Variance swaps with double exponential Ornstein-Uhlenbeck stochastic volatility

Abstract

Recent empirical studies find that with a high sampling frequency, stochastic volatility models can be differentiated from each other and the log-normal stochastic volatility model can capture the volatility of real market better than the Heston model. Unless certain restriction such as fast mean reversion or zero mean reversion is assumed, however, the log-normal stochastic volatility model does not allow an analytic solution for the characteristic function. When we consider derivative products such as variance swaps, the restriction may cause a significant gap between theoretical price and actual price. On the other hand, there are a good amount of studies showing that a multi-factor stochastic volatility model can capture the market more accurately than a single-factor stochastic volatility model. So, in this paper, we propose a two-factor log normal stochastic volatility model in which log volatility is given by an Ornstein-Uhlenbeck process which is a mean-reverting, temporally homogeneous, Markov Gaussian process. We derive an exact semi-analytic solution as well as an approximate analytic solution for the fair strike price of discretely sampled variance swap by using the first and second moments of the moment generating function. We check the validity of our solution through Monte Carlo simulation and show how to estimate the model parameters by implementing calibration to real market data and compare our two-factor model with the one-factor version and also the Heston and double Heston models. Furthermore, the sample discretization risk is examined for the four different models.