A general Multidimensional Monte Carlo Approach for Dynamic Hedging under stochastic volatility

TL;DR

This paper presents a Monte Carlo method for dynamic hedging of European options in multidimensional markets with stochastic volatility, enabling the computation of various hedging strategies in incomplete markets.

Contribution

It introduces a novel Monte Carlo approach based on martingale approximations for Galtchouk-Kunita-Watanabe decompositions, applicable to complex path-dependent options.

Findings

Effective in computing hedging strategies for stochastic volatility models

Applicable to quadratic, locally-risk minimizing, and mean-variance hedging

Demonstrated through numerical examples with various option types

Abstract

In this work, we introduce a Monte Carlo method for the dynamic hedging of general European-type contingent claims in a multidimensional Brownian arbitrage-free market. Based on bounded variation martingale approximations for Galtchouk-Kunita-Watanabe decompositions, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies w.r.t arbitrary square-integrable claims in incomplete markets. In particular, the methodology can be applied to quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate the method with numerical examples based on generalized Follmer-Schweizer decompositions, locally-risk minimizing and mean-variance hedging strategies for vanilla and path-dependent options written on local volatility and stochastic volatility models.