Asymptotic indifference pricing in exponential L\'evy models

Abstract

Financial markets based on L\'evy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the nonlinear partial integro-differential equation associated to the indifference price. In this work, we develop closed form approximations to exponential utility indifference prices in exponential L\'evy models. To this end, we first establish a new non-asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closed-form approximation of the indifference price by treating the L\'evy model as a perturbation of the Black-Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of L\'evy processes (A. \v{C}ern\'y, S. Denkl and J. Kallsen, arXiv:1309.7833) to nonlinear and non-smooth functionals. Our closed formula represents the indifference price as the linear combination of the Black-Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying L\'evy process, and the derivatives of the Black-Scholes price. As a by-product, we obtain a simple explicit formula for the spread between the buyer's and the seller's indifference price. This formula allows to quantify, in a model-independent fashion, how sensitive a given product is to jump risk in the limit of small jump size.