Utility indifference valuation for non-smooth payoffs with an application to power derivatives

TL;DR

This paper develops a framework for utility indifference valuation of non-smooth payoffs involving both traded and nontraded assets, with applications to power derivatives, using BSDEs and PDE techniques.

Contribution

It introduces a BSDE-based characterization of the utility indifference price for complex payoffs and applies it to energy market derivatives, including asymptotic expansions for small risk aversion.

Findings

Characterizes UIP as a viscosity solution of a PDE.

Provides regularity results for the hedging strategy Z.

Applies the framework to power derivatives in energy markets.

Abstract

We consider the problem of exponential utility indifference valuation under the simplified framework where traded and nontraded assets are uncorrelated but where the claim to be priced possibly depends on both. Traded asset prices follow a multivariate Black and Scholes model, while nontraded asset prices evolve as generalized Ornstein-Uhlenbeck processes. We provide a BSDE characterization of the utility indifference price (UIP) for a large class of non-smooth, possibly unbounded, payoffs depending simultaneously on both classes of assets. Focusing then on European claims and using the Gaussian structure of the model allows us to employ some BSDE techniques (in particular, a Malliavin-type representation theorem due to Ma (2002)) to prove the regularity of Z and to characterize the UIP for possibly discontinuous European payoffs as a viscosity solution of a suitable PDE with continuous space derivatives. The optimal hedging strategy is also identified essentially as the delta hedging strategy corresponding to the UIP. Since there are no closed-form formulas in general, we also obtain asymptotic expansions for prices and hedging strategies when the risk aversion parameter is small. Finally, our results are applied to pricing and hedging power derivatives in various structural models for energy markets.