# Heath-Jarrow-Morton Model

A framework to represent forward interest rates using an existing term structure of interest rates

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## What is the Heath-Jarrow-Morton Model?

The Heath-Jarrow-Morton Model – also known as the HJM Model – is a framework to represent forward interest rates using an existing term structure of interest rates. The model was created based on the work developed by David Heath, Robert A. Jarrow, and Andrew Morton during the late 1980s. Their research papers led to the establishment of the model that we know today.

The purpose of using the HJM Model is to predict forward interest rates so that the predictions can be used to calculate the prices of securities affected by interest rate movements, including securities such as bonds and options.

The model can be mathematically represented by the following general formula:

Where:

• α and σ are adapted
• W is a Brownian motion under the assumption of being risk-neutral
• df(t,T) represents the instantaneous forward interest rate with maturity at time T

### Assumptions of the Heath-Jarrow-Morton Model

There are several assumptions presented by the Heath-Jarrow-Morton Model, such as:

• The model assumes that the forward rate is driven by volatility because the volatility in the market for futures contracts can be predicted.
• Another assumption presented by the model is that the price of each security is observable. The security can be bought and sold at any quantity at the observed price.
• The model does not explain all the complexities that come from a changing term structure.

### Uses of the Heath-Jarrow-Morton Model

Investors use the Heath-Jarrow-Morton Mode to determine the prices of securities that are impacted by interest rate fluctuations. By being able to price securities, investors can engage in arbitrage opportunities to earn a riskless profit if there are differences between the price of the security in the market and the price of the security calculated based on the Heath-Jarrow-Morton Model.

In particular, the model can be used to price financial derivatives because the value of derivatives depends on the term structure of underlying assets. For example, the underlying asset for credit derivatives is the price of risky zero-coupon bonds. In addition to arbitrage seekers, it can also be used by asset-liability management.

### The Gaussian Heath-Jarrow-Morton Model and Short-Rate Models

When the drift and volatility of the instantaneous forward rate are assumed to be deterministic, it is known as the Gaussian Heath-Jarrow-Morton Model. In the mathematical formula, it is when σ becomes a deterministic function.

The Heath-Jarrow-Morton Model is often compared with other models when investors assess different strategies to price financial derivatives. They are often compared with short-rate models, but they are different from each other. The HJM Model represents the entire forward rate curve, but short-rate models only demonstrate a specific point on the curve.