American Options by Malliavin Calculus and Nonparametric Variance and Bias Reduction Methods

Abstract

This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f (St)/Ss] using the Malliavin calculus without localization. Then the variance of the estimator of E[f (St)/Ss] is reduced using closed formulas, techniques based on a conditioning and a judicious choice of the number of simulated paths. Finally, we perform the stopping times version of the dynamic programming algorithm to decrease the bias. On the one hand, we will develop the Malliavin calculus tools for exponential multi-dimensional diffusions that have deterministic and no constant coefficients. On the other hand, we will detail various nonparametric technics to reduce the variance. Moreover, we will test the numerical efficiency of our method on a heterogeneous CPU/GPU multi-core machine.

Toward a coherent Monte Carlo simulation of CVA

“…We also express CVA 0,T and ∂ S i 0 CVA 0,T using V t and ∂ S i 0 V t whose values are given by (2) and…”

“…where W is a multidimensional Brownian motion correlated to W as it is done in [2]. Then the conditional expectation in (7) is given by…”

“…In order to calculate V t using (2), one has to express the conditional expectation involved in each contract. Using Malliavin calculus for American contracts pricing, this conditional expectation was expressed as a ratio of two expectations (see for example [2,3]). We aim here to adapt the previous results to the CVA problem.…”

“…We have already seen that V t and ∂ S i 0 V t are given by (2) and (8) where the value of each contract is expressed using (3), (4) and (5). The only point that remains to be specified is the conditional expectation and its partial derivative in (7).…”

“…Inspired from papers [2,3] dedicated to American contracts, we rewrite (7) as a quotient of expectations that can be simulated by Monte Carlo. Going beyond the latter result, we express the backward conditional density and its gradient as a quotient of expectations that can be simulated by Monte Carlo.…”