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Introduction to Term Structure Models

Stochastic models for the future path of interest rates are complicated by the fact that they must consider a continuum of rates rather than just a single number, as is the case for stock prices and exchange rates. The most popular class of interest rate models are referred to as term-structure models, so called because the models are calibrated to be consistent with an observed set of zero-coupon rates.

In most cases, the value of derivative instruments that are dependent on the future evolution of the term structure cannot be determined using a closed-form solution. The typical valuation approach is instead based on a discrete-time representation of the range of possible interest rate movements. These rates are contained in a lattice structure, either a binomial or trinomial tree depending on the specific interest rate process (the 'model') that is under consideration. Once an appropriate tree has been constructed, a wide range of interest rate sensitive instruments can be valued using a standard two-step backward induction procedure. That procedure involves:

1.

Assessing the terminal payoff for the instrument and incorporating that value onto the appropriate node of the tree. For example, the value of zero-coupon bonds at maturity is equal to the stipulated face value, irrespective of the level of interest rates. Thus, the value of the bond at all nodes that correspond to the maturity date are set equal to the face value.

2.

Working back through the tree to the initial node, computing the price at each node as the discounted expected value. The expected value is based on the possible values of the instrument at the next branch in the tree.

This development will incorporate four of the best known term structure models. All four models are implemented in broadly the same manner, and only differ with respect to the assumed process that governs interest rate movements through time. The considered models are those due to:

 

Black-Derman-Toy (1990)

Hull and White (1993)

Ho and Lee (1986)

Black and Karasinski (1991)

Model Summary

A summary of the supported models is set out in the table below.

Some of the key features of the models from an implementation point of view include:

 

Only the Black-Derman-Toy models are implemented on a binomial tree. While construction of a binomial tree is generally much quicker that construction of a trinomial tree with the same number of timesteps, it is also more prone to stability problems. That is, the option price returned by the model could be quite sensitive to the chosen number of timesteps used to construct the tree.

 

The main reason for this instability is that the binomial tree is constructed with a constant timestep. Each branch in the tree spans a period of time equal to the total life of the underlying instrument divided by the total number of timesteps selected. There is therefore no guarantee that a branch in the constructed tree will fall on the relevant 'event' dates such as exercise dates and coupon payment dates. The impact on option value of building a tree with a branch that falls either just before or just after the event date can be significant, and can only be mitigated by increasing the number of tree branches to a 'sufficiently high' number.

 

In most cases, these stability difficulties can be overcome using a trinomial tree because these trees are built with a variable timestep. This means that the tree can be constructed such that a branch is guaranteed to fall on each of the deal's event dates, irrespective of the number of timesteps that is stipulated by the user.

 

Because of the greater stability of the models implemented on a trinomial tree, the Ho-Lee, Hull-White, and Black-Karasinski models may be favoured over the two Black-Derman-Toy models for this reason alone. However, examination of the stochastic differential equations presented in Table 3.1 reveals that Black-Derman-Toy 1 can be easily implemented on a trinomial tree as a modified version of Black-Karasinski, where the modification sets the mean reversion to 0.

Model Name

Short Rate Distribution

SDE for Short Rate

Tree Implementation

 

Black-Derman-Toy 1

Log-Normal

Equation Template

Binomial

Black-Derman-Toy 2

Log-Normal

Equation Template

Binomial

Hull-White

Normal

Equation Template

Trinomial

Ho-Lee

Normal

Equation Template

Trinomial

Black-Karansinski

Log-Normal

Equation Template

Trinomial

 

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