Stochastic Volatility: Modeling and Asymptotic Approaches to Option Pricing and Portfolio Selection

Asymptotic methods can be used to analyze and simplify pricing and portfolio optimization problems. Broadly speaking, there are two methods of setting up asymptotic expansions for option pricing and implied volatility. In contract asymptotics, one considers extreme regimes specific to the option contract. In the model asymptotics approach, one views the complicated incomplete market model as a perturbation around a more tractable model, often the Black-Scholes model. This chapter concentrates on model asymptotics as they are adaptable to other option contracts and, moreover, are amenable to nonlinear portfolio optimization problems. Model coefficient polynomial expansions can be used to find closed-form asymptotic approximations for option prices and implied volatilities in a general d-dimensional Markov setting. The chapter focuses on two simple cases, namely general two-dimensional local-stochastic volatility models, and general scalar Levy-type models. It also presents some new approximations for the Merton problem with stochastic volatility.