DETERMINATION OF THE LEVY EXPONENT IN ASSET PRICING MODELS

We consider the problem of determining the Levy exponent in a Levy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure P, consists of a pricing kernel {pi(t )}(t >= 0) together with one or more non-dividend-paying risky assets driven by the same Levy process. If {S-t}(t >= 0) denotes the price process of such an asset, then {pi S-t(t)}(t >= 0) is a P-martingale. The Levy process {xi(t)}(t >= 0 )is assumed to have exponential moments, implying the existence of a Levy exponent psi(alpha) = t(-1) log E(e(alpha xi t)) for alpha in an interval A subset of 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 = (zeta(T))(q) for some time T > 0, are given at time 0 for a range of values of q, where {zeta (t)}(t >= 0) is the so-called benchmark portfolio defined by zeta(t) = 1/pi (t) , then the Levy 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, H-T= (S-T)(q) for a general non-dividend-paying risky asset driven by a Levy process, and if we know that the pricing kernel is driven by the same Levy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Levy exponent up to a transformation psi(alpha) -> psi(alpha+mu) - psi(mu) + c alpha where c and mu are constants.