Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 537571, 17 pages
doi:10.1155/2010/537571
Research Article

Modeling and Pricing of Variance and Volatility Swaps for Local Semi-Markov Volatilities in Financial Engineering

Anatoliy Swishchuk1 and Raimondo Manca2

1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada
2Department of Mathematics, University of Rome ‘‘La Sapienza’’, Via del Castro, Laurenziano 9, 00161 Rome, Italy

Received 28 July 2010; Accepted 14 October 2010

Academic Editor: G. Rega

Copyright © 2010 Anatoliy Swishchuk and Raimondo Manca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a semi-Markov modulated security market consisting of a riskless asset or bond with constant interest rate and risky asset or stock, whose dynamics follow gemoetric Brownian motion with volatility that depends on semi-Markov process. Two cases for semi-Markov volatilities are studied: local current and local semi-Markov volatilities. Using the martingale characterization of semi-Markov processes, we find the minimal martingale measure for this incomplete market. Then we model and price variance and volatility swaps for local semi-Markov stochastic volatilities.