Pricing variance swaps under stochastic volatility with an Ornstein-Uhlenbeck process

Abstract

Pricing variance swaps under stochastic volatility has been an important subject pursued recently. Various approaches have been proposed, mainly due to the substantially increased trading activities of volatility-related derivatives in the past few years. In this note, the authors develop analytical method for pricing variance swaps under stochastic volatility with an Ornstein-Uhlenbeck (OU) process. By using Fourier transform algorithm, a closed-form solution for pricing variance swaps with stochastic volatility is obtained, and to give a comparison of fair strike value based on the discrete model, continuous model, and the Monte Carlo simulations.

An Analytical Approach for Variance Swaps With an Ornstein–uhlenbeck Process

“…At the end of this section, we will discuss some differences among the analytical pricing approaches in several recent articles. The same principles appeared in the work of Zhu and Lian [31], Little and plant [19], Jia et al [17] and our model. The first difference between them is that the models of the first two articles are based on the Heston model, which means that the volatility v t follows a noncentral chi-square distribution, while v t in the last two models is normally distributed.…”

“…The valuation problem for a variance swap is, therefore, reduced to calculating the expectation value of the future realized variance in the risk-neutral world. Jia et al [17] solved the analytical solution of the expectation with respect to (2.3), using the method of Little and Pant [19]. In this paper, we will focus on the expectation of log-return realized variance in (2.2).…”

“…On the other hand, in some formulas of our model including the derivative of C(ω, τ), D(ω, τ), E(ω, τ) such as (2.12), (2.13) and so on, theoretically there should be complex numbers, because is known that in our last formula the coefficient of i becomes 0 in the process of integrating polynomials. So, similar to the work of Zhu and Lian [32] and Jia et al [17], the parameters in our model are allowed to change randomly.…”

“…In this paper, the dimension-reduction technique is used to price variance swaps based on the mean-reverting Gaussian volatility model, that is, the Ornstein-Uhlenbeck process which was applied in the Heston model by Little and Pant [19] and Zhu and Lian [32], and was later used by Jia et al [17] to price actual-return realized variance swaps. This paper can be regarded as an extension of [17,19,32], in which the referring method is used to price log-return realized variance. In the meantime, we further clarify that this method can be used in many situations where the distribution of the variance or volatility can be calculated to price discretely sampled variance swaps.…”