Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T12:36:36.143Z Has data issue: false hasContentIssue false
  • English
  • Français

Pricing and Hedging of Quantile Options in a Flexible Jump Diffusion Model

Part of: Markov processes Integral transforms, operational calculus

Published online by Cambridge University Press:  14 July 2016

Ning Cai
Ning Cai*
Affiliation:
The Hong Kong University of Science and Technology
*
∗ Postal address: Department of Industrial Engineering and Logistics Management, Room 5521, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Jump diffusionoption pricinghyperexponentialquantile optionEuler inversion

MSC classification

Secondary: 60J75: Jump processes 44A10: Laplace transform
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 637 - 656
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–87.Google Scholar
[2] Akahori, J. (1995). Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383–388.Google Scholar
[3] 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.Google Scholar
[4] Bertoin, J., Chaumont, L. and Yor, M. (1997). Two chain transformations and their applications to quantiles. J. Appl. Prob. 34, 882–897.Google Scholar
[5] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
[6] Broadie, M. and Detemple, J. B. (2004). Option pricing: valuation models and applications. Manag. Sci. 50, 1145–1177.Google Scholar
[7] Cai, N. (2009). On first passage times of a hyper-exponential Jump diffusion process. Operat. Res. Lett. 37, 127–134.Google Scholar
[8] Cai, N. and Kou, S. G. (2011). Option pricing under a mixed-exponential Jump diffusion model. To appear in Mamag. Sci. Google Scholar
[9] 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.Google Scholar
[10] Carr, P. and Madan, D. B. (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73.Google Scholar
[11] 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.Google Scholar
[12] 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.Google Scholar
[13] 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.Google Scholar
[14] Dassios, A. (1996). Sample quantiles of stochastic processes with stationary and independent increments. Ann. Appl. Prob. 6, 1041–1043.Google Scholar
[15] Davydov, D. and Linetsky, V. (2001). Structuring, pricing and hedging double barrier step options. J. Comput. Finance 5, 55–87.Google Scholar
[16] Detemple, J. (2001). American options: symmetry properties. In Option Pricing, Interest Rates and Risk Management, eds Jouini, E., Cvitanic, J. and Musiela, M., Cambridge University Press, pp. 67–104.Google Scholar
[17] Embrechts, P., Rogers, L. C. G. and Yor, M. (1995). A proof of Dassios' representation of the α-quantile of Brownian motion with drift. Ann. Appl. Prob. 5, 757–767.Google Scholar
[18] Fusai, G. (2000). Corridor options and arc-sine law. Ann. Appl. Prob. 10, 634–663.Google Scholar
[19] Fusai, G. and Tagliani, A. (2001). Pricing of occupation time derivatives: continuous and discrete monitoring. J. Comput. Finance 5, 1–37.Google Scholar
[20] Glasserman, P. (2000). Monte Carlo Methods in Financial Engineering. Springer, New York.Google Scholar
[21] Hugonnier, J. (1999). The Feynman-Kac formula and pricing occupation time derivatives. Internat. J. Theoret. Appl. Finance 2, 153–178.Google Scholar
[22] 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.Google Scholar
[23] Kou, S. G. (2002). A Jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101.Google Scholar
[24] Kwok, Y. K. (1998). Mathematical Models of Financial Derivatives. Springer, Singapore.Google Scholar
[25] Kwok, Y. K. and Lau, K. W. (2001). Pricing algorithms for options with exotic path-dependence. J. Derivatives 9, 23–38.Google Scholar
[26] Leung, K. S. and Kwok, Y. K. (2007). Distribution of occupation times for constant elasticity of variance diffusion and the pricing of α-quantile options. Quant. Finance 7, 87–94.Google Scholar
[27] Linetsky, V. (1999). Step options. Math. Finance 9, 55–96.Google Scholar
[28] Miura, R. (1992). A note on look-back options based on order statistics. Hitotsubashi J. Commerce Manag. 27, 15–28.Google Scholar
[29] Pechtl, A. (1999). Some applications of occupation times of Brownian motion with drift in mathematical finance. J. Appl. Math. Decision Sci. 3, 63–73.Google Scholar
[30] Petrella, G. (2004). An extension of the Euler Laplace transform inversion algorithm with applications in option pricing. Operat. Res. Lett. 32, 380–389.Google Scholar
[31] Schiff, J. L. (1999). The Laplace Transform. Springer, New York.Google Scholar
[32] Schoroder, M. (1999). Changes of numeraire for pricing futures, forwards, and options. Rev. Financial Studies 12, 1143–1163.Google Scholar
[33] Shreve, S. E. (2004). Stochastic Calculus for Finance. II. Springer, New York.Google Scholar
[34] Yor, M. (1995). The distribution of Brownian quantiles. J. Appl. Prob. 32, 405–416.Google Scholar