Journal of Applied Mathematics

Unconditional Positive Stable Numerical Solution of Partial Integrodifferential Option Pricing Problems

M. Fakharany, R. Company, and L. Jódar

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Abstract

This paper is concerned with the numerical solution of partial integrodifferential equation for option pricing models under a tempered stable process known as CGMY model. A double discretization finite difference scheme is used for the treatment of the unbounded nonlocal integral term. We also introduce in the scheme the Patankar-trick to guarantee unconditional nonnegative numerical solutions. Integration formula of open type is used in order to improve the accuracy of the approximation of the integral part. Stability and consistency are also studied. Illustrative examples are included.

Article information

Source
J. Appl. Math., Volume 2015 (2015), Article ID 960728, 10 pages.

Dates
First available in Project Euclid: 15 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1429105033

Digital Object Identifier
doi:10.1155/2015/960728

Mathematical Reviews number (MathSciNet)
MR3312763

Zentralblatt MATH identifier
07000930

Citation

Fakharany, M.; Company, R.; Jódar, L. Unconditional Positive Stable Numerical Solution of Partial Integrodifferential Option Pricing Problems. J. Appl. Math. 2015 (2015), Article ID 960728, 10 pages. doi:10.1155/2015/960728. https://projecteuclid.org/euclid.jam/1429105033


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References

  • J. Y. Campbell, A. W. Lo, and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, Princeton, NJ, USA, 1997.
  • S. L. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” The Review of Financial Studies, vol. 6, no. 2, pp. 327–343, 1993.
  • R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
  • A. Pascucci, PDE and Martingale Methods in Option Pricing, vol. 2 of Bocconi & Springer Series, Springer, Milan, Italy, 2011.
  • S. G. Kou, “A jump-diffusion model for option pricing,” Management Science, vol. 48, no. 8, pp. 1086–1101, 2002.
  • R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” Journal of Financial Economics, vol. 3, no. 1-2, pp. 125–144, 1976.
  • I. Koponen, “Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process,” Physical Review E, vol. 52, no. 1, pp. 1197–1199, 1995.
  • D. B. Madan and F. Milne, “Option pricing with variance gamma martingale components,” Mathematical Finance, vol. 1, no. 4, pp. 39–55, 1991.
  • P. Carr, H. Geman, D. B. Madan, and M. Vor, “The fine structure of asset returns: an empirical investigation,” Journal of Business, vol. 75, no. 2, pp. 305–332, 2002.
  • F. Fang and C. W. Oosterlee, “A novel pricing method for European options based on Fourier-cosine series expansions,” SIAM Journal on Scientific Computing, vol. 31, no. 2, pp. 826–848, 2008.
  • A. Almendral and C. W. Oosterlee, “Numerical valuation of options with jumps in the underlying,” Applied Numerical Mathematics, vol. 53, no. 1, pp. 1–18, 2005.
  • A. Almendral and C. W. Oosterlee, “Accurate evaluation of European and American options under the CGMY process,” SIAM Journal on Scientific Computing, vol. 29, no. 1, pp. 93–117, 2007.
  • 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, pp. 231–262, 2000.
  • M.-C. Casabán, R. Company, L. Jódar, and J.-V. Romero, “Double discretization difference schemes for partial integrodifferential option pricing jump diffusion models,” Abstract and Applied Analysis, vol. 2012, Article ID 120358, 20 pages, 2012.
  • R. Company, L. Jódar, and M. Fakharany, “Positive solutions of European option pricing with CGMY process models using double discretization difference schemes,” Abstract and Applied Analysis, vol. 2013, Article ID 517480, 11 pages, 2013.
  • R. Cont and E. Voltchkova, “A finite difference scheme for option pricing in jump diffusion and exponential Lévy models,” SIAM Journal on Numerical Analysis, vol. 43, no. 4, pp. 1596–1626, 2005.
  • J. Toivanen, “Numerical valuation of European and American options under Kou's jump-diffusion model,” SIAM Journal on Scientific Computing, vol. 30, no. 4, pp. 1949–1970, 2008.
  • S. Salmi and J. Toivanen, “An iterative method for pricing American options under jump-diffusion models,” Applied Numerical Mathematics, vol. 61, no. 7, pp. 821–831, 2011.
  • I. R. Wang, J. W. Wan, and P. A. Forsyth, “Robust numerical valuation of European and American options under the CGMY process,” Journal of Computational Finance, vol. 10, pp. 31–69, 2007.
  • J. Lee and Y. Lee, “Tridiagonal implicit method to evaluate European and American options under infinite activity Lévy models,” Journal of Computational and Applied Mathematics, vol. 237, no. 1, pp. 234–243, 2013.
  • H. Burchard, E. Deleersnijder, and A. Meister, “A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations,” Applied Numerical Mathematics, vol. 47, no. 1, pp. 1–30, 2003.
  • B. M. Chen-Charpentier and H. V. Kojouharov, “An unconditionally positivity preserving scheme for advection-diffusion reaction equations,” Mathematical and Computer Modelling, vol. 57, no. 9–10, pp. 2177–2185, 2013.
  • S. V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, NY, USA, 1980.
  • A.-M. Matache, T. von Petersdorff, and C. Schwab, “Fast deterministic pricing of options on Lévy driven assets,” Mathematical Modelling and Numerical Analysis, vol. 38, no. 1, pp. 37–71, 2004.
  • R. Kangro and R. Nicolaides, “Far field boundary conditions for Black-Scholes equations,” SIAM Journal on Numerical Analysis, vol. 38, no. 4, pp. 1357–1368, 2000.
  • M. S. Milgram, “The generalized integro-exponential function,” Mathematics of Computation, vol. 44, no. 170, pp. 443–458, 1985.
  • N. M. Temme, Special Functions an Introduction to the Classical Functions of Mathematical Physics, John Wiley & Sons, New York, NY, USA, 1996.
  • P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, NY, USA, 2nd edition, 1984.
  • P. Linz, Analytical and Numerical Methods for Volterra Equations, vol. 7 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1985.
  • G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, UK, 3rd edition, 1985.
  • D. B. Madan, P. P. Carr, and E. C. Chang, “The variance gamma process and option pricing,” European Finance Review, vol. 2, no. 1, pp. 79–105, 1998. \endinput