Pricing Illiquid Options with $N+1$ Liquid Proxies Using Mixed Dynamic-Static Hedging

TL;DR

This paper develops a numerical method for pricing and hedging illiquid options using liquid proxies with a mixed dynamic-static hedging approach, applicable across various asset classes.

Contribution

It introduces an efficient numerical algorithm based on FGTs for solving the HJB equation in an indifference pricing framework with mixed hedging strategies.

Findings

The proposed method outperforms finite-difference schemes in efficiency and accuracy.

Application to a credit-equity Merton model demonstrates practical effectiveness.

Framework adaptable to different asset classes and illiquid derivatives.

Abstract

We study the problem of optimal pricing and hedging of a European option written on an illiquid asset $Z$ using a set of proxies: a liquid asset $S$, and $N$ liquid European options $P_i$, each written on a liquid asset $Y_i, i=1,N$. We assume that the $S$-hedge is dynamic while the multi-name $Y$-hedge is static. Using the indifference pricing approach with an exponential utility, we derive a HJB equation for the value function, and build an efficient numerical algorithm. The latter is based on several changes of variables, a splitting scheme, and a set of Fast Gauss Transforms (FGT), which turns out to be more efficient in terms of complexity and lower local space error than a finite-difference method. 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.