Random fields and interest-rate models: applications to pricing of interest-rate derivatives

Borrowing money comes at a cost and the charge that must be paid is called an interest rates. Interest rates are affected by both micro and macroeconomic factors; therefore, its fluctuations are difficult to predict. Unexpected changes in interest rates can amount to significant increases in borrowing costs and diminish investment returns. Therefore, investors rely on interest-rate derivatives to manage interest-rate risk.

A derivative is a financial product whose value is derived based on some underlying asset. In this thesis we focus on a particular class of derivatives called interest-rate derivatives. An interest-rate derivative is a useful financial tool that can be utilized as an insurance to manage interest-rate risks. However, a premium needs to paid, just like for any other insurance, to purchase interest-rate derivatives. Pricing interest-rate derivatives is a complex process and rely on tools from stochastic calculus and probability due to the random nature of interest-rate fluctuations.

We develop a model to price cross-currency interest-rate derivatives. In particular, we will construct a multi-economy market model which models interest rates and exchange rates of multiple economies. While there are existing models to price such derivatives, we take a more general approach by utilizing a random field to drive the volatility of our model. One of the key elements of derivative pricing models is that they should be arbitrage-free. In particular, it should not enable any opportunities to make profits without any undertaking any risk. We derive conditions to ensure arbitrage-free pricing of cross-currency interest rate derivatives.

Deriving exact formulas for cross-currency interest-rate derivatives are complex as models cannot focus only on interest rates but also need to factor in the effect of exchange rate fluctuations between countries. We develop exact and approximate pricing formulas for a variety of cross-currency derivatives using the random field model derived in our work by modeling the changes of interest rates and exchange rates.