Study On The Pricing Volatility Derivatives And Relatded Problems

Volatility derivatives are financial products where the volatility is the main un-derlying. These products are particularly important for market investors as they use them to have insight into the level of volatility to efficiently manage the volatility risk. These products are priced often by a continuously sampled approximation on simplify the computations. This thesis presents an analytical approach to efficiently and ac-curately price discretely sampled volatility derivatives, under the stochastic volatility model.The first issue is the pricing variance swap, which is discussed in Chapter3. Pa-pers focusing on analytically pricing discretely-sampled volatility derivatives are rare in literature, mainly due to the inherent difficulty associated with the nonlinearity in the pay-off function. An accurate approximation for the characteristic function of the discretely sampled realized variance is introduced. This characteristic function is then applied to derive analytical pricing formulae for volatility derivatives, and the closed-form solutions for the pricing volatility swap, variance swap and variance derivatives are given, under the framework of Heston (1993) stochastic volatility model.The second issue, which is discussed in Chapter4, is the pricing method for volatility derivatives, based on the stochastic volatility process with an OU process. We first present an approach to solve the partial differential equation (PDE), to obtain closed form solution to price variance swap with discrete sampling times. The ap-proach is also very versatile in terms of treating the pricing problem of variance swap with different definition of discretely-sampled realized variance in a highly unified. Secondly, pricing of volatility derivatives are studied, through research characteristic function of special variable, using the method of integral transform, the pricing formu-lae of volatility derivatives are derived, under this stochastic volatility model.The third issue, which is discussed in Chapter5, is pricing derivatives, under Markov skeleton process (abbreviated MSP) framework. Using the properties of Markov skeleton process, the characteristic function of the price process is given, the pricing formulae of derivatives including variance swap, volatility swap, and convertible bonds are given, under the Markov skeleton process framework.The last issue, which is discussed in the Chapter6, is pricing of path-dependent options, where the strike time is the random variable. A series of closed-form formulas for a variety of path-dependent options (such as look back option, barrier option and accumulator) are derived, under a double exponential jump diffusion process.