International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 435145, 27 pages
doi:10.1155/2011/435145
Research Article

Pricing Variance Swaps for Stochastic Volatilities with Delay and Jumps

Anatoliy Swishchuk1 and Li Xu2

1Mathematical and Computational Finance Laboratory, Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada
2H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, GA 30332, USA

Received 16 December 2010; Revised 9 March 2011; Accepted 16 March 2011

Academic Editor: Kambiz Farahmand

Copyright © 2011 Anatoliy Swishchuk and Li Xu. 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 study the valuation of the variance swaps under stochastic volatility with delay and jumps. In our model, the volatility of the underlying stock price process not only incorporates jumps, which are found to be active empirically, but also exhibits past dependence: the behavior of a stock price right after a given time 𝑡 depends not only on the situation at 𝑡 but also on the whole past (history) of the process 𝑆 ( 𝑡 ) up to time 𝑡 as well. The jump part in our model is finally represented by a general version of compound Poisson processes. We provide some analytical closed forms for the expectation of the realized variance for the stochastic volatility with delay and jumps. We also present a lower bound for delay as a measure of risk. As applications of our analytical solutions, a numerical example using S&P60 Canada Index (1998–2002) is then provided to price variance swaps.