Computational Methods for Lévy Driven Russian Options

Jimmy Olsson

Handledare: Sören Assmusen

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

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.