Analytical Solutions of the SABR Stochastic Volatility Model

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 b