Asymptotic indifference pricing in Lévy models

Peter Tankov (Université Paris Diderot)
Thursday, February 12, 2015 - 5:00pm
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Financial markets based on Lévy processes are typically incomplete and option prices depend on risk preferences of individual agents. In this context, the notion of utility indifference price has gained a certain 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 non-linear partial integro-differential equation associated to the indifference price. In this work we develop closed form approximations to exponential utility indifference prices in exponential Lévy 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 Lévy model as a perturbation of the Black-Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes [Aleš Černý, Stephan Denkl, and Jan Kallsen. Hedging in Lévy models and the time step equivalent of jumps. ArXiv, September 2013] to nonlinear and non-smooth functionals. Our closed formula represents the indifference price as a linear combination of the Black-Scholes price and correction terms which depend on tractable characteristics of the underlying Lévy process, such as skewness and kurtosis 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 the jump risk in the limit of small jump size.