Finance, Economics, and Mathematics

Chapter 5Introduction to Part II

Interest rates are a function of time and term (time to maturity). As a function of time, rates behave as stochastic processes. As a function of term, interest rates on a given date form a yield curve. Term structure models describe the behavior of interest rates of different maturities as a joint stochastic process.

Term structure models are a necessary tool for valuation and risk management of interest rate contingent claims—that is, securities or transactions whose payoff depends on future values of interest rates, such as callable or putable bonds, swaps, swaptions, caps, and floors. For instance, a bond will be called if its value on the call date is greater than the call price. To determine the current value of the bond, it is necessary to know the subsequent behavior of interest rates. The same is true for all debt securities subject to prepayment, such as mortgages with refinancing options.

The immediate acceptance and application of term structure models in banking and investment practice is due to the fact that there are few financial instruments whose value is not in some degree dependent on future interest rates. Even stock options such as calls and puts depend on the development of interest rates. Interest rate models enter into valuation of firms and their liabilities. Besides valuation, term structure models are necessary for interest rate risk measurement, management, and hedging.

Interest rates of different maturities behave ...