Exponential Hedging with Optimal Stopping and Application to ESO Valuation

Abstract

We study the problem of hedging early exercise (American) options with respect to exponential utility within a general incomplete market model. This leads us to construct a duality formula involving relative entropy minimization and optimal stopping. We further consider claims with multiple exercises, and static-dynamic hedges of American claims with other European and American options. The problem is important for accurate valuation of employee Stock Options (ESOs), and we demonstrate this in a standard diffusion model. We find that incorporating static hedges with market-traded options induces the holder to delay exercises, and increases the ESO cost to the firm.

Keywords: Utility maximization, optimal stopping, employee stock options, static hedging, dynamic hedging, financial mathematics, utility indifference pricing, American options

JEL Classification: M41, M44, J33, G13, G40, B28, B70, E20