The Annals of Applied Probability

Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation

Abstract

In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(\Delta t)$ with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of $\epsilon$ from $O(\epsilon^{-3})$ to $O(\epsilon^{-2})$. However, in general, to obtain a rate of strong convergence higher than $O(\Delta t^{1/2})$ requires simulation, or approximation, of Lévy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of Lévy areas and still achieve an $O(\Delta t^{2})$ multilevel correction variance for smooth payoffs, and almost an $O(\Delta t^{3/2})$ variance for piecewise smooth payoffs, even though there is only $O(\Delta t^{1/2})$ strong convergence. This results in an $O(\epsilon^{-2})$ complexity for estimating the value of European and Asian put and call options.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 4 (2014), 1585-1620.

Dates
First available in Project Euclid: 14 May 2014

https://projecteuclid.org/euclid.aoap/1400073658

Digital Object Identifier
doi:10.1214/13-AAP957

Mathematical Reviews number (MathSciNet)
MR3211005

Zentralblatt MATH identifier
1373.65007

Citation

Giles, Michael B.; Szpruch, Lukasz. Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Probab. 24 (2014), no. 4, 1585--1620. doi:10.1214/13-AAP957. https://projecteuclid.org/euclid.aoap/1400073658

References

• [1] Burkholder, D. L., Davis, B. J. and Gundy, R. F. (1972). Integral inequalities for convex functions of operators on martingales. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) 2 223–240. Univ. California Press, Berkeley, CA.
• [2] Clark, J. M. C. and Cameron, R. J. (1980). The maximum rate of convergence of discrete approximations for stochastic differential equations. In Stochastic Differential Systems (Proc. IFIP–WG 7/1 Working Conf., Vilnius, 1978). Lecture Notes in Control and Information Sci. 25 162–171. Springer, Berlin.
• [3] Duffie, D. and Glynn, P. (1995). Efficient Monte Carlo simulation of security prices. Ann. Appl. Probab. 5 897–905.
• [4] Gaines, J. G. and Lyons, T. J. (1994). Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 1132–1146.
• [5] Giles, M. (2008). Improved multilevel Monte Carlo convergence using the Milstein scheme. In Monte Carlo and Quasi-Monte Carlo Methods 2006 (A. Keller, S. Heinrich and H. Niederreiter, eds.) 343–358. Springer, Berlin.
• [6] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
• [7] Giles, M. B., Higham, D. J. and Mao, X. (2009). Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff. Finance Stoch. 13 403–413.
• [8] Giles, M. B. and Reisinger, C. (2012). Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIAM J. Financial Math. 3 572–592.
• [9] Giles, M. B. and Waterhouse, B. J. (2009). Multilevel quasi-Monte Carlo path simulation. In Advanced Financial Modelling. Radon Ser. Comput. Appl. Math. 8 165–181. Walter de Gruyter, Berlin.
• [10] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York.
• [11] Heston, S. I. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6 327–343.
• [12] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
• [13] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
• [14] Milstein, G. N. and Tretyakov, M. V. (2004). Stochastic Numerics for Mathematical Physics. Scientific Computation. Springer, Berlin.
• [15] Müller-Gronbach, T. (2002). Strong approximation of systems of stochastic differential equations. Habilitation thesis, TU, Darmstadt.
• [16] Neuenkirch, A. and Szpruch, L. (2014). First order strong approximations of scalar SDEs with values in a domain. Numer. Math. To appear. arXiv preprint, available at arXiv:1209.0390.
• [17] Rydén, T. and Wiktorsson, M. (2001). On the simulation of iterated Itô integrals. Stochastic Process. Appl. 91 151–168.
• [18] Wiktorsson, M. (2001). Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions. Ann. Appl. Probab. 11 470–487.