Pricing Discretely Sampled Variance Swaps Under Stochastic Volatility And Stochastic Interest Rates

A variance swap is a financial derivative arose in the 1990s,whose value depends on the underlying assets volatility level in the future.A variance swap is essentially a forward contract on annualized variance.The long position of a variance swap pays a fixed delivery price at expiration and receives the floating amounts of annualized realized variance.The short position is just the opposite.It offers a more direct and more pure risk exposure of the underlying asset.Variance swaps,therefore,are not only used as an important way to gain the benefits,but also as an effective tool to hedge volatility risk.In the financial markets,the calculation of the realized variance of a swap variance should be based on discrete sampling times,however,in order to avoid the difficulty of the discrete sampling in mathematics calculation,most of the pricing methods are based on discrete sampling times,so the resulting price will inevitably lead to error.Even though some pricing methods are based on discrete sampling times,the stochastic volatility models are also used under the stochastic interest rate.This thesis price discretely sampled variance swaps under the mixed model(MRG-Vasicek model)of mean reversion Gaussian volatility model and Vasieek stochastic interest rate model.We first present an closed-form pricing formula for discretely sampled proportional variance swaps under the above MRG-Vasicek model.Then compared with the Monte Carlo(MC)simulations and the continuously sampled realized variance,our closed-form formula shows substantial advantage,in terms of both accuracy and efficiency.We also present an closed-form pricing formula for discretely sampled logarithmic variance swaps under the above MRG-Vasicek model.Then,assuming that the MRG-Vasicek model with full correlation,a semi-closed form pricing formula is obtained.Finally,the pricing problems of proportional variance swaps with discrete sampling times are studied,where the stochastic volatility and the stochastic intorest rate are both with regime switching,a semi-closed form pricing formula under the MRG-Vasicek model with regime switching is obtained.