Abstract
We investigate the extension of the multilevel Monte Carlo path simulation method to jump-diffusion SDEs. We consider models with finite rate activity using a jump-adapted discretisation in which the jump times are computed and added to the standard uniform discretisation times. The key component in multilevel analysis is the calculation of an expected payoff difference between a coarse path simulation and a fine path simulation with twice as many timesteps. If the Poisson jump rate is constant, the jump times are the same on both paths and the multilevel extension is relatively straightforward, but the implementation is more complex in the case of state-dependent jump rates for which the jump times naturally differ.
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References
R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall, 2004.
S. Dereich. Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. The Annals of Applied Probability, 21(1):283–311, 2011.
S. Dereich and F. Heidenreich. A multilevel Monte Carlo algorithm for Lévy driven stochastic differential equations. Stochastic Processes and their Applications, 121(7):1565–1587, 2011.
M.B. Giles. Improved multilevel Monte Carlo convergence using the Milstein scheme. In A. Keller, S. Heinrich, and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2006, pages 343–358. Springer-Verlag, 2007.
M.B. Giles. Multilevel Monte Carlo path simulation. Operations Research, 56(3):607–617, 2008.
P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer, New York, 2004.
P. Glasserman and N. Merener. Convergence of a discretization scheme for jump-diffusion processes with state-dependent intensities. Proc. Royal Soc. London A, 460:111–127, 2004.
M.B. Giles and B.J. Waterhouse. Multilevel quasi-Monte Carlo path simulation. Radon Series Comp. Appl. Math., 8, 1–18, 2009.
R.C. Merton. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1–2):125–144, 1976.
E. Platen. A generalized Taylor formula for solutions of stochastic equations. Sankhyā: The Indian Journal of Statistics, Series A, 44(2):163–172, 1982.
N. Platen and E. Bruti-Liberati. Numerical Solution of Stochastic Differential Equations with Jumps in Finance, volume 64 of Stochastic Modelling and Applied Probability. Springer-Verlag, 1st edition, 2010.
Y. Xia. Multilevel Monte Carlo method for jump-diffusion SDEs. Technical report. http://arxiv.org/abs/1106.4730
Y. Xia and M.B. Giles. Numerical analysis of multilevel Monte Carlo for scalar jump-diffusion SDEs. Working paper in preparation, 2011.
Acknowledgements
Xia is grateful to the China Scholarship Council for financial support, and the research has also been supported by the Oxford-Man Institute of Quantitative Finance. The authors are indebted to two anonymous reviewers for their invaluable comments.
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Xia, Y., Giles, M.B. (2012). Multilevel Path Simulation for Jump-Diffusion SDEs. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_41
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DOI: https://doi.org/10.1007/978-3-642-27440-4_41
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