Indifference Pricing of American Option Underlying Illiquid Stock under Exponential Forward Performance

TL;DR

This paper develops a mathematical framework for indifference pricing of American options on illiquid stocks using exponential forward performance, characterizing the value function via PDEs and deriving optimal strategies.

Contribution

It introduces a novel PDE-based approach to indifference pricing under exponential forward measures for illiquid stocks, including a dual representation and optimal strategy existence.

Findings

Characterization of the value function as a PDE solution

Existence of optimal trading strategies

Application to employee stock options

Abstract

This work focuses on the indifference pricing of American call option underlying a non-traded stock, which may be partially hedgeable by another traded stock. Under the exponential forward measure, the indifference price is formulated as a stochastic singular control problem. The value function is characterized as the unique solution of a partial differential equation in a Sobolev space. Together with some regularities and estimates of the value function, the existence of the optimal strategy is also obtained. The applications of the characterization result includes a derivation of a dual representation and the indifference pricing on employee stock option. As a byproduct, a generalized Ito's formula is obtained for functions in a Sobolev space.