DETERMINATION OF THE LÉVY EXPONENT IN ASSET PRICING MODELS

Abstract

We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure [Formula: see text], consists of a pricing kernel [Formula: see text] together with one or more non-dividend-paying risky assets driven by the same Lévy process. If [Formula: see text] denotes the price process of such an asset, then [Formula: see text] is a [Formula: see text]-martingale. The Lévy process [Formula: see text] is assumed to have exponential moments, implying the existence of a Lévy exponent [Formula: see text] for [Formula: see text] in an interval [Formula: see text] containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is [Formula: see text] for some time [Formula: see text], are given at time [Formula: see text] for a range of values of [Formula: see text], where [Formula: see text] is the so-called benchmark portfolio defined by [Formula: see text], then the Lévy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if [Formula: see text] for a general non-dividend-paying risky asset driven by a Lévy process, and if we know that the pricing kernel is driven by the same Lévy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Lévy exponent up to a transformation [Formula: see text], where [Formula: see text] and [Formula: see text] are constants.