Malliavin calculus method for asymptotic expansion of dual control problems

TL;DR

This paper introduces a Malliavin calculus-based technique for asymptotic expansion of dual control problems in incomplete markets, enabling approximation of indifference prices and identification of minimal entropy measures.

Contribution

It develops a novel Malliavin-Bismut calculus approach for asymptotic analysis of dual control problems involving measure changes and quadratic penalties.

Findings

Approximate indifference prices in low risk aversion limit.

Identify minimal entropy martingale measure as a perturbation.

Apply method to stochastic volatility models.

Abstract

We develop a technique based on Malliavin-Bismut calculus ideas, for asymptotic expansion of dual control problems arising in connection with exponential indifference valuation of claims, and with minimisation of relative entropy, in incomplete markets. The problems involve optimisation of a functional of Brownian paths on Wiener space, with the paths perturbed by a drift involving the control. In addition there is a penalty term in which the control features quadratically. The drift perturbation is interpreted as a measure change using the Girsanov theorem, leading to a form of the integration by parts formula in which a directional derivative on Wiener space is computed. This allows for asymptotic analysis of the control problem. Applications to incomplete It\^o process markets are given, in which indifference prices are approximated in the low risk aversion limit. We also give an application to identifying the minimal entropy martingale measure as a perturbation to the minimal martingale measure in stochastic volatility models.