Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift

TL;DR

This paper develops a probabilistic framework and integration by parts formulas for certain stochastic volatility models with unbounded drift, enabling efficient Monte Carlo methods for option pricing and sensitivities.

Contribution

It introduces a novel perturbation technique and Malliavin calculus approach for stochastic volatility models, facilitating unbiased Monte Carlo simulations for complex financial derivatives.

Findings

Efficient unbiased Monte Carlo simulation method for option pricing.

Accurate computation of Greeks like Delta and Vega.

Numerical results demonstrate the method's effectiveness.

Abstract

In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.