In this thesis, pricing formulas for various discretely sampled variance derivatives are derived under the 3/2-model and the time-changed Lévy model. In the first part, we consider pricing of various types of exotic discrete variance swaps, like the gamma swaps and corridor swaps, under the 3/2-stochastic volatility models with jumps as this model has been shown to fit the market data better than the popular Heston model. By using the partial integro-differential equation formulation, we manage to derive quasi-closed form pricing formulas for the fair strike values of various types of discrete variance swaps. Pricing properties of these exotic discrete variance swaps under different market conditions are explored. The second part suggests a numerical method of pricing options...[ Read more ]

In this thesis, pricing formulas for various discretely sampled variance derivatives are derived under the 3/2-model and the time-changed Lévy model. In the first part, we consider pricing of various types of exotic discrete variance swaps, like the gamma swaps and corridor swaps, under the 3/2-stochastic volatility models with jumps as this model has been shown to fit the market data better than the popular Heston model. By using the partial integro-differential equation formulation, we manage to derive quasi-closed form pricing formulas for the fair strike values of various types of discrete variance swaps. Pricing properties of these exotic discrete variance swaps under different market conditions are explored. The second part suggests a numerical method of pricing options on discretely sampled variance swaps under the time-changed Lévy processes. We construct numerical algorithm that relies on the computation of the moment generating function of the realized variance under the time-changed Lévy models. By using the randomization of the Laplace transform of the discrete log return with a standard normal random variable, recursive quadrature algorithm can be derived to compute the moment generating function. The option prices are computed by inverse Laplace transform method. The fair strikes of discrete volatility swaps are also obtained by a similar method.