# 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 Ko (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
EQ(f (K*) | Ft ) = EQ(f (Ko) · w | Ft ) . 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.