Determination of the Lévy Exponent in Asset Pricing Models

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 P, consists of a pricing kernel {π_t} together with one or more non-dividend-paying risky assets driven by the same Lévy process. If {S_t} denotes the price process of such an asset then {π_t S_t} is a P-martingale. The Lévy process {ξ_t} is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α) = 1/t log E(e^{α ξ_t}) for α in an interval A ⊂ R containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is H_T = (ζ_T )^q for some time T > 0, are given at time 0 for a range of values of q, where {ζ_t} is the so-called benchmark portfolio defined by ζ_t = 1/π_t, 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 H_T = (S_T )^q 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 ψ(α) → ψ(α + μ) − ψ(μ) + cα, where c and μ are constants.