| Option pricing is an active area in financial industry. The development of option pricing problems has resulted in the employment of many mathematical techniques. I apply the ray method of geometrical optics and matched asymptotic expansions to some option pricing models. This includes the constant elasticity of variance (CEV) process and the Black-Scholes (BS) model.;We consider the CEV model with knock-out barriers, in the limit of small volatility. We show that the asymptotic structure of the CEV model can lead to caustics, caustic boundaries and interesting "corner" and "transition" layers, where the solutions are expressed in terms of Airy functions.;We also examine American up-and-out put barrier options and American floating strike lookback put options under the BS model, which correspond to moving boundary problems for partial differential equations (PDEs). We consider the PDEs in the limit of a low interest rate and/or large volatility. This limit is particularly relevant to today's financial crisis. We show the importance of separately analyzing the different space-time scales. |
