Computational Methods for Lévy Driven Russian Options

Jimmy Olsson

Handledare: Sören Assmusen

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2002:E26

Abstract:

In this master thesis we present some computational methods for the Russian option, where the priceprocess of the underlying stock is modeled as an exponential Lévy process with phase-type distributedjumps. In addition, we derive an analytic expression for the expected waiting time of the optimalstopping strategy of the Russian option in the case of general phase-type jumps. By using the denseness property of phase-type distributions, we obtain an approximate value of theexpected waiting time of a Russian option driven by an arbitrary Lévy process in the following way:By minimizing the distance between the first cumulants of the given Lévy model, in our case a normalinverse Gaussian Lévy model, and a phase-type Lévy model in the sense of least squares, we transferthe properties of the given Lévy model to the phase-type model. The derived expression for thewaiting time is then evaluated with the adjusted phase-type Lévy model parameters, and bycomparing the obtained value with values coming from simulation we study the efficiency of themethod.

Key words:

option pricing in mathematical finance, incomplete markets, Lévy processes, phase-type, normal inverse Gaussian (NIG) Lévy processes, reflection, expected waiting time for Russian option, the Kella-Asmussen martingale, martingale methods, cumulants, cumulant fitting, simulated NIG processes, discretization error.