Asymptotic methods applied to finance: Equity and volatility derivatives

There has been a growing interest in more sophisticated models for pricing derivatives that go beyond the Black-Scholes-Merton model to better account for real-world market behavior. Pricing of equity and volatility derivatives are studied when the volatility of the underlying asset is no longer constant but may be deterministic or stochastic. The Constant Elasticity of Variance (CEV) model is studied in which the local volatility is deterministic function of the underlying asset. Simple exact and asymptotic pricing formulas for derivatives, such as the variance swap, log contract and European options are constructed using risk-neutral pricing. The probability density function of the asset price is needed and constructed exactly and approximately using the asymptotic ray method near expiration. A detailed boundary layer analysis is needed to take into account the possibility of default. The stochastic CEV or Stochastic Alpha Beta Rho (SABR) model is also studied in which the underlying asset follows the CEV process but with stochastic volatility. The ray method is extended to the SABR model to derive approximations to the density function of the asset price near expiration and used to price various derivatives. A detailed boundary layer analysis is presented leading to a more complete understanding of the SABR model. The results for both the CEV and SABR models are shown to be accurate by comparing with known exact results or numerical computations. The simple approximate pricing formulas for theses models provide a useful balance between mathematical theory and practical implementations given that short term expiration for equity and volatility contracts are in general the most liquid contracts.