# Short Rate Model

A mathematical model used in the evaluation of interest rate derivatives

## What is a Short Rate Model?

A short rate model is a mathematical model used in the evaluation of interest rate derivatives to illustrate the evolution of interest rates over time by determining the evolution of the short rate r(t) over time.

In short rate modeling, the stochastic state variable is the spot rate at a particular time (instantaneous interest rate). Therefore, the short rate r(t) is the continuously compounded, annualized interest rate at which money can be borrowed by an entity for an infinitesimally short period of time on the yield curve.

### Summary

• Short rate models are mathematical models used in the evaluation of interest rate derivatives to illustrate the evolution of interest rates over time by identifying the evolution of the short rate r(t) over time.
• The purpose of short rate modeling is to price interest rate derivatives.
• One-factor short rate models work under the assumption that the future evolution of the interest rates is dependent on only one stochastic factor. Multiple-factor short rate models work under the assumption that more than one stochastic factor affect the future evolution of the interest rates.

### Short Rate Model Formula Where:

• dr is the change in interest rates
• dt is the time interval
• σ is the variance of the rate changes
• dZ is the random variable
• r is the risk-less interest rate

### Importance of Short Rate Modeling

Short rate models are usually used in the evaluation of interest rate derivatives. They are particularly used to price mortgages, credit instruments, bonds, and other derivatives that are sensitive to changes in interest rates.

### One-Factor Short Rate Models vs. Multi-Factor Short Rate Models

Generally, interest rate modeling is seen to be quite complex because interest rates are affected by a number of factors that cause uncertainty in an interest rate model. Some of these factors include political decisions, economic state, laws of demand and supply, government intervention, etc. As a result, different interest rate models account for different characteristics of interest rates.

#### One-factor short rate models

One-factor short rate models work under the assumption that the future evolution of the interest rates is dependent on only one stochastic factor. Although unrealistic, the models provide good estimates of the term structure of interest rates if the different factors that affect the interest rates are highly interrelated.

The following are some of the common one-factor short rate models:

• Merton’s model
• Vasicek model
• Rendleman-Bartter model
• Cox-Ingersoll-Ross model
• Ho-Lee model
• Hull-While model
• Black-Derman-Toy model
• Black-Karasinski model
• Kalotay-Williams-Fabozzi model

#### Multiple-factor short rate models

Multiple-factor short rate models work under the assumption that more than one stochastic factor affect the future evolution of the interest rates. The accuracy of the models increases as more factors are incorporated. Multiple-factor models are typically very complex; to solve them, numerical optimization methods are required.

Two-factor short rate models, such as the Longstaff–Schwartz model, include two sources of uncertainty. The Chen model is an example of a three-factor short rate model. For the purpose of risk management, multi-factor short rate models are at times favored over one-factor short rate models. It is because they produce scenarios that are generally more consistent with the actual movements of the yield curve, thus creating realistic interest rate simulations.

### Types of Short Rate Models

Short rate models come in two types:

#### 1. Arbitrage-free short rate models

Arbitrage-free models (also known as no-arbitrage models) are short rate models that use real market data to estimate the actual short rate generating process. One instrument is priced by relating it to the prices of other instruments.

Arbitrage-free models make the assumption that market prices of underlying securities are accurate. The models are not appropriate for relative analysis where prices of securities are compared, and during shocks or liquidation, when a large surplus of securities is issued.

#### 2. Equilibrium short rate models

Equilibrium short rate models represent a balance of supply and demand. Certain assumptions about the actual short rate generating process are made to estimate the right theoretical term structure. The models can be used for relative analysis to compare the value of two securities. With equilibrium short rate models, there is no restriction that the securities need to be given an accurate price.

### More Resources

CFI is the official provider of the Certified Banking & Credit Analyst (CBCA)™ certification program, designed to transform anyone into a world-class financial analyst.

To keep learning and developing your knowledge of financial analysis, we highly recommend the additional resources below:

• Equilibrium Term Structure Models
• Market Efficiency
• Public Securities
• Yield Curve

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