Dyson type formula for pure jump Lévy processes with some applications to finance

Sixian Jin, Henry Schellhorn and Josep Vives

Stochastic Processes and their Applications, 2020, vol. 130, issue 2, 824-844

Abstract: In this paper we obtain a Dyson type formula for integrable functionals of a pure jump Lévy process. We represent the conditional expectation of a FT-measurable random variable F at a time t≤T as an exponential formula involving Malliavin derivatives evaluated along a frozen path. The series representation of this exponential formula turns out to be useful for different applications, and in particular in quantitative finance, as we show in the following examples: the first one is the pricing of options in the Poisson–Black–Scholes model; the second one is the pricing of discount bonds in the Lévy quadratic model. We also obtain, for the conditional expectation of a function of a finite number of the process values, a backward Taylor expansion, that turns out to be useful for numerical calculations.

Keywords: Lévy processes; Malliavin calculus; Clark–Ocone formula; Dyson type formula; Backward Taylor expansion (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2019.03.019

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