Forward indifference valuation of American options

Abstract

We analyse the valuation of American options under the forward performance criterion introduced by Musiela and Zariphopoulou [Quant. Finance 9 (2008), pp. 161–170]. In this framework, the performance criterion evolves forward in time without reference to a specific future time horizon, and may depend on the stochastic market conditions. We examine two applications: the valuation of American options with stochastic volatility and the modelling of early exercises of American-style employee stock options. We work with the assumption that forward indifference prices have sufficient regularity to be solutions of variational inequalities, and provide a comparative analysis between the classical and forward indifference valuation approaches. In the case of exponential forward performance, we derive a duality formula for the forward indifference price. Furthermore, we study the marginal forward performance price, which is related to the classical marginal utility price introduced by Davis (Mathematics of Derivatives Securities, Cambridge University Press, 1997, pp. 227–254). We prove that, under arbitrary time-monotone forward performance criteria, the marginal forward indifference price of any claim is always independent of the investor's wealth and is represented as the expected discounted pay-off under the minimal martingale measure.