Singular Perturbations on Non-smooth Boundary Problems in Finance

Han, Chuan-Hsiang. Singular Perturbations on Non-Smooth Boundary Problems in Finance. (Under the direction of Jean-Pierre Fouque.) 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 perturbation techniques are primarily used. A singular-regular perturbation technique is applied on Asian option problems. Epsilon-Martingale decompositions are employed to the pricing and hedging of volatility contracts. First, 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 for the application of singular and regular perturbation techniques for pricing the European options, proposed by Fouque-Papanicolaou-Sircar-Solna [16, 18], in the context of stochastic volatility environment. Second, we apply the singular perturbation technique to evaluate option prices 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 lacking European options. Third, the pricing problem for arithmetic-average Asian options with stochastic volatility models is considered. We utilize a dimensional reduction technique to deduce two one-dimensional pricing partial differential equations [12], in contrast to the usual two two-dimensional PDEs [13]. In addition, a singular-regular perturbation is performed to deal with the fast and slow volatility factors. Last, the pricing and hedging of variance or volatility contracts are considered. We use the Epsilon-Martingale decomposition [14] to deal with these problems in a unified way. A case study for the corridor swap [7] is presented. In particular, the local and occupation times appear in our analysis. This is due to the discontinuity in the payoff of contracts. The conclusions and future work are described in the end.