Journal of Applied Probability

Pricing and hedging of quantile options in a flexible jump diffusion model

Ning Cai

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This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.

Article information

J. Appl. Probab., Volume 48, Number 3 (2011), 637-656.

First available in Project Euclid: 23 September 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J75: Jump processes
Secondary: 44A10: Laplace transform

Jump diffusion option pricing hyperexponential quantile option Euler inversion


Cai, Ning. Pricing and hedging of quantile options in a flexible jump diffusion model. J. Appl. Probab. 48 (2011), no. 3, 637--656. doi:10.1239/jap/1316796904.

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  • Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–87.
  • Akahori, J. (1995). Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383–388.
  • Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.
  • Bertoin, J., Chaumont, L. and Yor, M. (1997). Two chain transformations and their applications to quantiles. J. Appl. Prob. 34, 882–897.
  • Borodin, A. N. and Salminen P. (2002). Handbook of Brownian Motion–-Facts and Formulae, 2nd edn. Birkhäuser, Basel.
  • Broadie, M. and Detemple, J. B. (2004). Option pricing: valuation models and applications. Manag. Sci. 50, 1145–1177.
  • Cai, N. (2009). On first passage times of a hyper-exponential jump diffusion process. Operat. Res. Lett. 37, 127–134.
  • Cai, N. and Kou, S. G. (2011). Option pricing under a mixed-exponential jump diffusion model. To appear in Mamag. Sci.
  • Cai, N., Chen, N. and Wan, X. (2010). Occupation times of jump-diffusion processes with double exponential jumps and the pricing of options. Math. Operat. Res. 35, 412–437.
  • Carr, P. and Madan, D. B. (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73.
  • Choudhury, G. L., Lucantoni, D. M. and Whitt, W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Prob. 4, 719–740.
  • Craddock, M., Heath, D. and Platen, E. (2000). Numerical inversion of Laplace transforms: a survey of techniques with applications to derivative pricing. J. Comput. Finance 4, 57–81.
  • Dassios, A. (1995). The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options. Ann. Appl. Prob. 5, 389–398.
  • Dassios, A. (1996). Sample quantiles of stochastic processes with stationary and independent increments. Ann. Appl. Prob. 6, 1041–1043.
  • Davydov, D. and Linetsky, V. (2001). Structuring, pricing and hedging double barrier step options. J. Comput. Finance 5, 55–87.
  • Detemple, J. (2001). American options: symmetry properties. In Option Pricing, Interest Rates and Risk Management, eds E. Jouini, J. Cvitanic and M. Musiela, Cambridge University Press, pp. 67–104.
  • Embrechts, P., Rogers, L. C. G. and Yor, M. (1995). A proof of Dassios' representation of the $\alpha$-quantile of Brownian motion with drift. Ann. Appl. Prob. 5, 757–767.
  • Fusai, G. (2000). Corridor options and arc-sine law. Ann. Appl. Prob. 10, 634–663.
  • Fusai, G. and Tagliani, A. (2001). Pricing of occupation time derivatives: continuous and discrete monitoring. J. Comput. Finance 5, 1–37.
  • Glasserman, P. (2000). Monte Carlo Methods in Financial Engineering. Springer, New York.
  • Hugonnier, J. (1999). The Feynman-Kac formula and pricing occupation time derivatives. Internat. J. Theoret. Appl. Finance 2, 153–178.
  • Jeannin, M. and Pistorious, M. (2010). A transform approach to compute prices and Greeks of barrier options driven by a class of Lévy processes. Quant. Finance 10, 629–644.
  • Kou, S. G. (2002). A jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101.
  • Kwok, Y. K. (1998). Mathematical Models of Financial Derivatives. Springer, Singapore.
  • Kwok, Y. K. and Lau, K. W. (2001). Pricing algorithms for options with exotic path-dependence. J. Derivatives 9, 23–38.
  • Leung, K. S. and Kwok, Y. K. (2007). Distribution of occupation times for constant elasticity of variance diffusion and the pricing of $\alpha$-quantile options. Quant. Finance 7, 87–94.
  • Linetsky, V. (1999). Step options. Math. Finance 9, 55–96.
  • Miura, R. (1992). A note on look-back options based on order statistics. Hitotsubashi J. Commerce Manag. 27, 15–28.
  • Pechtl, A. (1999). Some applications of occupation times of Brownian motion with drift in mathematical finance. J. Appl. Math. Decision Sci. 3, 63–73.
  • Petrella, G. (2004). An extension of the Euler Laplace transform inversion algorithm with applications in option pricing. Operat. Res. Lett. 32, 380–389.
  • Schiff, J. L. (1999). The Laplace Transform. Springer, New York.
  • Schoroder, M. (1999). Changes of numeraire for pricing futures, forwards, and options. Rev. Financial Studies 12, 1143–1163.
  • Shreve, S. E. (2004). Stochastic Calculus for Finance. II. Springer, New York.
  • Yor, M. (1995). The distribution of Brownian quantiles. J. Appl. Prob. 32, 405–416.