The Analytical Pricing Of Variance Swap Under Stochastic Volatility Models

As the demand for risk control and hedge rising continuously in the financial market, volatility trading becomes increasingly important. Volatility derivatives which are new type of financial derivatives make direct trading of volatility possible. Since the risk exposure of volatil-ity can be efficiently provided, volatility and variance swaps are among the most active trading products. This thesis mainly studies the pricing issue of variance swaps which are forward contracts in essence. The existing researches are primarily to obtain simple pricing formulas under the situation of continuous sampling, but this will not generally be the case. Focusing on discretely sampling, this thesis mainly aims to obtain analytical solutions of variance swap-s pricing through the application of a new method under several different kinds of stochastic volatility models. There are five parts in this thesis:In the first part, we introduce the background and research significance of volatility deriva-tives, explain the essence of variance swap and it's trading conditions, summarize the previous pricing solutions and introduce the main content of this thesis.In the second part, basic knowledge of stochastic analysis is introduced, including some properties of Ito integrals, the concept of Ito process and Ito formula, Feynman-Kac formula, the Martingale representation theorem, the Girsanov theorem and compound Poisson process.In the third part, we mainly focus on the pricing problems of variance swaps under the Heston model and a kind of generalized model. We firstly introduce the Heston model, prove the transformation between physical probability measure and risk neutral probability measure, show the new expression of Heston model under the risk neutral probability measure, and obtain pricing formulas under both continuous and discrete sampling situations through the existing methods. Then we propose a new kind of method which doesn't depend on the specific distri-bution of volatility and is more suitable for different stochastic volatility models. By solving the associated PDE, we prove the analytical pricing formula. Finally we propose a general-ized model of Heston model by introducing an elastic parameter and prove the fair strike price under continuous-sampling-time situation. When the sampling time is discrete, we obtain an approximate analytical pricing formula by means of the Taylor expansion.In the fourth part, we propose a stochastic volatility model called the generalized double exponential jump-diffusion model on the basis of Heston model. This volatility model can efficiently reflect the leptokurtic, fat tailed and left skewed characteristics of logarithmic return in the real financial market. According to this model, we prove analytical pricing formulas of variance swap under both continuous and discrete sampling situations by the new method proposed in the third part. Besides, we also explain the relationship between the analytical pricing formulas and the associated pricing formulas obtained according to the Heston model.In the fifth part, using pricing formulas of variance swap under different models introduced in the previous sections, we make detailed numerical tests to analyze the changes of pricing result with respect to the model parameters, the sensitivity of strike price and the influence on pricing differences between the continuous and discrete sampling exerted by each parameter. In the end, by means of Monte Carlo simulations and empirical analysis, we show the reasonability of different models and the correctness of associated analytical pricing formulas.