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 } t≥0 together with one or more non-dividend-paying risky assets driven by the same Lévy process. If {S t } t≥0 denotes the price process of such an asset then {π t S t } t≥0 is a P-martingale. The Lévy process {ξ t } t≥0 is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α) = t −1 log E(e αξt ) 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 } t≥0 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.

We present an overview of the broad class of financial models in which the prices of assets are Lévy-Ito processes driven by an n-dimensional Brownian motion and an independent Poisson random measure. The Poisson random measure is associated with an n-dimensional Lévy process. Each model consists of a pricing kernel, a money market account, and one or more risky assets. We show how the excess rate of return above the interest rate can be calculated for risky assets in such models, thus showing the relationship between risk and return when asset prices have jumps. The framework is applied to a variety of asset classes, allowing one to construct new models as well as interesting generalizations of familiar models.

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We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a Lévy-Ito market, where assets are driven jointly by an n-dimensional Brownian motion and an independent Poisson random measure on an n-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of an expected least squared-error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.

We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a Lévy-Ito market, where assets are driven jointly by an n-dimensional Brownian motion and an independent Poisson random measure on an n-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of an least expected squared error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.