Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T18:11:09.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

Published online by Cambridge University Press:  28 May 2015

Hong-Kui Pang  and
Hai-Wei Sun
Hong-Kui Pang*
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, China
Hai-Wei Sun*
Affiliation:
Department of Mathematics, University of Macau, Macao, China
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Get access

Abstract

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

Keywords

91B2862P0535K1565F1065M0691B7047B35Stochastic volatility jump diffusionEuropean optionbarrier optionpartial integro-differential equationmatrix exponentialshift-invert Arnoldimatrix splittingmultigrid method
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 1 , February 2014 , pp. 52 - 68
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Andersen, L. and Andreasen, J., Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Derivatives Res. 4, 231–262 (2000).Google Scholar
[2]Barndorff-Nielsen, O., Process of normal inverse Gaussian type, Finance Stoch. 2, 41–68 (1998).CrossRefGoogle Scholar
[3]Bates, D., Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options, Rev. Financial Stud. 9, 69–107 (1996).Google Scholar
[4]Black, F. and Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ. 81, 637–654 (1973).Google Scholar
[5]Briggs, W.L., Henson, V.E., and McCormick, S.F., A Multigrid Tutorial, 2nd edition, SIAM, Philadelphia, 2000.Google Scholar
[6]Carr, P., Geman, H., Madan, D., and Yor, M., The fine structure of asset returns: An empirical investigation, J. Business 75, 305–332 (2002).Google Scholar
[7]Carr, P. and Wu, L., Finite moment log stable process and option pricing, J. Finance 58, 753–777 (2003).Google Scholar
[8]Chan, R. and Jin, X., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.CrossRefGoogle Scholar
[9]d'Halluin, Y., Forsyth, P., and Vetzal, K., Robust numerical methods for contingent claims under jump diffusion, IMA J. Numer. Anal. 25, 87–112 (2005).Google Scholar
[10]Duffie, D., Pan, J., and Singleton, K., Transform analysis and asset pricing for affine jump diffusions, Econometrica 68, 1343–1376 (2000).Google Scholar
[11]Eberlein, E., Keller, U., and Prause, K., New insights into smile, mispricing, and value at risk: The hyperbolic model, J. Business 71, 371–405 (1998).Google Scholar
[12]Eiermann, M. and Ernst, O., A restarted Krylov subspace method for the evaluation of matrix functions, SIAM J. Numer. Anal. 44, 2481–2504 (2006).CrossRefGoogle Scholar
[13]Eraker, B., Johannes, M., and Polson, N., The impact of jumps in volatilityand returns, J. Finance 58, 1269–1300 (2003).Google Scholar
[14]Feng, L. and Linetsky, V., Pricing options in jump diffusion models: An extrapolation approach, Oper. Res. 56, 304–325 (2008).Google Scholar
[15]Heston, S., Aclosed form solution for options with stochastic volatility with applications to bond and currencyoptions, Rev. Financial Stud. 6, 327–343 (1993).Google Scholar
[16]Higham, N., Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.Google Scholar
[17]Hochbruck, M. and Lubich, C., On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 34, 1911–1925 (1997).Google Scholar
[18]Hull, J. and White, A., The pricing of options on assets with stochastic volatilites, J. Finance 42, 281–300 (1987).CrossRefGoogle Scholar
[19]Kou, S., A jump diffusion model for option pricing, Management Sci. 48, 1086–1101 (2002).Google Scholar
[20]Lee, S., Liu, X., and Sun, H., Fast exponential time integration scheme for option pricing with jumps, Numer. Linear Algebra Appl. 19, 87–101 (2012).Google Scholar
[21]Lee, S., Pang, H. and Sun, H., Shift-invert Arnoldi approximation to the Toeplitz matrix exponential, SIAM J. Sci. Comput. 32, 774–792 (2010).Google Scholar
[22]Lopez, L. and Simoncini, V., Analysis of projection methods for rational function approximation to the matrix exponential, SIAM J. Numer. Anal. 44, 613–635 (2006).Google Scholar
[23]Madan, D., Carr, P., and Chang, E., The variance gamma process and option pricing, Eur. Finance Rev. 2 79–105 (1998).Google Scholar
[24]Merton, R., Option pricing when underlying stock returns are discontinuous, J. Financial Econom. 3, 125–144 (1976).CrossRefGoogle Scholar
[25]Moret, I. and Novati, P., RD-rational approximations of the matrix exponential, BIT, 44 (2004), pp. 595–615.Google Scholar
[26]Pang, H. and Sun, H., Shift-invert Lanczos method for the symmetric positive semidefinite Toeplitz matrix exponential, Numer. Linear Algebra Appl., 18 (2011), pp. 603–614.Google Scholar
[27]Popolizio, M. and Simoncini, V., Acceleration techniques for approximating the matrix exponential operator, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 657–683.CrossRefGoogle Scholar
[28]Rambeerich, N., Tangman, D., Gopaul, A., and Bhuruth, M., Exponential time integration for fast finite element solutions of some financial engineering problems, J. Comput. Appl. Math. 224, 668–678 (2009).Google Scholar
[29]Rubinstein, M., Implied binomial trees, J. Finance 49, 771–818 (1994).Google Scholar
[30]Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 29, 209–228 (1992).Google Scholar
[31]Shampine, L., Vectorized adaptive quadrature in Matlab, J. Comput. Appl. Math. 211,131-140 (2008).Google Scholar
[32]Tangman, D., Gopaul, A., and Bhuruth, M., Exponential time integration and Chebychev discretisation schemes for fast pricing of options, Appl. Numer. Math. 58, 1309–1319 (2008).CrossRefGoogle Scholar
[33]van den Eshof, J. and Hochbruck, M., Preconditioning Lanczos approximations to the matrix exponential, SIAM J. Sci. Comput. 27, 1438–1457 (2006).Google Scholar
[34]Zhang, Y., Pang, H., Feng, L., and Jin, X., Quadratic finite element and preconditioning for options pricing in the SVCJ model, to appear in J. Comput. Finance.Google Scholar