Pricing options on illiquid assets with liquid proxies using utility indifference and dynamic-static hedging

TL;DR

This paper develops a method for pricing and hedging European options on illiquid assets by combining dynamic and static hedging strategies with utility indifference pricing, providing analytical solutions in a correlated asset framework.

Contribution

It introduces an analytical approach using asymptotic expansion for pricing and hedging illiquid assets with liquid proxies, applicable across various asset classes.

Findings

Derived a HJB equation for the indifference price

Solved the equation analytically using asymptotic expansion

Applicable to multiple asset classes beyond credit-equity models

Abstract

This work addresses the problem of optimal pricing and hedging of a European option on an illiquid asset Z using two proxies: a liquid asset S and a liquid European option on another liquid asset Y. We assume that the S-hedge is dynamic while the Y-hedge is static. Using the indifference pricing approach we derive a HJB equation for the value function, and solve it analytically (in quadratures) using an asymptotic expansion around the limit of the perfect correlation between assets Y and Z. While in this paper we apply our framework to an incomplete market version of the credit-equity Merton's model, the same approach can be used for other asset classes (equity, commodity, FX, etc.), e.g. for pricing and hedging options with illiquid strikes or illiquid exotic options.