Determination of the L\'evy Exponent in Asset Pricing Models

We consider the problem of determining the L\'evy exponent in a L\'evy 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 $\{\pi_t\}_{t\geq0}$ together with one or more non-dividend-paying risky assets driven by the same L\'evy process. If $\{S_t\}_{t\geq0}$ denotes the price process of such an asset then $\{\pi_t S_t\}_{t\geq0}$ is a $\mathbb P$-martingale. The L\'evy process $\{ \xi_t \}_{t\geq0}$ is assumed to have exponential moments, implying the existence of a L\'evy exponent $\psi(\alpha) = t^{-1}\log \mathbb E(\rm e^{\alpha \xi_t})$ for $\alpha$ 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 $H_T = (\zeta_T)^q$ for some time $T>0$, are given for a range of values of $q$, where $\{\zeta_t\}_{t\geq0}$ is the so-called benchmark portfolio defined by $\zeta_t = 1/\pi_t$, then the L\'evy 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 L\'evy process, and if we know that the pricing kernel is driven by the same L\'evy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the L\'evy exponent up to a transformation $\psi(\alpha) \rightarrow \psi(\alpha + \mu) - \psi(\mu) + c \alpha$, where $c$ and $\mu$ are constants.