R-Minimizing Hedging in an Incomplete Market: Malliavin Calculus Approach

Abstract

In this paper we derive the locally risk-minimizing hedging for a general contingent claim in an incomplete market via the generalized Clark-Ocone formula. Using this result in a stochastic volatility model, we study its connection with the hedge obtained via PDE approach. We see these hedging strategies, under weak conditions, are the same as the ones generated by the PDE approach. Within the same model we establish the pricing and the locally risk-minimizing hedging formulas for the path-dependent options, particularly pronouncing an impact from the correlation between stock and volatility on pricing and hedging. These formulas provides a basis to derive the approximations, which are much more computationally accessible.

Keywords: R-minimizing hedging, Delta hedging, incomplete markets, stochastic volatility model, Clark-Ocone formula, Malliavin Calculus, path-dependent options

JEL Classification: G13, C61, C63