# Heath-Jarrow-Morton Model – HJM Model Definition ## What is the Heath-Jarrow-Morton model – HJM model?

The Heath-Jarrow-Morton model (HJM model) is used to model forward interest rates. These rates are then modeled using an existing forward interest rate structure to determine the appropriate prices for securities sensitive to interest rates.

## The formula of the HJM model is

In general, the HJM model and those built on its frame follow the formula:

## What does the Heath-Jarrow-Morton model tell you?

A Heath-Jarrow-Morton model is very theoretical and is used at the most advanced levels of financial analysis. It is mainly used by arbitrators looking for arbitrage opportunities, as well as by analysts evaluating derivative products. The HJM model predicts forward interest rates, the starting point being the sum of so-called drift terms and diffusion terms. Forward rate drift is due to volatility, known as the HJM drift condition. In the basic sense, an HJM model is any interest rate model driven by a finite number of Brownian movements.

The HJM model is based on the work of economists David Heath, Robert Jarrow and Andrew Morton from the 1980s. The trio wrote two notable articles in the late 1980s that laid the groundwork for the framework, including “The pricing of bonds and the forward structure of interest rates: a new methodology ”.

There are various additional models based on the HJM framework. They all seek to predict the entire forward rate curve, not just the short rate or the point on the curve. The biggest problem with HJM models is that they tend to have infinite dimensions, which makes calculation almost impossible. There are different models that seek to express the HJM model as a finite state.

### Key points to remember

• The Heath-Jarrow-Morton model (HJM model) is used to model forward interest rates using a differential equation that allows for randomness.
• These rates are then modeled according to an existing forward interest rate structure to determine the appropriate prices for securities sensitive to interest rates such as bonds or swaps.
• Today, it is mainly used by arbitrageurs looking for arbitrage opportunities, as well as by analysts evaluating derivatives.

## Price of HJM model and options

The HJM model is also used in option pricing, which refers to finding the fair value of a derivative contract. Trading institutions can use models to assess options as a strategy for finding undervalued or overvalued options.

Option pricing models are mathematical models that use known inputs and predicted values, such as implied volatility, to find the theoretical value of options. Traders will use certain models to determine the price at a certain point, updating the calculation of value based on changing risk.

For an HJM model, to calculate the value of an interest rate swap, the first step is to form a discount curve based on the current option prices. From this discount curve, forward rates can be obtained. From there, the volatility of forward interest rates must be entered, and if the volatility is known, the drift can be determined.