Proxy simulation schemes for generic robust Monte Carlo sensitivities, process-oriented importance sampling and high-accuracy drift approximation

Christian P. Fries, Jörg Kampen

Abstract

ABSTRACT

We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme K^o (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting the measure (via likelihood ratio) to match the distribution of K* such that
E^Q(f (K*) | F_t ) = E^Q(f (K^o) · w | F_t ) . This is done numerically in every time step, on every path. This makes the approach independent of the product (the function f in the above equation) and even of the model; it only depends on the numerical scheme. The approach is essentially a numerical version of the likelihood ratio method (Broadie and Glasserman 1996) and Malliavin’s calculus (Fournié et al 1999; Malliavin 1997) reconsidered on the level of the discrete numerical simulation scheme. As the numerical scheme represents a time-discrete stochastic process sampled on a discrete probability space, the essence of the method may be motivated without a deeper mathematical understanding of the time continuous theory (eg, Malliavin’s calculus). The framework is completely generic and may be used for: high-accuracy drift approximations; process-oriented importance sampling; and the robust calculation of partial derivatives of expectations with respect to model parameters (ie, sensitivities, also known as Greeks) by applying finite differences by re-evaluating the expectation with a model with shifted parameters. We present numerical results using a Monte Carlo simulation of the LIBOR Market Model for benchmarking.