# Introduction

Growth in the derivatives markets has brought with it an ever-increasing volume and range of interest rate dependent derivative products. To allow profitable, efficient trading in these products, accurate and mathematically sound valuation techniques are required to make pricing, hedging and risk management of the resulting positions possible.

The value of vanilla European contingent claims such as caps, floors and swaptions depends only on the level of the yield curve. These types of instruments are priced correctly using the simple model developed by Black [ ** 5 ** ]. This model makes several simplifying assumptions which allow closed-form valuation formulae to be derived. This class of vanilla contingent claims has become known as ˜first-generation' products.

These instruments expose investors to the level of the underlying yield curve at one point in time. They reflect the investors' view of the future changes in the level of the yield curve, not their view of changes in the slope of the curve. ˜Second-' and ˜third-generation' derivatives, such as path -dependent and barrier options, provide exposure to the relative levels and correlated movements of various portions of the yield curve. Rather than hedging these exotic options with the basic underlying instrument, i.e. the bond, the ˜first generation' instruments are used. Therefore, the Black model prices of these ˜first generation' instruments are taken as given. This does not necessarily imply a belief in the intrinsic correctness of the Black model. Distributional assumptions which are not included in the Black model, such as mean reversion and skewness , are incorporated by adjusting the implied volatility input.

The more sophisticated models developed allow the pricing of instruments dependent on the changing level and slope of the yield curve. A crucial factor is that these models must price the exotic derivatives in a manner that is consistent with the pricing of vanilla instruments. When assessing the correctness of any more sophisticated model, its ability to reproduce the Black prices of vanilla instruments is vital . It is not a model's * a priori * assumptions, but rather the correctness of its hedging performance that plays a pivotal role in its market acceptance.

The calibration of the model is an integral part of its specification, so the usefulness of a model cannot be assessed without considering the reliability and robustness of parameter estimation.

## General framework

The pricing of interest rate contingent claims has two parts . Firstly, a finite number of pertinent economic fundamentals are used to price all default-free zero coupon (discount) bonds of varying maturities. This gives rise to an interest rate term structure, which attempts to explain the relative pricing of zero coupon bonds of various maturities. Secondly, taking these zero coupon bond prices as given, all interest rate derivatives may be priced.

As with asset prices, the movement of interest rates is assumed to be determined by a finite number of random shocks, which feed into the model through stochastic processes. Assuming continuous time and hence also continuous interest rates, these sources of randomness are modelled by Brownian motions (Wiener processes).

When modelling interest rates we do not have a finite set of assets, but rather a one-parameter family of assets: the discount bonds, with the maturity date as the parameter. The risk-free rate of interest (short-term interest rate) is not specified exogenously (as in stock price models), but is the rate of return on a discount bond with instantaneous maturity. Also, unlike in asset pricing theory, the fundamental assets - the discount bonds, may themselves be viewed as derivatives. Hence the modelling of the interest rate term structure may be viewed as tantamount to interest rate derivative pricing.

The theory of interest rate dynamics relies on a degree of abstraction in that the fundamental assets (the discount bonds) are assumed to be perfect assets, that is default-free and available in a continuum of maturities.