A regression-based Monte Carlo method to solve backward stochastic differential equations

Abstract

We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2005, Vol. 15, No. 3, 2172-2202. This reprint differs from the original in pagination and typographic detail. 1 2 E. GOBET, J.-P. LEMOR AND X. WARINApplications of BSDEs. Such equations, first studied by Pardoux and Peng [26] in a general form, are important tools in mathematical finance. We mention some applications and refer the reader to [10,12] for numerous references. In a complete market, for the usual valuation of a contingent claim with payoff Φ(S), Y is the value of the replicating portfolio and Z is related to the hedging strategy. In that case, the driver f is linear w.r.t. Y and Z. Some market imperfections can also be incorporated, such as higher interest rate for borrowing [4]: then, the driver is only Lipschitz continuous w.r.t. Y and Z. Related numerical experiments are developed in Section 6. In incomplete markets, the Föllmer-Schweizer strategy [14] is given by the solution of a BSDE. When trading constraints on some assets are imposed, the super-replication price [13] is obtained as the limit of nonlinear BSDEs. Connections with recursive utilities of Duffie and Epstein [11] are also available. Peng has introduced the notion of g-expectation (here g is the driver) as a nonlinear pricing rule [28]. Recently he has shown [27] the deep connection between BSDEs and dynamic risk measures, proving that any dynamic risk measure (E t ) 0≤t≤T (satisfying some axiomatic conditions) is necessarily associated to a BSDE (Y t ) 0≤t≤T (the converse being known for years). The least we can say is that BSDEs are now inevitable tools in mathematical finance. Another indirect application may concern variance reduction techniques for the Monte Carlo computations of expectations, say E(Φ) taking f ≡ 0. Indeed, T 0 Z s dW s is the so-called martingale control variate (see [24], for instance). Finally, for applications to semi-linear PDEs, we refer to [25], among others.The mathematical analysis of BSDE is now well understood (see [23] for recent references) and its numerical resolution has made recent progresses. However, even if several numerical methods have been proposed, they suffer of a high complexity in terms of computational time or are very costly in terms of computer memory. Thus, their uses in practice on real problems are difficult. Hence, it is still topical to devise more efficient algorithms. This article contributes in this direction by developing a simple approach, based on Monte Carlo regression on function bases. It is in the vein of the general regressi...

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“…Let R t 1 denote the remainder after the projection P t 1 so that for any X ∈ F t 1 , X = P t 1 (X) + R t 1 (X), and the two latter terms are orthogonal in L 2 . The following theorem is similar to Theorem 2 in Gobet et al (2005). It shows the relationship between the error at the k-th layer,…”

“…Several canonical choices have been proposed, including the Laguerre polynomials in the original paper of Longstaff and Schwartz (2001), the indicator functions of a partition of the domain of {X t } in Gobet et al (2005), and the logistic basis in Haugh and Kogan (2004). However, there is now widespread consensus that the numerical precision can be greatly improved by customizing the basis (for example, the customization advantage is documented by Andersen and Broadie (2004)).…”

“…The basic TvR scheme (32) is essentially a simple algorithm for solving discrete-time reflected BSDE's, a topic that has been a very active area of research, see e.g. Chevance (1997), Bouchard and Touzi (2004), Gobet et al (2005). Empirical evidence strongly suggests that the convergence properties of the TvR scheme are not as good as of the LS scheme, however the former is much more amenable to analysis.…”

“…Looking closely the sampling error has three components that are closely intertwined-error due to using α Nexisting results, even in simpler settings, are still incomplete. To summarize the current state-ofthe-art in this field, let us mention the work of Clément et al (2002) and Egloff (2005) on the Longstaff-Schwartz scheme and Bouchard and Touzi (2004), Bally and Pagès (2003) and Gobet et al (2005) on numerical methods for backward SDEs. For the LS scheme it has been shown that the use of optimal-stopping-time iteration in (29) is consistent, namely that for a fixed N B the asymptotic sampling error as N → ∞ has mean zero and Gaussian distribution (Clément et al 2002).…”

“…However, the involved computational level is an order of magnitude higher. Alternatively, Gobet et al (2005) obtain the O(N −1/2 ) L 2 -convergence rate when using a single set of paths but no reflection in (43). We conjecture that the sampling L 2 -error is on the order O(∆t −1 N −1/2 ) for our problem as well, since this is the general convergence rate for Monte Carlo methods.…”