The binomial weave is a new technique for modeling the term structure of interest rates. It will handle American- as well as European-style structures. The weave is a multivariate technique that is flexible, intuitive and transparent.
The modeling of the term structure of interest rates has generated a great deal of interest in past years. Interest rates are especially difficult to model because one must model many related assets evolving through time together without allowing arbitrage to occur between these assets or derivatives of these assets. Additionally, calibration of term-structure models can be difficult because of the number of variables involved and the often convoluted and indirect nature of the required calibration.
Many methods have been applied to term structure of interest rate modeling including binomial trees, trinomial trees, differential equations, integral solutions, Monte Carlos and closed analytic or partially analytic solutions. Often, proposed term structure models are non-intuitive, unwieldy, difficult to calibrate to the observable market and difficult to extend or enhance. Many are of the nature as to be difficult or impossible for anyone but the model's creator to understand deeply. Trees of some sort or Monte Carlos are the most popular implementation techniques for generic and flexible multifactor modeling. One reason for the popularity of the tree or Monte Carlo approach is that these techniques are accessible to intuition and therefore easier to implement and debug.
Tree-based models are appropriate to model and manage the risk associated with a large variety of financial structures. Recombining binomial trees are used extensively in equity, currency and commodity modeling when interest rates can be assumed to be non-random. Until now, it has been impossible to set up a recombining process for a general, N- factor, multivariate term-structure model.
THE BINOMIAL WEAVE MODEL
The model is a multi factor term structure model. It assumes N pseudo futures contracts and a full covariance structure that underlie an interest rate market. Each contract corresponds to a random factor. So it is an N- factor model. These contracts are not required to be real but rather are created for analytical purposes. Yet they are close enough to 'real' tradable objects so as to allow the use of intuition. One always has to assume some underlying structure from which to derive the dynamics. Futures contracts used as random factors are the most intuitive approach and easy to implement and work with because of the lack of drift.
Thus we begin by assuming futures contracts at as fine a grid as we require. Each of these futures settles into a cash rate, or more accurately, a range of cash rates in a binomial framework. These cash rates are spliced together in a binomial weave tree that contains many more paths than a standard binomial. The binomial weave transition probabilities are the key element and they are determined through a straightforward derivation. The transition from cash rate in a given period to a cash rate in the next period requires a transition probability based on only two trees. In order to calculate the full transition probability matrix, one only needs to focus on a sequence of two factor binomial processes. Discounting through the tree is done by discounting with the tree-generated risk-free rates and probability weighting with the appropriate transition probabilities. This produces a full N- factor term structure model on a binomial-like recombining tree foundation.
APPLYING THE WEAVE
The binomial weave is a tree-based implementation and has several applications including risk management, derivatives structuring, pricing and hedging, and proprietary trading. It was designed with transparency and intuition in mind. When something does go wrong, if a model is transparent and intuitive, it is far easier and faster to find the problem than if the model is convoluted and difficult to follow or decipher.
The derivatives business ultimately requires a high level of multifactor modeling because the business is centered on the idea of creating complex, customized, risk-mitigation structures that can be broken down into simpler, directly tradable components that can be traded off or managed through hedging. One of the risks which is sometimes overlooked in putting on a long-term deal is model risk. One form of model risk is a model in which the term structure is inadequately modeled. This may show up in hedging losses. In larger scope, however, model risk is the risk that the model used initially to mark the deal was inappropriate and before the deal is off the books it will have to be marked to another model. This inevitably produces mark-to-market losses. In a sense, when new and innovative structures are modeled, the trader has to guess the future modeling philosophy of the market for a time in the future when the deal will become commonplace and spreads will tighten. They are forced to speculate on model evolution. If they are wrong about the way the market will evolve in terms of modeling, they may take a hit.
In addition to the motivations for developing the binomial weave discussed above, there is the additional attraction of the weave as a research tool. One can ask questions of the model for which answers would not be obvious without such a tool. For example, one might be interested in the process followed by the volatility of the spot rate as time passes. This quantity would be of interest in the case of pricing structures sensitive to assumptions as to how the volatility evolves, like in the case of chooser-range notes. This information could be easily extracted from the weave model.
Additionally, one can ask questions on correlation structure with a full N-factor model that cannot be asked with lower order models. If detailed correlation patterns produce significant changes in value then such a model provides a powerful view.
All the above concerns have been taken into account in developing the binomial weave. Moreover, in order for the derivatives markets to develop it needs standardized, widely accepted and rational risk management tools. Wide acceptance of non-intuitive models is unlikely to happen. The binomial weave was designed to fill this void.
This week's Learning Curve was written by Don Goldman, president of the DGAnalytics Corp., a Ridgewood, N.J.-based consulting company.
