DETERMINATION OF THE LÉVY EXPONENT IN ASSET PRICING MODELS

George Bouzianis and Lane P. Hughston ()
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George Bouzianis: Department of Computing, Goldsmiths College, University of London, New Cross, London SE14 6NW, UK
Lane P. Hughston: Department of Computing, Goldsmiths College, University of London, New Cross, London SE14 6NW, UK

International Journal of Theoretical and Applied Finance (IJTAF), 2019, vol. 22, issue 01, 1-18

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 ℙ, consists of a pricing kernel {πt}t≥0 together with one or more non-dividend-paying risky assets driven by the same Lévy process. If {St}t≥0 denotes the price process of such an asset, then {πtSt}t≥0 is a ℙ-martingale. The Lévy process {ξt}t≥0 is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α) = t−1log𝔼(eαξt) 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 HT = (ζT)q for some time T > 0, are given at time 0 for a range of values of q, where {ζt}t≥0 is the so-called benchmark portfolio defined by ζt = 1/πt, 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 HT = (ST)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.

Keywords: Asset pricing; Lévy models; Lévy processes; Lévy exponent; exponential moments; option pricing; option replication; power payoffs (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (3)

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DOI: 10.1142/S0219024919500080

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