This is the first article in a two-part series looking at the characteristics and applications of market models, and their advantage over traditional approaches in pricing interest-rate derivatives. This article reviews the literature and covers the relationship between the well-known Heath, Jarrowand Morton(HJM) approach and the market models approach. Next week's Learning Curve will examine application issues, namely the calibration of the market models to caps and swaptions, closed form solutions useful for calibration and pricing of Bermudan options with Monte Carlo in the context of the market models.
The LIBOR and swap market models are the latest generation of interest-rate derivatives valuation models. As the name suggests, they are particularly adept at addressing LIBOR and swaps. But the models can also be used for other securities, such as callable bonds. They can price complex derivatives, such as Bermudan swaptions or constant maturity swap spread options in a manner consistent with market prices of liquid instruments.
Review Of Interest-Rate Derivative Valuation Models
In the 1980s, the main application of interest-rate models was bonds with embedded options. The challenge was to evaluate them in a manner consistent with non-callable bonds, i.e., calibration of the yield curve. To this end, earlier models of the short-term instantaneous interest-rate, such as the Vasicek and the Cox, Ingersoll and Ross (CIR) models were modified appropriately. Specifically, various "drift" parameters of these models were allowed to depend on time in such a fashion that non-callable bonds were priced consistently with the market. Among the most popular of such models is the Black and Karasinski (BK) model, in which the short-term rate follows a mean-reverting lognormal process. Special "forward-induction" techniques are available whereby the required time-dependent drift can be computed quickly and efficiently.
Model parameters being forced to depend on time was troubling aesthetically, as it imposed constraints on the evolution of the short rate that could be unintuitive or difficult to justify empirically or economically. Yet, the framework was convenient, because not only did the model calibrate to the yield curve, but it also enabled the valuation of American and Bermudan callable bonds in an efficient way by the classical technique of "backward induction" on a tree or a grid. In the late 1980s the issue of proper evolution of the whole yield curve was satisfactorily addressed in the HJM model, but the valuation of derivatives was numerically formidable in that framework.
The valuation challenge acquired a new dimension with the prominence of interest-rate swaps in the 1990s, because it needed to be calibrated to caps and European swaptions in addition to the yield curve. Swap traders had already developed their own models for evaluating caps and European swaptions, which were radically different from the aforementioned term-structure model. They applied the Black-Scholes formula to forward LIBOR rates for caplets and to forward swap rates for swaptions as the underlying, but treated discounting, or deflating, as though the discount rates were deterministic. While convenient and robust, such an approach seemed inconsistent with term-structure theory. More importantly, without a proper theoretical foundation, it was very difficult to extend such an approach to more complex derivatives, such as Bermudan swaptions or path-dependent cap products. Different traders tried different ad-hoc techniques, but they were all ridden with conceptual shortcomings. One thing they took comfort in was that their valuation was calibrated to market prices of liquid caps or swaptions.
Market models were developed to justify traders' Black-Scholes formulas for caps and swaptions, and embed them in a consistent and general term-structure theory. The "lognormal LIBOR (respectively swap) market model" is the market model in which forward LIBOR (respectively swap) rates have deterministic volatilities; it yields Black-Scholes formula for caplets (respectively swaptions), consistent with market quotation. These models also provide a framework for valuation of other derivatives.
Relationship To The HJM Model
Market models, like the HJM model, describe the evolution of the whole yield curve allowing multiple factors, and achieve this by determining "drifts" of forward rates in terms of forward-rate volatilities and covariances. Whereas the HJM model concentrates on instantaneous and continuously compounded forward rates, market models consider discrete and simple compound rates, namely forward LIBOR and forward swap rates. At first this may appear as a change of emphasis from an abstract quantity to more observable quantities, but the difference is profound, and a proper theoretical development of market models requires a different framework from the HJM model.
In the market model the instantaneous rate, and its corresponding "numeraire" of "continuously compounded money market" play no significant role. One can easily construct a perfectly valid market model in which such a numeraire does not even exist, in the sense that there exists no self-financing trading strategy with a continuous and finite variation price. Such a setting would not fit in the HJM framework. Yet one can still price swaps using the market model, because dynamic hedging strategies that market models provide do not involve instantaneous rates and money markets. Instead they involve the finite number of zero-coupon bonds, or floating rate agreements, corresponding to the cash flow dates of the underlying. Because of this market model theory is developed in a framework distinct from HJM. In place of instantaneous money market numeraire and its corresponding "risk-neutral measure" which are irrelevant in market model theory, there are other numeraires and corresponding measures that play roles, such as the forward, spot LIBOR and swap measures.
Conclusion
The market model has certain advantages over the HJM model for the practitioner. It enjoys certain analytical solutions, which are crucial for fast calibration. For numerical valuation, the market model is generally more efficient, as it is faster and easier to compute discount factors from forward LIBOR rates than from forward instantaneous rates.
This week's Learning Curve was written by Farshid Jamshidian, head of model research at Cygnifi in London.
