What is the commonly used mathematical tool for describing the movement of interest rates that can incorporate the properties of an interest-rate model?

What will be an ideal response?

The commonly used mathematical tool for describing the movement of interest rates (that can incorporate the properties of drift, volatility and mean reversion) is stochastic differential equations (SDEs). A rigorous treatment of interest-rate modeling requires an understanding of this specialized topic in mathematics. It is also worth noting that SDEs are used in the pricing of options. More details are given below.

The most common interest-rate model used to describe the behavior of interest rates assumes that short-term interest rates follow some statistical process and that other interest rates in the term structure are related to short-term rates. The short-term interest rate (i.e., short rate) is the only one that is assumed to drive the rates of all other maturities. Hence, these models are referred to as one-factor models. The other rates are not randomly determined once the short rate is specified. Using arbitrage arguments, the rate for all other maturities is determined.

There are also multi-factor models that have been proposed in the literature. The most common multi-factor model is a two-factor model where a long-term rate is the second factor. In practice, however, one-factor models are used because of the difficulty of applying a multi-factor models. The high correlation between rate changes for different maturities provides some support for the use of a one-factor model as well as empirical evidence that supports the position that a level shift in interest rates accounts for the major portion of the change in the yield curve.

Although the value of the short rate at some future time is uncertain, the pattern by which it changes over time can be assumed. In statistical terminology, this pattern or behavior is called a stochastic process. Thus, describing the dynamics of the short rate means specifying the stochastic process that describes the movement of the short rate. It is assumed that the short rate is a continuous random variable and therefore the stochastic process used is a continuous-time stochastic process. There are different types of continuous-time stochastic processes. In all of these models because time is a continuous variable, the letter d is used to denote the "change in" some variable. Terms used in these models are defined below.

r = the short rate and therefore dr denotes the change in the short rate
t = time and therefore dt denotes the change in time or equivalently the length of the time interval (dt is a very small interval of time)
z = a random term and dz denotes a random process