An analytic solution and an approximate solution for log‐return variance swaps under double‐mean‐reverting volatility

Abstract

Variance swaps is a kind of financial instrument that plays an important role in volatility risk management. In this paper, we study the pricing problem of log‐return variance swaps under the double mean reversion DMR (Heston‐CIR) model. Compared with Kim's work, we introduce the square‐root process into the diffusion term of the long‐term mean and present a stochastic approach that greatly simplify the solution of the problem without solving PDEs. An analytical solution and approximate solution are obtained. Some numerical examples show that the exact solution and MC simulation fit well. It is worth mentioning that the difference between the approximate solution and the exact solution is small when the parameters are selected appropriately. By the mean time, the parameter of the long‐term mean has an important impact on the solution, which implies that the introduction of a multi‐factor model is necessary.