Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T07:20:50.988Z Has data issue: false hasContentIssue false
  • English
  • Français

SIMULATION OF MULTI-ASSET OPTION GREEKS UNDER A SPECIAL LÉVY MODEL BY MALLIAVIN CALCULUS

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  17 February 2016

YONGZENG LAI  and
HAIXIANG YAO
YONGZENG LAI
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada email [email protected]
HAIXIANG YAO*
Affiliation:
School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China email [email protected]
*
✉[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.

We discuss simulation of sensitivities or Greeks of multi-asset European style options under a special Lévy process model: that is, the subordinated Brownian motion model. The Malliavin calculus method combined with Monte Carlo and quasi-Monte Carlo methods is used in the simulations. Greeks are expressed in terms of the expectations of the option payoff functions multiplied by the weights involving Malliavin derivatives for multi-asset options. Numerical results show that the Malliavin calculus method is usually more efficient than the finite difference method for options with nonsmooth payoffs. The superiority of the former method over the latter is even more significant when both are combined with quasi-Monte Carlo methods.

Keywords

subordinated Brownian motionsmulti-asset optionsoption sensitivities or GreeksMalliavin calculusquasi-Monte Carlo methods

MSC classification

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60G51: Processes with independent increments; L'evy processes 65C05: Monte Carlo methods
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 3 , January 2016 , pp. 280 - 298
Copyright
© 2016 Australian Mathematical Society 

References

Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions (Dover Publications, New York, 1965).Google Scholar
Asmussen, S. and Glynn, P. W., Stochastic simulation (Springer, New York, 2007).Google Scholar
Bally, V., “An elementary introduction to Malliavin calculus” Research Report, 2003;http://www.inria.fr/rrrt//RR-4718.pdf.Google Scholar
Boyle, P. P., “Options: a Monte Carlo approach”, J. Financial Eco. 4.3 (1977) 323–338 ; doi:10.1016/0304-405x(77)90005-8.CrossRefGoogle Scholar
Boyle, P., Lai, Y. and Tan, K. S., “Pricing options using lattice rules”, N. Am. Actuar. J. 9 (2005) 50–76 ; doi:10.1080/10920277.2005.10596211.CrossRefGoogle Scholar
Cont, R. and Tankov, P., Financial modelling with jump processes (Chapman & Hall, Boca Raton, FL, 2004).Google Scholar
Davis, M. H. A. and Johansson, M. P., “Malliavin Monte Carlo greeks for jump diffusions”, Stochastic. Process. Appl. 116 (2006) 101–129 ; doi:10.1016/j.spa.2005.08.002.CrossRefGoogle Scholar
Dick, J. and Pillichshammer, F., Digital nets and sequences: discrepancy theory and quasi-Monte Carlo integration (Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
Di Nunno, G., Øksendal, B. and Proske, F., Malliavin calculus for Lévy processes with applications to finance (Springer, Berlin, Heidelberg, 2009).CrossRefGoogle Scholar
Eberlein, E. and Prause, K., “The generalized hyperbolic model: Financial derivatives and risk measures”, in: Mathematical finance-bachelier congress 2000 (Springer, Berlin, 2002) 245–267. doi:10.1007/978-3-662-12429-1_12.CrossRefGoogle Scholar
Fajardo, J. and Farias, A., “Generalized hyperbolic distributions and Brazillian data”, Braz. Rev. Econ. 24 (2004) 249–271 ; doi:10.2139/ssrn.338283.Google Scholar
Fournié, E., Larsy, J. M., Lebuchoux, J., Lions, P. L. and Touzi, N., “Applications of Malliavin calculus to Monte Carlo methods in finance I”, Finance Stoch. 3 (1999) 391–412 ; doi:10.1007/s007800050068.Google Scholar
Fournié, E., Larsy, J. M., Lebuchoux, J., Lions, P. L. and Touzi, N., “Applications of Malliavin calculus to Monte Carlo methods in finance II”, Finance Stoch. 5 (2001) 201–236 ; doi:10.1007/pl00013529.CrossRefGoogle Scholar
Glasserman, P., Monte Carlo methods for financial engineering (Springer, New York, 2003).CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R., Matrix analysis, 2nd edn (Cambridge University Press, New York, 1985).CrossRefGoogle Scholar
Hua, L. and Wang, Y., Applications of number theory in numerical analysis (Springer, Rio de Janeiro, 1980).Google Scholar
Joe, S. and Kuo, F., “Remark on algorithm 659: Implementing Sobol’s quasirandom sequence generator”, ACM Trans. Math. Software 29 (2003) 49–57 ; doi:10.1145/641876.641879.CrossRefGoogle Scholar
Kawai, R. and Kohatsu-Higa, A., “Computation of Greeks and multidimensional density estimation for asset price models with time-changed Brownian motion”, Appl. Math. Finance 17 (2010) 301–321 ; doi:10.1080/13504860903336429.CrossRefGoogle Scholar
Lai, Y., “Intermediate rank lattice rules and applications to finance”, Appl. Numer. Math. 59 (2009) 1–20 ; doi:10.1016/j.apnum.2007.11.024.CrossRefGoogle Scholar
L’Ecuyer, P., “A unifed view of the IPA, SF, and LR gradient estimation techniques”, Management Sci. 36 (1990) 1364–1383 ; doi:10.1287/mnsc.36.11.1364.CrossRefGoogle Scholar
L’Ecuyer, P., “Quasi-Monte Carlo methods with applications in finance”, Finance Stoch. 13 (2009) 307–349 ; doi:10.1007/s00780-009-0095-y.CrossRefGoogle Scholar
L’Ecuyer, P. and Perron, G., “On the convergence rates of IPA and FDC derivative estimators”, Oper. Res. 42 (1994) 643–656 ; doi:10.1287/opre.42.4.643.CrossRefGoogle Scholar
Michael, J., Schucany, W. and Haas, R., “Generating random variates using transformations with multiple roots”, Amer. Statist. 30 (1976) 88–90 ; doi:10.2307/2683801.Google Scholar
Montero, M. and Kohastu-Higa, A., “Malliavin calculus applied to finance”, Physica A 320 (2002) 548–570 ; doi:10.1016/s0378-4371(0)201531-5.CrossRefGoogle Scholar
Niederreiter, H., Random number generation and quasi-Monte Carlo methods (SIAM, Philadelphia, 1992).CrossRefGoogle Scholar
Nualart, D., The Malliavin calculus and related topics (Springer, Berlin, 1995).CrossRefGoogle Scholar
Øksendal, B., An introduction to Malliavin calculus with applications in economics, Unpublished Lecture Notes (1997) available at: http://www.citeulike.org/user/alexv/article/2366596.Google Scholar
Ribeiro, C. and Webber, N., “Valuing path-dependent options in the variance-gamma model by Monte Carlo with a gamma bridge”, J. Comput. Finance 7 (2004) 81–100 ;http://wrap.warwick.ac.uk/1808/1/WRAP_Riveiro_fwp02-04.pdf.CrossRefGoogle Scholar
Sato, K., Lévy processes and infinitely divisible distributions (Cambridge University Press, Cambridge, 1999).Google Scholar
Sloan, I. and Joe, S., Lattice methods for multiple integration (Clarendon Press, Oxford, 1994).CrossRefGoogle Scholar
Tuffin, B., “On the use of low discrepancy sequences in Monte Carlo methods”, Monte Carlo Methods Appl. 2 (1996) 295–320 ; doi:10.1515/mcma.1996.2.4.295.CrossRefGoogle Scholar
Wichurat, M. J., “Algorithm AS 241: The percentage points of the normal distribution”, Appl. Stat. 37 (1988) 477–484 ; doi:10.2307/2347330.CrossRefGoogle Scholar
Xu, Y., Lai, Y. and Xi, X., Efficient simulations for exotic options under NIG model, IEEE CS Proceedings of The 4th International Conference on Computational Sciences and Optimization, Kunming, China, 15–19 April 2011 1271–1275; doi:10.1109/cso.2011.123.CrossRefGoogle Scholar
Xu, Y., Lai, Y. and Yao, H., “Efficient simulation of Greeks of multiasset European and Asian style options by Malliavin calculus and quasi-Monte Carlo methods”, Appl. Math. Comput. 236 (2014) 493–511 ; doi:10.1016/j.amc.2014.03.057.Google Scholar