Introduction

One-factor interest rate models simplify the complex dynamics of interest rates by assuming a single source of uncertainty. In this chapter, we’ll delve into the concept of one-factor interest rate models, explore their significance, and identify common examples of interest rate factors. This knowledge forms the basis for comprehending the calculations and concepts discussed later in the module.

One-Factor Interest Rate Models

A one-factor interest rate model, often referred to as a single-factor model, simplifies interest rate movements by considering a single source of uncertainty. The primary assumption is that all changes in interest rates can be attributed to this single factor. The most common example of a one-factor model is the Vasicek model.

Vasicek Model Example

The Vasicek model assumes that the short-term interest rate, denoted as “r”, follows a stochastic differential equation:

$$dr=a(b-r) dt+\sigma dW$$

Where:

• $a$ is the speed of reversion
• $b$ is the long-term mean of the interest rate
• $\sigma$ is the volatility of interest rates
• $dW$ is a Wiener process (Brownian motion) In this model, interest rates tend to revert towards the long-term mean, “b”, at a rate determined by “a”. The parameter $\sigma$ represents the randomness or volatility of interest rate movements.

Examples of Interest Rate Factors

One-factor models like Vasicek capture the essence of interest rate movements driven by a single factor. Common examples of these factors include:

• Short-Term Rate: In many models, the short-term interest rate is considered the primary factor driving interest rate changes. It impacts a wide range of financial instruments, including government bonds, corporate bonds, and loans.

• Yield Curve Slope: The steepness or flatness of the yield curve, often measured as the difference between long-term and short-term interest rates, can be a significant factor in interest rate models. Changes in the yield curve slope influence various fixed-income securities.

• Economic Indicators: Macroeconomic variables, such as inflation rates, GDP growth, and central bank policies, can also be viewed as one-factor models. A change in these indicators can have a profound impact on interest rates.