## Interest Rate Models. An introductions

*in*Asset Liability Management

# Interest Rate Models.

In the introduction to this course we will cover interest rate models, features of a good practical model, importance of calibrating a model and the criteria for model selection. We also briefly look at the features of equilibrium and no-arbitrage models and one-factor and multifactor models.

Interest rates tend to fluctuate on a day to day basis as well as occasionally when there is a regime shift. These changes can signify a significant risk to those portfolios/ instruments whose values are derived from movements in the interest rate. Over the years many tools and products have been created that seek to hedge against or reduce and /or control such risk.

Models are usually employed in order to value instruments which are dependent on interest rates as well as to value the new hedge instruments.

Models are defined by state variables and their processes. The values taken by the state variables that constitute a model give the position or state of the item being model. The processes determine how the state variables change over time. Interest rate processes or changes in state variables are usually stochastic processes, i.e. they incorporate an element of randomness. These processes can usually be divided into a non-random deterministic component, called drift and a random, noise term called volatility.

The purpose of interest rate models therefore is firstly to understand interest rate behaviour. By being able to study the distributional and statistically properties of interest rate movements such as the width of the distribution, its shape, the likelihood of reaching certain levels, etc, we are better able to discern interest rate movements, determine the likely range of future values and estimate the probabilities of price movements beyond a certain threshold. Understanding rate movements helps in setting acceptable limits; helps in setting economic policy; helps in valuing financial instruments more reliably or hedging them more effectively.

The general features of a good model are:

- They should be able to accurately value simple and liquid market instruments
- They should be easy to calibrate to market data
- They must be robust
- They must be able to be extended to value and hedge new instruments

Specifically an interest rate model must contain the following features in order to have practical application.

- A statistical description for the state variable processes
- A process of how interest rate derivatives will be priced from these statistical descriptions

An additional aspect of interest rate modelling is that the model would need to be calibrated by fitting the model to existing good prices. Once the model is calibrated it will be used to value instruments similar to the ones used in the construction and calibration of the model.

There are a number of families of interest rate models to choose from. The criteria for choosing a particular model should be based on:

The fit the model has with the market data, which includes a fit to the:

- current yield curve,
- current prices and
- current volatility structure inherent in the yield curve.

Note however, that it may not always be possible to fit all of this data.

- Matching certain dynamical features that are observed in the way the prices change e.g.
- The time varying reversion level, rate and volatility structure of short rates
- The number and shapes of the principal components of a whole yield curve term structure
- The dynamics of selected instruments

- The tractability of the model, i.e. whether it is able to provide computable solutions for simple instruments and simple numerical methods for complicated instruments.

## a. Equilibrium vs. No-Arbitrage models

There are two main categories of term structure models; Equilibrium models and No-Arbitrage models.

Equilibrium models use assumptions about various economic variables and derive a process for determining the short term risk free rate. The initial term structure of rates is an output of such a model and may not automatically be consistent with today’s term structure.

On the other hand No-Arbitrage models are exactly consistent with today’s term structure as the initial term structure is taken as an input in the model, The model is built so as to admit no arbitrage possibilities. The model’s output is calibrated to this initial term structure to ensure a good fit to the current term structure.

## b. One- factor vs. multiple-factor models

One-factor models, such as short rate models like the Cox Ingersoll and Ross (CIR) and Black Derman and Toy (BDT) models, assume that the entire term structure of interest rates can be inferred with reference to the process underlying a single factor (e.g. the short rate). For example, short rate models implicitly assume that the changes in the longer maturity rates are perfectly correlated to the uncertainty in short rates.

Multiple factor models on the other hand include the impacts of other factors, such as slope and volatilities of the yields of different maturities, into the model. For example, the Heath Jarrow and Merton (HJM) model is a multiple factor model that models the yield curve as a whole.

In this post we have considered the components of interest rate models, the features of good models, and the criteria employed when selecting models. We have considered the differences between various types of term structure models, such as equilibrium and no-arbitrage models and one-factor and multifactor models. In the posts that follow we look in more detail at the one-factor equilibrium CIR model, the one-factor no-arbitrage Black-Derman-Toy (BDT) model and the multi-factor no-arbitrage Heath-Jarrow-Merton (HJM) model.