign exchange option that was establishedin a recent paper by Ahlip and Rutkowski [1].

The paper is organized as follows. In Sect. 2, we set the foreign exchange modelconsidered in this paper (see also Ahlip and Rutkowski [1]). The forward start op-tion pricing problem is introduced in Sect. 3. In Sect. 4, we recall valuation formulaefor zero-coupon bonds in the CIR short-term rate model. In Sect. 5, we introduceauxiliary probability measures and we examine the dynamics of relevant processesunder these measures. Section 6 furnishes some preliminary results that are subse-quently used in Sect. 7 to derive the main results, Theorems 1 and 2, that providetwo alternative pricing formulae for the forward start foreign exchange call option.The paper concludes by deriving the put-call parity relationship for forward startforeign exchange options within the postulated setup.

2 Foreign Exchange Model

Let (Ω,F ,P) be an underlying probability space. We postulate that the dynam-ics of the exchange rate Q

=(Qt )t

∈[0,T

], its instantaneous squared volatility v

=(vt )t ∈[0,T ], the domestic short-term interest rates r = (rt )t ∈[0,T ], and the foreignshort-term interest rate r = (rt )t ∈[0,T ] are governed by the stochastic differentialequations

dQt =

rt −rt Qt d t + Qt √ vt d W Qt ,dvt =

θ − κ vt

dt + σ v√ vt d W vt ,

drt =

ad − bd rt

dt + σ d √ rt d W d t ,d rt = af − bf rt dt + σ f

√

rt d W f t .(1)

We work throughout under the following standing assumptions:

(A.1) W Q = (W Qt )t ∈[0,T ] and W v = (W vt )t ∈[0,T ] are correlated Brownian motionswith a constant correlation coefficient, so that the quadratic covariation of W Q and W v satisfies d [W Q, W v]t = ρ dt for some constant ρ ∈ [−1, 1],

(A.2) W d = (W d t )t ∈[0,T ] and W f = (W f t )t ∈[0,T ] are independent Brownian mo-tions and they are also independent of the Brownian motions W Q and W v

(hence, in particular, the processes v, r and

r are independent),

(A.3) the model’s parameters satisfy the stability conditions (see, e.g., Wong andHeyde [20])

2θ

σ 2v> 1,

2ad σ 2d

> 1,2af σ 2f

> 1.


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It is worth stressing again that we postulate here that the squared volatility pro-cess v, the domestic short-term interest rate, denoted as r , and its foreign coun-terpart, denoted as

r , are independent CIR processes. As argued in Ahlip and

Rutkowski [1], this assumption is indeed crucial and thus it cannot be relaxed.

In our computations, we will usually adopt the domestic perspective, which willbe sometimes represented by the subscript d . Similarly, we will use the subscript f when referring to a foreign denominated variable.

3 Forward Start Foreign Exchange Options

The forward start foreign exchange option is a contract in which the holder receives(at no additional cost) at the strike determination time T

0 < T an option with expiry

date T and some F T 0 -measurable strike K . Typically, we have that K = kQT 0 forsome positive constant k. For any strike K , the terminal payoff at expiry of theforward start foreign exchange call option is given by the following expression

CT (T , K) = (QT − K)+ = QT 1D − K1Dwhere we denote D = {QT > K}.

We denote by F the filtration generated by the Brownian motions W Q, W v , W d ,W f and we write EPt ( · ) and Pt ( · ) to denote the conditional expectation and theconditional probability under P with respect to the σ -fieldF t , respectively.Let the process B represent the domestic savings account, that is, d Bt = rt Bt d t with B0 = 1. The underlying probability measure P is interpreted as the domesticmartingale measure. Hence the price of the option at time t equals, for all t ∈ [0, T ],

Ct (T , K) = Bt EPt B−1T CT (T , K)= Bt EPt B−1T (QT − K)+or, equivalently,

Ct (T , K) = Bt EPt (B−1T QT 1D) − Bt EPt (B−1T K1D).Formula above is valid for any strike K . However, in what follows it will be al-ways assumed that K = kQT 0 . Since the process Q is governed under P by (1), therandom variable Qt satisfies, for all t ∈ [0, T ],

Qt = Q0 exp t

0

√ vu dW

Qu +

t 0

ru −ru − (1/2)vudu. (2)

4 Bond Pricing and Forward Exchange Rate

We make the standard assumption that the zero-coupon bond prices discountedby the domestic spot rate are martingales under P, that is, the bond price equals


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Bd ( t , T ) = Bt EPt (B−1T ) for all t ∈ [0, T ]. An analogous formula holds for the priceprocess Bf ( t , T ) of the foreign discount bond under the foreign spot martingalemeasure (see, e.g., Chap. 14 in Musiela and Rutkowski [14]).

We recall the well-known pricing result for zero-coupon bonds (see, e.g., Cox et

al. [4] or Chap. 10 in Musiela and Rutkowski [14]). It is worth stressing that we usehere, in particular, the postulated independence of Brownian motions W Q and W f

driving the foreign interest rater and the exchange rate Q. Under Assumption (A.2),the dynamics of the foreign bond price Bf ( t , T ) under the domestic spot martingalemeasure P can thus be obtained from formula (14.3) in Musiela and Rutkowski [14].

Proposition 1 The prices at date t of a domestic and foreign discount bonds ma-

turing at time T ≥ t in the CIR model are given by

Bd ( t , T ) = expmd ( t , T ) − nd (t,T)rt ,Bf ( t , T ) = exp

mf ( t , T ) − nf ( t , T )rt ,

where for i ∈ {d, f }

mi ( t , T ) = 2ai

σ 2ilog

γ i e

12 bi (T −t)

γ i cosh(γ i (T − t)) + 12 bi sinh(γ i (T − t))

,

ni ( t , T ) = sinh(γ i (T − t))γ i cosh(γ i (T − t )) + 12 bi sinh(γ i (T − t ))

,

and

γ i = 1

2

b2i + 2σ 2i .

The dynamics of the domestic and foreign bond prices under the domestic spot mar-

tingale measure P are given by

dBd ( t , T ) = Bd ( t , T )

rt d t − σ d nd ( t , T )√

rt d W d t

,

dBf ( t , T ) = Bf ( t , T )rt d t − σ f nf ( t , T ) rt d W f t .

The following result is also well known (see, for instance, Sect. 14.1.1 in Musielaand Rutkowski [14]).

Lemma 1 The forward exchange rate F ( t , T ) at time t for settlement date T

equals, for all t ∈ [0, T ],

F ( t , T ) = Bf ( t , T )Bd ( t , T )

Qt . (3)


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5 Auxiliary Probability Measures

Since manifestly QT = F ( T , T ), the option’s payoff at its expiration can also beexpressed as follows

CT (T , K) = F ( T , T )1{F(T,T )>K} − K1{F(T,T)>K}.Hence the option’s price admits the following representation, for all t ∈ [0, T ],

Ct (T , K) = EPt exp− T t

ru du

F ( T , T )1{F(T,T)>K}

−EPt

K exp

− T

t

ru du

1{F(T,T )>

K}

.

When pursuing the probabilistic approach to the valuation of foreign exchangeoptions, we are going to employ several auxiliary probability measures equivalentto the domestic spot martingale measure P. Let us first recall the classical conceptof the domestic forward martingale measure PT .

Definition 1 The domestic forward martingale measure PT is the probabilitymeasure equivalent to P on (Ω,F T ) with the Radon-Nikodým derivative processη = (ηt )t ∈[0,T ] given by

ηt = d PT d P F t = exp− t

0σ d nd (u,T)√ ru dW d u − 12 t 0 σ 2d n2d (u, T )ru du.

Under our assumptions, the process η can be checked to be a (true) martingale;one can use to this end the arguments given in the appendix in Kruse and Nögel [11].Hence it follows from the Girsanov theorem that the process W T = (W T t )t ∈[0,T ],which is given by the equality

W T t = W d t +

t

0σ d nd (u,T)

√ ru du,

is the standard Brownian motion under the domestic forward martingale mea-sure PT . It is also clear that the dynamics of r under PT are

drt =

ad −bd (t)rt dt + σ d √ rt d W T t (4)where the functionbd : [0, T ] →R equalsbd (t ) = bd + σ 2d nd ( t , T ). The followingresult is borrowed from Ahlip and Rutkowski [1].

Lemma 2 Under Assumptions (A.1) – (A.3), the dynamics of the forward exchange

rate F (t , T ) under the domestic forward martingale measure PT are given by thestochastic differential equation

d F ( t , T ) = F ( t , T )√

vt d W Qt + σ d nd ( t , T )

√ rt d W

T t − σ f nf ( t , T )

rt d W f t


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or , equivalently,

F ( T , T ) = F ( t , T ) exp

T

t σ F (u,T) · d

W T u −

1

2 T

t

σ F (u,T)2 du

where the dot · represents the inner product in R3, by (σ F (t,T))t ∈[0,T ] we denotethe R3-valued process given by

σ F ( t , T ) = √ vt , σ d nd ( t , T )√ rt , −σ f nf ( t , T ) rt and W T = (W T t )t ∈[0,T ] stands for the three-dimensional standard Brownian motionunder PT that is given by

W T = [W Q, W T , W f ]∗.

Using the classical change of a numéraire technique, one can check that underthe probability measure PT the time t price of the forward start foreign exchangecall option equals, for all t ∈ [T 0, T ],

Ct (T , K) = Bd ( t , T )EPT t F ( T , T )1{F(T,T)>K}− KBd ( t , T )EPT t 1{F(T,T)>K}.After the strike determination date the forward start foreign exchange call optionbecomes a plain-vanilla foreign exchange call option and thus it can be dealt withas in Ahlip and Rutkowski [1]. To compute the first term in the right-hand side in

the formula above, we introduce an auxiliary probability measure PT .Definition 2 The probability measurePT , equivalent to PT on (Ω ,F T ), is definedby the Radon-Nikodým derivative processη = (ηt )t ∈[0,T ] where

ηt = d PT d PT

F t

= exp t

0σ F (u,T) · d W T u − 12

t 0

σ F (u,T)2 du.As a first step towards general valuation results presented in Sect. 7, we will now

derive some preliminary results related to the pricing of the forward start foreignexchange call option prior to the strike determination date. In what follows, wepresent two alternative pricing methods. We will argue that each of them has someadvantages, but also certain drawbacks.

5.1 Bond Price Numéraire

We define the process ξ = (ξ t )t ∈[0,T ] by setting ξ t = ξ T 0 for all t ∈ [T 0, T ] and

ξ t = Qt Bf (t,T 0)

Q0Bf (0, T 0)Bt , ∀ t ∈ [0, T 0]. (5)


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In view of the postulated independence of processesr and r , the foreign bond priceBf (t,T 0) satisfies under the domestic martingale measure P are (see Proposition 1)

dBf (t,T 0)

=rt Bf (t,T 0) dt − Bf (t,T 0)σ f nf ( t , T ) rt d W f t .

By combining this formula with the dynamics of the exchange rate Q, we obtain thefollowing result.

Lemma 3 The process (ξ t )t ∈[0,T ] is a positive martingale under P stopped at T 0.Specifically,

ξ t = exp t ∧T 0

0

√ vu dW

Qu −

1

2

t ∧T 00

vu du

× exp− t ∧T 0

0σ f nf (u,T) ru dW f u − 12 t ∧T 00 σ 2f n2f (u,T)ru du.

Due to Lemma 3, we are in the position to define the probability measure PN ,equivalent to P on (Ω,F T ), by postulating that the Radon-Nikodým density processof PN with respect to P equals ξ .

Definition 3 The probability measure PN is equivalent to P on (Ω,F T ) with theRadon-Nikodým density process with respect to P given by the formula

ξ t = d PN

d P

F t

= exp t ∧T 0

0

√ vu dW

Qu −

1

2

t ∧T 00

vu du

× exp

− t ∧T 0

0σ f nf (u,T)

ru dW f u − 12 t ∧T 0

0σ 2f n

2f (u,T)ru du.

Note that the process W Q = (W Qt )t ∈[0,T ] that is given byW Qt = W Qt − t ∧T 00 √ vu du

is the standard Brownian motion under the auxiliary probability measure PN . Thefollowing useful result is an immediate consequence of the Girsanov theorem andAssumptions (A.1)–(A.3).

Lemma 4 The processes W v, W f and W d that are given by the equalities, for allt ∈ [0, T ],

W vt = W vt − ρ t ∧T 00

√ vu du,

W f t = W f t + t ∧T 00

σ f nf (u,T 0) ru du,


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W d t = W d t ,are independent standard Brownian motions under PN . The processes v, r and

r , with dynamics under P given by (1), are governed under PN by the followingstochastic differential equations, for all t ∈ [0, T 0],dvt =

θ −κ vt dt + σ v√ vt d W vt ,

drt =

ad − bd rt

dt + σ d √

rt d W d t , (6)d rt = af −bf (t)rt dt + σ f rt d W f t ,

where we denote

κ = κ − ρσ v and we set

bf (t) = bf + σ 2f nf (t,T 0) for all t ∈

[0, T 0

].

Our next goal is to show that by changing the probability from P to PN wecan essentially simplify the pricing formula for the forward start foreign exchangeoption. Let the auxiliary process (Qt )t ∈[T 0,T ] be given by

Qt = Qt QT 0

= exp t

T 0

√ vu dW

Qu +

t T 0

ru −ru − (1/2)vudu.

Equivalently, the process (

Qt )t ∈[T 0,T ] is the unique solution to the stochastic dif-

ferential equationd Qt = rt −rt Qt d t + Qt √ vt d W Qt (7)

with the initial condition QT 0 = 1. The following lemma underpins the computationof the price of the forward start foreign exchange call option in Theorem 1.

Lemma 5 The price of the forward start foreign exchange call option equals, for all t ∈ [0, T 0],

Ct (T , K) = Qt Bf (t,T 0)EPN t BT 0 EPT 0B−1T (QT − k)+ .Consequently,

Ct (T , K) = Qt Bf (t,T 0)EPN t CT 0 (T,k) (8)where we denote

CT 0 (T,k) = BT 0 EPT 0

B−1T (

QT − k)+

. (9)

Proof Recall that K = kQT 0 . Using the Bayes formula and recalling that ξ t = ξ T 0for t ∈ [T 0, T ], we obtain, for all t ∈ [0, T 0],Ct (T , K) = Bt EPt B−1T (QT − K)+


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= ξ t Bt EPN t

ξ −1T B−1T (QT − K)+

= Qt Bf (t,T 0)Q0Bf (0, T 0)

EPN t

ξ −1T 0 B

−1T (QT −

K)+

= Qt Bf (t,T 0)EPN t Q−1T 0 BT 0 B−1T (QT − kQT 0 )+= Qt Bf (t,T 0)EPN t

BT 0 B

−1T (

QT − k)+= Qt Bf (t,T 0)EPN t

BT 0 E

PN T 0

B−1T (QT − k)+.

In view of the definition of the probability measure PN and Lemma 4, we have that

BT 0 EPN T 0 B−1T (QT − k)+= BT 0 EPT 0B−1T (QT − k)+= CT 0 (T,k)

and thus formula (8) is established.

5.2 Savings Account Numéraire

Let the process Bf represent the foreign savings account, so that dBf t =rt B

f t dt

with Bf 0 = 1. We define the processξ = (ξ t )t ∈[0,T ] by settingξ t =ξ T 0 for t ∈ [T 0, T ]and

ξ t = Qt Bf t Q0Bt

, ∀ t ∈ [0, T 0]. (10)

By combining formula (10) with the dynamics of the exchange rate Q under P, weobtain, for all t ∈ [0, T 0],

d

ξ t =

ξ t

√ vt d W

Qt

and thus we arrive at the following explicit representation for the processξ ξ t = exp t ∧T 0

0

√ vu dW

Qu −

1

2

t ∧T 00

vu du

.

The processξ is a positive martingale under P stopped at time T 0, and thus it can beused to define an equivalent probability measure, denoted asPN .Definition 4 The probability measure PN is equivalent to P on (Ω,F T ) with theRadon-Nikodým density process with respect to P given by the formula

ξ t = d PN d P

F t

= exp T 0

0

√ vu dW

Qu −

1

2

T 00

vu du

.


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It is clear that the process W Q = (W Qt )t ∈[0,T ] given by the equality

W

Qt = W Qt −

t ∧T 00

√ vu du

is the standard Brownian motion under PN . In view of Assumptions (A.1)–(A.3),the following counterpart of Lemma 4 is rather obvious.

Lemma 6 The processes W v, W f and W d that are given by the equalities, for allt ∈ [0, T ],

W vt = W vt − ρ

t ∧T 00

√ vu du,

W f t = W f t ,W d t = W d t ,are independent standard Brownian motions under PN . The processes v, r and r ,with dynamics given by (1), are governed under PN by the following stochasticdifferential equations, for all t ∈ [0, T 0],

dvt =

θ −

κ vt

dt + σ v√ vt d W vt ,drt

= ad − bd rt dt + σ d √

rt d W d t , (11)

d rt = af − bf rt dt + σ f rt d W f t ,whereκ = κ − ρσ v .

The following result will be used in the proof of Theorem 2.

Lemma 7 The price of the forward start foreign exchange call option at time t

equals, for all t ∈ [0, T 0],

Ct (T , K) = Qt Bf t EPN t (Bf T 0 )−1BT 0 EPT 0B−1T (QT − K)+.Consequently, we have that

Ct (T , K) = Qt Bf t EPN t (Bf T 0 )−1CT 0 (T,k) (12)where we denote

CT 0 (T,k) = BT 0 EPT 0

B−1T (

QT − k)+

.

Proof Recall that K = kQT 0 . Using the abstract Bayes formula, we obtain, for allt ∈ [0, T 0],

Ct (T , K) = Bt EPt B−1T (QT − K)+


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=ξ t Bt EPN t ξ −1T B−1T (QT − K)+= Q−10 Qt B

f t E

PN t

ξ −1T 0 B

−1T (QT −

K)+

= Qt Bf t EPN t (QT 0 Bf T 0 )−1BT 0 B−1T (QT − kQT 0 )+so that

Ct (T , K) = Qt Bf t EPN t (Bf T 0 )−1BT 0 EPN T 0 B−1T (QT − k)+.The definition of the probability measurePN and Lemma 6 yield

BT 0 EPN T 0 B

−1T (QT − k)+= BT 0 EPT 0B

−1T (QT − k)+= CT 0 (T,k).

This completes the proof of the lemma.

6 Preliminary Results

We will need the following auxiliary lemma borrowed from Ahlip and Rutkowski[1] (see also Duffie et al. [5]). Note that the dynamics of the exchange rate processQ are not relevant for this result. Let us set τ = T − t . For any complex numbersµ, λ, µ,λ, µ andλ, we denote by F (τ , vt , rt ,rt ) the conditional expectationEPt

exp

−λvT − µ

T t

vu du −λrT −µ T t

ru du −λrT −µ T t

ru du

.

Lemma 8 Let the dynamics of processes v, r and r under the probability mea-sure P be given by stochastic differential equations (1) with independent standard Brownian motions W v, W d and W f . Then

F (τ , vt , rt ,rt ) = exp −G1(τ,λ,µ)vt − G2(τ ,λ,µ)rt − G3(τ ,λ,µ)rt − θ H 1(τ,λ,µ) − ad H 2(τ ,λ,µ) − af H 3(τ ,λ,µ)

where

G1(τ,λ,µ) = λ[(γ + κ) + eγ τ (γ − κ)] − 2µ(1 − eγ τ )σ 2v λ

eγ τ − 1+ γ − κ + eγ τ (γ + κ) ,

H 1

(τ,λ,µ) = − 2σ 2v ln 2γ e(γ +κ)

τ

2

σ 2v λ eγ τ − 1+ γ − κ + eγ τ (γ + κ) ,G2(τ ,λ,µ) =λ[(γ + bd ) + eγ τ (γ − bd )] − 2µ(1 − eγ τ )

σ 2d λ eγ τ − 1+γ − bd + eγ τ (γ + bd ) ,


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H 2(τ ,λ,µ) = − 2σ 2d

ln

2γ e (γ +bd )τ 2σ 2d

λ

eγ τ − 1+γ − bd + eγ τ (

γ + bd )

,G3(τ ,λ,µ) =λ[(γ + bf ) + eγ τ (γ − bf )] − 2µ(1 − eγ τ )

σ 2f λ eγ τ − 1+γ − bf + eγ τ γ + bf ,

H 3(τ ,λ,µ) = − 2σ 2f

ln

2γ e (γ +bf )τ 2σ 2f λ eγ τ − 1+γ − bf + eγ τ γ + bf

,where we denote γ =

κ2 + 2σ 2v µ,

γ =

b2d + 2σ 2d

µ and

γ =

b2f + 2σ 2f

µ.

Note that Lemma 8 yields, in particular, alternative (but equivalent to formulaeof Proposition 1) representations for the bond prices Bd ( t , T ) and Bf ( t , T ), specif-ically,

Bd ( t , T ) = exp−ad H 2(τ , 0, 1) − G2(τ , 0, 1)rt , (13)

Bf ( t , T ) = exp−af H 3(τ , 0, 1) − G3(τ , 0, 1)rt . (14)

Recall that the dynamics of the auxiliary process (

Qt )t ∈[T 0,T ] under P are given by

equation (7). Hence the next result is a straightforward consequence of Theorem 4.1

in Ahlip and Rutkowski [1]. For the sake of conciseness, we write here τ 0 = T − T 0.Recall also that the bond prices Bd (T 0, T ) and Bf (T 0, T ) are given in Proposition 1.

Proposition 2 Assume that the foreign exchange model is given by stochastic dif-

ferential equations (1) under Assumptions (A.1) – (A.3). Then the conditional expec-tation CT 0 (T,k) defined by (9) is given by the following expression

CT 0 (T,k) = Bf (T 0, T ) P 1

T 0, vT 0 , rT 0 ,

rT 0 , k

− kBd (T 0, T ) P 2

T 0, vT 0 , rT 0 ,

rT 0 , k

.

The functions P 1 and P 2 are given by

P j

T 0, vT 0 , rT 0 ,rT 0 , k= 12 + 1π ∞

0Re

f j (φ)

exp(−iφ ln k)iφ

dφ

where the F T 0 -conditional characteristic functions

f j (φ) = f j (φ,T 0, vT 0 , rT 0 ,

rT 0 ), j = 1, 2,

of the random variable ln QT under the probability measures PT (see Definition 2)and PT (see Definition 1), respectively, satisfyln(f 1(φ)) = iφ

mf (T 0, T ) − md (T 0, T )

− (1 + iφ) ρσ v

vT 0 + θ τ 0


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− iφ T

T 0

ad nd (u ,T)du + nf (T 0, T )rT 0+

(1+

iφ ) T

T 0

af nf (u ,T)du

− G1(τ 0, s1, s2)vT 0 − G2(τ 0, s3, s4)rT 0 − G3(τ 0, s5, s6)rT 0− θ H 1(τ 0, s1, s2) − ad H 2(τ 0, s3, s4) − af H 3(τ 0, s5, s6) (15)

and

ln(f 2(φ)) = iφ

mf (T 0, T ) − md (T 0, T )− iφρ

σ v

vT 0 + θ τ 0

+ (1 − iφ)× T

T 0

ad nd (u ,T)du + nd (T 0, T ) rT 0 + iφ T T 0

af nf (u ,T)du

− G1(τ 0, q1, q2)vT 0 − G2(τ 0, q3, q4)rT 0 − G3(τ 0, q5, q6)rT 0− θ H 1(τ 0, q1, q2) − ad H 2(τ 0, q3, q4) − af H 3(τ 0, q5, q6) (16)

where the functions G1, G2, G3, H 1, H 2, H 3 are defined in Lemma 8. The constantss1, s2, s3, s4, s5, s6 are given by

s1 = −(1

+iφ)ρ

σ v,

s2 = −(1 + iφ)2(1 − ρ2)

2 − (1 + iφ)ρκ

σ v+ 1 + iφ

2 , (17)

s3 = 0, s4 = −iφ, s5 = 0, s6 = 1 + iφ,

and the constants q1, q2, q3, q4, q5, q6 equal

q1 = −iφρ

σ v ,

q2 = − iφρκσ v

− (iφ)2(1 − ρ2)

2 + iφ

2 , (18)

q3 = 0, q4 = 1 − iφ, q5 = 0, q6 = iφ .

7 Valuation of Forward Start Foreign Exchange Options

In this section, we establish the main results of this work, Theorems 1 and 2. Beforestating these results, we need to introduce some notation. For the sake of brevity,in what follows we write τ = T 0 − t and τ 0 = T − T 0. Recall also that we denote


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κ = κ − ρσ v . Assume that the functions H 1, G1, H 2, G2 solve the following ODEs∂

G1(τ,λ)

∂τ = −1

2σ 2v

G21(τ,λ) −

κ

G1(τ,λ),

∂ H 1(τ,λ)∂τ

= G1(τ,λ),∂G2(τ,λ)

∂τ = −1

2σ 2d G22(τ,λ) − bd G2(τ,λ),

∂ H 2(τ,λ)∂τ

= G2(τ,λ),with initial conditions: G1(0, λ) = λ, G2(0,λ) =λ and H 1(0, λ) = H 2(0,λ) = 0.From the proof of Lemma 8, which is given in Ahlip and Rutkowski [1], it iseasy to deduce that the functions H 1, G1, H 2, G2, H 3 are given by Lemma 8 withµ =µ =µ = 0 and κ replaced byκ = κ − ρσ v . More explicitly,

G1(τ,λ) = 2λκσ 2v λ

eκτ − 1+ 2κeκτ ,

G2(τ,

λ) = 2

λbd σ 2d

λ

ebd τ − 1

+ 2bd ebd τ

,

H 1(τ,λ) = − 2σ 2v

ln 2κe2κτ σ 2v λ

eκτ − 1+ 2κ eκτ ,

H 2(τ,λ) = − 2σ 2d

ln

2bd ebd τ

σ 2d λebd τ − 1+ 2bd ebd τ

.

(19)

7.1 Options Pricing Formula in the Bond Numéraire

We are in the position to prove the first main result of this work. According to themethod developed in Sect. 5.1, the price of this option prior to the strike determina-tion date T 0 can be expressed in terms of the foreign zero-coupon bond Bf (t,T 0)and the exchange rate Qt , as well as a certain conditional expectation (see for-mula (8)) that we will now evaluate in Heston’s stochastic volatility model for theexchange rate combined with independen

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