## Annals of Applied Probability

### Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing

Ahmed Kebaier

#### Abstract

We study the approximation of $\mathbb{E}f(X_{T})$ by a Monte Carlo algorithm, where X is the solution of a stochastic differential equation and f is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps δ and δβ,0<β<1. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal β (which turns out to be β=1/2), we establish a central limit type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.

#### Article information

Source
Ann. Appl. Probab., Volume 15, Number 4 (2005), 2681-2705.

Dates
First available in Project Euclid: 7 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1133965776

Digital Object Identifier
doi:10.1214/105051605000000511

Mathematical Reviews number (MathSciNet)
MR2187308

Zentralblatt MATH identifier
1099.65011

#### Citation

Kebaier, Ahmed. Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 (2005), no. 4, 2681--2705. doi:10.1214/105051605000000511. https://projecteuclid.org/euclid.aoap/1133965776

#### References

• Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331.
• Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 43–60.
• Boyle, P., Broadie, M. and Glasserman, P. (1997). Monte Carlo methods for security pricing. J. Econom. Dynam. Control 21 1267–1321.
• Broadie, M. and Detemple, J. (1997). Recent advances in numerical methods for pricing derivative securities. In Numerical Methods in Finance 43–66. Cambridge Univ. Press.
• Duffie, D. and Glynn, P. (1995). Efficient Monte Carlo simulation of security prices. Ann. Appl. Probab. 5 897–905.
• Faure, O. (1992). Simulation du mouvement Brownien et des diffusions. Ph.D. thesis, Ecole Nationale des Ponts et Chaussées.
• Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655 232–246. Springer, Berlin.
• Jacod, J., Kurtz, T. G., Méléard, S. and Protter, P. (2005). The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. H. Poincaré. Special Issue Devoted to the Memory of P. A. Meyer 41 523–558.
• Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
• Kebaier, A. (2004). Romberg extrapolation, variance reduction and applications to option pricing. Preprint.
• Kurtz, T. G. and Protter, P. (1991). Wong–Zakai corrections, random evolutions, and simulation schemes for SDEs. In Stochastic Analysis 331–346. Academic Press, Boston, MA.
• Kurtz, T. G. and Protter, P. (1999). Weak error estimates for simulation schemes for SDEs.
• Lamberton, D. and Lapeyre, B. (1997). Introduction au calcul stochastique appliqué à la finance, 2nd ed. Ellipses, Paris.
• Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
• Rényi, A. (1963). On stable sequences of events. Sankhyā Ser. A 25 293–302.
• Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 483–509.
• Temam, E. (2001). Couverture approché d'options exotiques. Pricing des options Asiatiques. Ph.D. thesis, Univ. Paris VI.