Pricing variance swaps under stochastic volatility and stochastic interest rate

In this thesis, we study the issue of pricing discretely-sampled variance swaps under stochastic volatility and stochastic interest rate. In particular, our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, whereas the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) model. We first extend the framework of [119] by incorporating the CIR interest rate into their Heston model for pricing discretely-sampled variance swaps. We impose partial correlation between the asset price and the volatility, and derive a semiclosed form pricing formula for the fair delivery price of a variance swap. Several numerical examples and comparisons are provided to validate our pricing formula, as well as to show the effect of stochastic interest rate on the pricing of variance swaps. In addition, the pricing of discretely-sampled variance swaps with full correlation among the asset price, interest rate as well as the volatility is investigated. This offers a more realistic model with practical importance for pricing and hedging. Since this full correlation model is incompliant with the analytical tractability property, we determine the approximations for the non-affine terms by following the approach in [55] and present a semiclosed form approximation formula for the fair delivery price of a variance swap. Our results confirm that the impact of the correlation between the stock price and the interest rate on variance swaps prices is very crucial. Besides that, the impact of correlation coefficients becomes less apparent as the number of sampling frequencies increases for all cases. Finally, the issue of pricing discretely-sampled variance swaps under stochastic volatility and stochastic interest rate with regime switching is also discussed. This model is an extension of the corresponding one in [34] and is capable of capturing several macroeconomic issues such as alternating business cycles. Our semi-closed form pricing formula is proven to achieve almost the same accuracy in far less time compared with the Monte Carlo simulation. Through numerical examples, we discover that prices of variance swaps obtained from the regime switching Heston-CIR model are significantly lower than those of its non-regime switching counterparts. Furthermore, when allowing the Heston-CIR model to switch across three regimes, it is observable that the price of a variance swap is cheapest in the best economy, and most expensive in the worst economy among all.