# Equilibrium models on interest rates

The equilibrium models on interest rates are equilibrium models based on a Brownian geometric process and on the risk neutrality of short-term interest rates. In other words, equilibrium interest rate models use shorter-term interest rates to calculate future interest rates taking into account the term structure of interest rates.

As a reference for short-term interest rates, we will use the interest rates of zero coupon bonds . An example would be Spanish Treasury Bills issued in the short term.

Recommended items: zero coupon bond, option, and mean reversion.

The time structure of zero coupon bond prices is obtained from the Brownian geometric process that captures infinitesimal changes in short-term interest rates.

Zero coupon bond prices are used to value the price of zero coupon bond options and coupon bond options.

So, to calculate future zero-coupon bond prices, we need short-term zero-coupon interest rates. In this way, we can also build the curve or time structure of zero coupon interest rates. Once we have the curve, we can determine the evolution of long-term interest rates given the short-term interest rates.

## Equilibrium model assumptions about interest rates

The assumptions of the model are:

• Risk neutrality.

We assume neutral risk as the classic assumption for asset valuation in financial markets. This assumption is key to obtaining the price of a bond using Monte Carlo simulation.

• Log-normal distribution of bonds and interest rates.

We assume the log-normal distribution since we pose interest rates as a positive variable like bond prices. It would not make sense to evaluate negatively priced bonds. Assuming log-normal distribution of interest rates, we can say that interest rates will follow a Brownian geometric process. If the distribution of interest rates were a normal distribution, then we would say that interest rates follow a Brownian arithmetic process.

## Single factor equilibrium models

Single-factor equilibrium models are models for calculating the term structure of interest rates from short-term interest rates.

We say of a single factor since the risk or uncertainty is given by a single factor: the volatility of interest rates. There are two-factor equilibrium models that provide more possibilities for interest rate movements.

Mathematically, we define a single-factor equilibrium model of the form:

Where,

• r (t): short-term interest rates at an instant of time t.
• dr: change in interest rates (r) over time (dt).
• dt: passage of time = evolution of time.
• m (r) dt: direction or trend (m) taken by interest rates (r) over time (dt).
• s (r): standard deviation of interest rates (r).
• dZ: random component or disturbance that follows a normal distribution with mean 0 and variance 1.

The above expression is known as a stochastic differential equation expressed by the Itô process.

## Model types

The most common one-factor equilibrium models are:

• Rendleman and Bartter model.
• Vasicek model.
• Cox, Ingresoll and Ross model.