Esscher transform for Lévy processes in a separable Hilbert space

by Aksel Holler Jonassen

Institution: University of Oslo
Year: 1000
Keywords: VDP::410
Record ID: 1296049
Full text PDF: https://www.duo.uio.no/handle/10852/37700



In this master’s thesis we define a pricing model for futures contracts by using Lévy processes in an infinite dimensional Hilbert space H. The elements in H are assumed to be absolutely continuous functions on the positive real line. Let L(t,⋅) be a Lévy-process in H. The price at time t, of a futures contract delivering at time t+x, is modeled by functions of the form exp(L(t,x)). By using the Itô formula for H-valued semimartingales, we derive the dynamics of exp(L(t,x)) and sufficient conditions for when the process exp(L(t,x)) is a martingale. Moreover, we define the Esscher transform for H-valued Lévy processes, and prove that processes of the form exp(L(t,x)) can be turned into martingales under relatively mild conditions on L. Furthermore, we generalize the normal inverse Gaussian (NIG) distribution to random variables in H, and derive the dynamics of H-valued NIG processes under the Esscher transform. Similarly to the one-dimensional case, the Esscher transform preserves the NIG properties of Lévy processes in H. However, for H-valued NIG processes, both the skewness and steepness parameters are modified. This differs from the one-dimensional case where only the skewness parameter changes.