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.
About the Author
Simona Svoboda is a Quant on the interest rates structuring desk, Rand Merchant Bank, South Africa.