The first swap agreement took place between IBM and the World Bank in the early 1980s. Since then sophisticated products have been developed to meet the needs of capital markets players searching for higher returns for investors and more efficient hedging tools as well as offering new financing means for international companies. In order to handle these new complex products and to manage their risk, a range of pricing models have been introduced over the last few years.
Which Model?
There are many issues one has to consider when selecting a model for valuing derivatives. First, one should examine the effectiveness of the hedging performance of a model. Models that exhibit better hedging performance are more likely to predict accurate future market prices of vanilla derivatives such as at-the-money caps and swaptions. These models enable risk managers to measure risk exposure.
Secondly, calibration is an integral part of any interest-rate model. For a given class of securities the ease and robustness of parameter estimation procedures often leads to a narrow list of choices among the known models.
Thirdly, the choice of a particular model should reflect the specific features of the securities being analyzed. For example, some short-rate models such as BDT (Black-Derman-Toy) are not suitable for valuing a portfolio of mortgage-backed securities (MBS). This is because a long-term Treasury par yield, such as the 10-year yield, is a major factor that drives the mortgage market and these short-rate models do not explicitly provide information on future yield curves as needed to model MBS redemptions.
All of the well-known interest rate models in the market may be classified either as short-rate models or as forward-rate models.
A short-rate model posits a short-term rate random process. The short-rate process is typically Markovian so that there is a partial differential equation associated with each model. The class of short-rate models includes the Vasicek model, the Cox-Ingersoll Ross (CIR) model, the Hull-White (HW) model, the Black-Derman-Toy (BDT) model and the Black-Karasinski (BK) model.
Forward-Rate Models
The Ho-Lee model is an early example of arbitrage-free modeling of the forward-rate dynamics. A generalized framework for forward-rate modeling was subsequently developed by Heath-Jarrow-Morton. The general HJM model is non-Markovian; an up move for the yield curve followed by a down move does not generate the same outcome as a down move followed by an up move. As a result, a general HJM process cannot be mapped to a recombining tree. This makes the HJM model more computationally intensive than some Markovian models, such as the Ho-Lee and Hull-White models.
In forward-rate models, Brace Garacek Musiala (BGM) has recently become a popular model among traders as it provides users with a market model working directly on forward or swap rates, in opposition to models such as Black Derman Toy or Hull White which work on short term rates. The intuitive character of the model and the calibration process that involves forward or swap rate volatility and correlations have undoubtedly contributed to the success and popularity of this model. In fact the BGM model is perfectly designed for path dependent products such as, revolver caps, ratchet caps (or swaps), cumulative caps, Bermuda swaptions and so on.
Conclusion
The development of exotics has also acted as a catalyst for the reinterpretation of internal trading roles; risk management has moved from a primarily defensive role to a more proactive and business orientated function. The highly competitive nature of the market will continue to generate more complex structured instruments, requiring increasingly robust and flexible technology to support this evolution.
This week's Learning Curve was written by Anis Kraiem, senior consultant at Ubitradein Paris.
