Determination of the Lévy Exponent in Asset Pricing ModelsTools Bouzianis, George and Hughston, Lane P.. 2019. Determination of the Lévy Exponent in Asset Pricing Models. International Journal of Theoretical and Applied Finance, 22(1), p. 1950008. ISSN 02190249 [Article]
Official URL: https://www.worldscientific.com/doi/abs/10.1142/S0...
Abstract or DescriptionWe 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 realworld measure P, consists of a pricing kernel {π_t} together with one or more nondividendpaying risky assets driven by the same Lévy process. If {S_t} denotes the price process of such an asset then {π_t S_t} is a Pmartingale. The Lévy process {ξ_t} is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α) = 1/t log E(e^{α ξ_t}) for α in an interval A ⊂ R containing the origin as a proper subset. We show that if the prices of powerpayoff 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} is the socalled 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 nondividendpaying 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 powerpayoff 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.
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