Pricing Options in Jump Diffusion Models Using Mellin Transforms

Robert Frontczak

Landesbank Baden-Wuerttemberg (LBBW), Stuttgart, Germany.

DOI: 10.4236/jmf.2013.33037                

Abstract

This paper is concerned with the valuation of options in jump diffusion models. The partial integro-differential equation (PIDE) inherent in the pricing problem is solved by using the Mellin integral transform. The solution is a single integral expression independent of the distribution of the jump size. We also derive analytical expressions for the Greeks. The results are implemented and compared to other approaches.

Keywords

Jump Diffusion; Mellin Transform; European Option

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 125-144. doi:10.1016/0304-405X(76)90022-2
[2]	C. Ahn and H. Thompson, “Jump-Diffusion Processes and the Term Structure of Interest Rates,” Journal of Finance, Vol. 43, No. 1, 1988, pp. 155-174. doi:10.1111/j.1540-6261.1988.tb02595.x
[3]	D. Bates, “Jumps and Stochastic Volatility: Exchange Rate Process Implicit in Deutsche Mark Options,” Review of Financial Studies, Vol. 9, No. 1, 1996, pp. 69-107. doi:10.1093/rfs/9.1.69
[4]	D. Madan, P. Carr and E. Chang, “The Variance Gamma Process and Option Pricing,” European Finance Review, Vol. 2, 1998, pp. 79-105.
[5]	L. Andersen and J. Andreasen, “Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing,” Review of Derivatives Research, Vol. 4, No. 3, 2000, pp. 231-262. doi:10.1023/A:1011354913068
[6]	D. Duffie, J. Pan and K. Singleton, “Transform Analysis and Option Pricing for Affine Jump-Diffusions,” Econometrica, Vol. 68, No. 6, 2000, pp. 1343-1376. doi:10.1111/1468-0262.00164
[7]	S. Kou, “A Jump-Diffusion Model for Option Pricing,” Management Science, Vol. 48, No. 8, 2002, pp. 1086-1101. doi:10.1287/mnsc.48.8.1086.166
[8]	S. Kou and H. Wang, “Option Pricing under a Double Exponential Jump Diffusion Model,” Management Science, Vo. 50, No. 9, 2004, pp. 1178-1192. doi:10.1287/mnsc.1030.0163
[9]	A. Sepp, “Analytical Pricing of Double-Barrier Options under a Double-Exponential Jump-Diffusion Process: Applications of Laplace Transform,” International Journal of Theoretical and Applied Finance, Vol. 7, No. 2, 2004, pp. 151-175. doi:10.1142/S0219024904002402
[10]	K. Amin, “Jump Diffusion Option Valuation in Discrete Time,” Journal of Finance, Vol. 48, No. 5, 1993, pp. 1833-1863. doi:10.1111/j.1540-6261.1993.tb05130.x
[11]	J. Hilliard and A. Schwartz, “Pricing European and American Derivatives under a Jump-Diffusion Process: A Bivariate Tree Approach,” Journal of Financial and Quantitative Analysis, Vol. 40, No. 3, 2005, pp. 671-691. doi:10.1017/S0022109000001915
[12]	Y. D’Halluin, P. Forsyth and K. Vetzal, “Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes,” IMA Journal of Numerical Analysis, Vol. 25, No. 1, 2005, pp. 87-112. doi:10.1093/imanum/drh011
[13]	P. Carr and A. Mayo, “On the Numerical Evaluation of Option Prices in Jump Diffusion Processes,” European Journal of Finance, Vol. 13, No. 4, 2007, pp. 353-372.
[14]	A. Mayo, “Methods for the Rapid Solution of the Pricing PIDEs in Exponential and Merton Models,” Journal of Computational and Applied Mathematics, Vol. 222, No. 1, 2008, pp. 128-143. doi:10.1016/j.cam.2007.10.017
[15]	S. Clift and P. Forsyth, “Numerican Solution of Two Asset Jump Diffusion Models for Option Valuation,” Applied Numerical Mathematics, Vol. 58, No. 6, 2008, pp. 743-782. doi:10.1016/j.apnum.2007.02.005
[16]	R. Panini and R. P. Srivastav, “Option Pricing with Mellin Transforms,” Mathematical and Computer Modelling, Vol. 40, No. 1-2, 2004, pp. 43-56. doi:10.1016/j.mcm.2004.07.008
[17]	R. Frontczak, “Valuing Options in Heston’s Stochastic Volatility Model: Another Analytical Approach,” Journal of Applied Mathematics, Vol. 2011, 2011, Article ID: 198469. doi:10.1155/2011/198469
[18]	R. Cont and P. Tankov, “Financial Modelling with Jump Processes,” Chapman & Hall/CRC, Boca Raton, 2004.
[19]	B. Eraker, M. Johannes and N. Polson, “The Impact of Jumps in Volatility and Returns,” Journal of Finance, Vol. 58, No. 3, 2003, pp. 1269-1300.
[20]	I. Gradshteyn and I. Ryzhik, “Table of Integrals, Series, and Products,” 7th Edition, Elsevier Academic Press, Amsterdam, 2007.