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 $\mathbb P$, consists of a pricing kernel $\{π_t\}_{t\geq0}$ together with one or more non-dividend-paying risky assets driven by the same Lévy process. If $\{S_t\}_{t\geq0}$ denotes the price process of such an asset then $\{π_t S_t\}_{t\geq0}$ is a $\mathbb P$-martingale. The Lévy process $\{ ξ_t \}_{t\geq0}$ is assumed to have exponential moments, implying the existence of a Lévy exponent $ψ(α) = t^{-1}\log \mathbb E(\rm e^{αξ_t})$ for $α$ in an interval $A \subset \mathbb R$ containing the origin as a proper subset. We show that if the initial prices of power-payoff derivatives, for which the payoff is $H_T = (ζ_T)^q$ for some time $T>0$, are given for a range of values of $q$, where $\{ζ_t\}_{t\geq0}$ 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 $ψ(α) \rightarrow ψ(α+ μ) - ψ(μ) + c α$, where $c$ and $μ$ are constants.