Title: Analytical Solutions of the SABR Stochastic Volatility Model
Description: This thesis studies a mathematical problem that arises in modeling the prices of option contracts in an important part of global financial markets, the fixed income option market. Option contracts, among other derivatives, serve an important function of transferring and managing financial risks in today's interconnected financial world. When options are traded, we need to specify what the underlying asset an option contract is written on. For example, is it an option on IBM stock or on precious metal? Is it an option on Sterling-Euro exchange rate or on US dollar interest rates? Usually option markets are organized according to their underlying assets and they can be traded either on exchanges or over-the-counter. The scope of this thesis is the option markets on currency exchange rates and interest rates, which are less familiar to the general public than those of equities and commodities, and are mostly traded over-the-counter as bi-lateral agreements among large financial institutions such as investment banks, central banks, commercial banks, government agencies, and large corporations.

Since early 1970's, the Black-Scholes-Merton option model has become the market standard of buying and selling standard option contracts of European style, namely calls and puts. Of particular importance is this ever more quantitative approach to the practice of option trading, in which the volatility parameter of the Black-Scholes-Merton's model has become the market "language'' of quoting option prices. Despite its tremendous success, the Black-Scholes-Merton model has exhibited a few well-known deficiencies, the most important of which are first, the assumption that the underlying asset is lognormally distributed and second, the volatility of the underlying asset's return is constant. In reality, the return distribution of an underlying asset can exhibit various level of tail behavior, ranging from "sub''-normal to normal, from lognormal to "super''-lognormal. Also the implied volatilities of liquidly traded options generally vary with both option strikes and option maturity. This variation with strike is termed the "volatility skew'' or the "volatility smile''.

Naturally as market evolves, so does the model. People then start to look for the new standard. Among various successful extensions, models with constant elasticity of variance (CEV) prove to be able to generate enough range of return distributions while models with volatility itself being stochastic start to become popular in terms fitting the "smile'' or "skew'' phenomenon of option implied volatilities. In 2002, the combination of CEV model with stochastic volatility, particulary the SABR model, became the new market standard in fixed income option market. This is the starting point of this thesis. However, being the market standard also poses new challenges, which are speed and accuracy. Three mathematical aspects of the model prevent one from obtaining a strictly speaking closed form solution of its joint transition density, namely the nonlinearity from the CEV type local volatility function, the coupling between the underlying asset process and the volatility process, and finally the correlation between the two driving Brownian motions.

We look at the problem from a PDE perspective where the joint transition density follows a linear second order equation of parabolic type in non-divergence form with coordinate-dependent coefficients. Particularly, we construct an expansion of the joint density through a hierarchy of parabolic equations after applying a financially justified scaling and a series of well designed transformations. We then derive accurate asymptotic formulas in both free-boundary conditions and absorbing-boundary conditions. We further establish an existence result to characterize the truncation error and examined extensively the derived formulas through various numerical examples. Finally we go back to the fixed income market itself and use our result to examine empirically whether today's option prices traded at different expiries contain information on predicting future levels of option prices, using ten-year over-the-counter FX option data from a major investment bank dealer desk. Our theoretical results for the joint density of the SABR model serve as a basis for banks and dealers to manage the forward smile risk of their fixed income option portfolio.

Our empirical studies extend the forward concept from interest rate term structure modeling to interest rate volatility term structure modeling and examine the relationship between today's forward implied volatility and future spot implied volatility.