Nonparametric Malliavin–Monte Carlo Computation of Hedging Greeks

Maria Elvira Mancino,Simona Sanfelici

doi:10.3390/risks8040120

Abstract

We propose a way to compute the hedging Delta using the Malliavin weight method. Our approach, which we name the λ-method, generally outperforms the standard Monte Carlo finite difference method, especially for discontinuous payoffs. Furthermore, our approach is nonparametric, as we only assume a general local volatility model and we substitute the volatility and the other processes involved in the Greek formula with quantities that can be nonparametrically estimated from a given time series of observed prices.

Highlights

In finance, the numerical computation of the option price sensitivities, named Greeks, has attracted much attention in the last few years, as the knowledge of the Greeks and their implementation as risk management tools have paramount importance when trading financial derivatives
The numerical computation of the option price sensitivities, named Greeks, has attracted much attention in the last few years, as the knowledge of the Greeks and their implementation as risk management tools have paramount importance when trading financial derivatives. Their numerical computation using the Monte Carlo finite difference method is inefficient in the case when the option payoff is represented by a discontinuous function of the underlying asset, as it is often the case
Our approach is nonparametric, as we only assume a general local volatility model and we substitute the volatility and the other processes involved in the Greek formula with quantities that can be nonparametrically estimated given the time series of observed prices

Summary

Introduction

The numerical computation of the option price sensitivities, named Greeks, has attracted much attention in the last few years, as the knowledge of the Greeks and their implementation as risk management tools have paramount importance when trading financial derivatives. Our approach is nonparametric, as we only assume a general local volatility model and we substitute the volatility and the other processes involved in the Greek formula with quantities that can be nonparametrically estimated given the time series of observed prices This result can be achieved by expressing the weight using the rescaled variation which has been introduced in Barucci et al (2003); Malliavin and Mancino (2002b). An important advantage of expressing the Malliavin weight in terms of the rescaled variation relies in the fact that this function satisfies an ordinary differential equation instead of a stochastic differential equation This result has an impact on the numerical efficiency of the Malliavin–Monte Carlo method. The Appendix A resumes the mathematical computation of the rescaled variation and its model-free expression

Greeks Computation via Malliavin Calculus and the Rescaled Variation

The Black–Scholes Model

The CEV Model

Simulation Study

Computation of the Weight

Efficiency of the Method

Conclusions