Nash equilibria for game contingent claims with utility-based hedging

TL;DR

This paper develops a framework for analyzing game contingent claims in incomplete markets by constructing Nash equilibria through a stochastic game approach involving utility indifference values, extending existing literature.

Contribution

It introduces a method to solve the stochastic game for GCCs with continuous stopping and trading, providing a new existence result for optimal exercise times under utility indifference.

Findings

Constructed Nash equilibria for GCCs using exponential utility indifference values.

Extended the analysis of American claims to include utility-based optimal exercise times.

Linked the optimal exercise problem to a nonlinear Snell envelope, proving existence results.

Abstract

Game contingent claims (GCCs) generalize American contingent claims by allowing the writer to recall the option as long as it is not exercised, at the price of paying some penalty. In incomplete markets, an appealing approach is to analyze GCCs like their European and American counterparts by solving option holder's and writer's optimal investment problems in the underlying securities. By this, partial hedging opportunities are taken into account. We extend results in the literature by solving the stochastic game corresponding to GCCs with both continuous time stopping and trading. Namely, we construct Nash equilibria by rewriting the game as a non-zero-sum stopping game in which players compare payoffs in terms of their exponential utility indifference values. As a by-product, we also obtain an existence result for the optimal exercise time of an American claim under utility indifference valuation by relating it to the corresponding nonlinear Snell envelope.