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 $\mathbb{P}$, consists of a pricing kernel $\{{\pi }_t{\}}_{t\ge 0}$ together with one or more non-dividend-paying risky assets driven by the same Lévy process. If $\{S_t{\}}_{t\ge 0}$ denotes the price process of such an asset then $\{{\pi }_t{S}_t{\}}_{t\ge 0}$ is a $\mathbb{P}$-martingale. The Lévy process $\{{\xi }_t{\}}_{t\ge 0}$ is assumed to have exponential moments, implying the existence of a Lévy exponent $\psi (\alpha )=t^{-1}\log \mathbb{E}({\mathrm{e}}^{\alpha {\xi }_{\mathrm{t}}})$ 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 $\{{\zeta }_t{\}}_{t\ge 0}$ is the so-called benchmark portfolio defined by ${\zeta }_t=1/{\pi }_t$, 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 $\psi (\alpha )\to \psi (\alpha +\mu )-\psi (\mu )+c\alpha$, where $c$ and $\mu$ are constants.

International Journal of Theoretical and Applied Finance, Vol. 22, No. 1 (2019) 1950008:1-18