Modeling and Pricing of Swaps for Stochastic Volatilities with Delay and Jumps

Abstract:     
The jumps in stock market volatility are found to be so active that this discredits many recently proposed stochastic volatility models without jumps (Bollerslev et al (2008)). The most convincing evidence comes from recent nonparametric work using high-frequency data as in Barndorff-Nielsen and Shephard (2007) and Aït-Sahalia and Jacod (2008) among others. From the other side, some statistical studies of stock prices (see Sheinkman and LeBaron (1989), and Akgiray (1989)) indicate the dependence on past returns.

In this paper, we incorporate a jump part into the stochastic volatility model with delay and without jumps proposed by Swishchuk (2005). Our model of stochastic volatility exhibits jumps and also past-dependence: the behavior of a stock price right after a given time t not only depends on the situation at t, but also on the whole past (history) of the process S(t) up to time t. This draws some similarities with fractional Brownian motion models (see Mandelbrot (1997)) due to a long-range dependence property. Another advantage of this model is mean-reversion. This model is also a continuous-time version of GARCH(1,1) model (see Bollerslev (1986)) with jumps.

The valuation of the variance swaps for stochastic volatility with delay and jumps is discussed in this paper. A variance swap is a forward contract on realized variance, the square of the realized volatility. We provide some analytical closed forms for the expectation of the realized variance for the stochastic volatility with delay and jumps. Besides, we also present a lower bound for delay as a measure of risk. We also discuss the approaches for calculating of other swaps such as volatility, covariance, correlation swaps. As applications of our analytical solutions, a numerical example using S&P60 Canada Index (1998-2002) is then provided to price variance swaps with delay and jumps.

Number of Pages in PDF File: 2

Keywords: stochastic volatility with delay and jumps, variance and volatility swaps, covariance and correlation swaps

JEL Classification: G13, C61

working papers series