In this work we apply asymptotic analysis on compound options, American options, Asian options, and variance (or volatility) contracts in the context of stochastic volatility models. Singular perturbations are used mainly. A singular-regular perturbation is applied on Asian option problems. Epsilon-Martingale decompositions are employed to the pricing and hedging of volatility contracts. Firstly, we begin by presenting some applicable concepts in probability theory, stochastic differential equations, and the risk-neutral evaluation for pricing derivatives. Stochastic volatility models are introduced. A statistical tool, variogram analysis, is used to justify time-scale factors in mean reverting stochastic volatility models. The effect of jumps in addition to diffusion models is also analyzed. A brief review is presented to the application of singular and regular perturbation techniques for pricing the European options, proposed by Fouque-Papanicolaou-Sircar-Solna cite [FPSS, FPSSmultiscale], in the context stochastic volatility environment. Secondly, we apply the singular perturbation to evaluate options defined on non-smooth payoffs, which may include unobservable volatilities. A special case, namely a European-type compound option, is considered. We then consider an approximation for American options and propose proxies for the "implied American volatility." It is useful when the market is lack of European options. Thirdly, the pricing problem for arithmetic-average Asian options with the stochastic volatility is considered. We utilize a dimensional reduction technique to deduce two one-dimensional pricing partial differential equations cite [FH], in contrast to the usual two two-dimensional PDEs cite [FPS]. In addition, a singular-regular perturbation is preformed to deal with the fast and slow volatility factors. Lastly, the pricing and hedging of variance or volatility contracts are considered. We use the Epsilon-Martingale decomposition cite [FPSeps] to deal with these problems in a unified way. A case study for the corridor swap cite [CarrLewis] is presented. In particular, the local time and occupation time appear in our analysis. This is duce to the discontinuity in the payoff of contracts. The conclusions and future work are described in the end.
