Modeling the term structure of interest rates has always been linked with complex probability theory and technical jargon that can act as a deterrent to exploring the world of derivative pricing. Advanced mathematics helps in the complete understanding of the theory behind interest rate models. However, in my experience a simple application of the theory helps in initiating students of derivative pricing to a vast and exciting world. This article takes the Black Derman Toy one factor interest rate model and explains how it could be applied to an excel spreadsheet in a few hours.
Consider the binomial tree in the figure. Each node is defined by the time period 't' and the shock level 'I'. The one factor or variable in the BDT model is the short rate-the one-period interest rate whose changes drive all security prices. The short rate at t=0 is known and it evolves in a series of up and down moves to populate the nodes of the tree. The two constraints that determine the values the nodes can take are:
* The mean interest rate term structure predicted by the tree has to conform to the current term structure of interest rates as exhibited by the eurodollar futures and bond markets.
* The mean volatility of interest rates at each time period, i.e. the term structure of volatility has to conform with the current term structure of volatility as exhibited by the price of caps, floors etc. in the market.
Let Rt(I) be the short rate at node (t,I) and let At and Bt be constants that change with time. Then the evolution of the short rate can be described by Rt(I) = At*exp(Bt*I). The Green function G signifies the present value (at t=0) of a dollar at any node and helps simplify the calculations. The formulae for the green function (they look technical but are actually quite simple) are:
The steps needed to apply this on a spreadsheet are:
* Put in two rows going out six periods for At and Bt and populate the fields with constants say 0.05 and 0.001 respectively.
* Using the constraints above build a binomial tree using the formula for the short rate Rt.
* Construct a similar binomial tree using the Green functions to get the 'Green value' at each node.
* Sum all the nodes of the Green tree to get a set of sums (one for each t). This array gives you the mean interest rate term structure predicted by the short rate tree. The array is in discount factors.
* Below the predicted term structure input the current term structure as predicted by the euro dollar futures and bond markets (This is easily constructed or obtained from a news service)
* The difference between the actual and predicted values should be zero at each t. To enforce this condition the solver can be used in excel to constraint each of the differences to zero by changing At and Bt.
* This will give you a short rate tree which conforms to the current term structure.
To apply the Second Constraint
* Using the values of the short rate tree and the Green tree a price of a caplet with a given strike can be deduced. The payoff at a node for a caplet with strike X and period (t to t-1) is Max(Rt(I)-X,0)*Gt(I)*exp(-Rt(I)). The payment for a caplet is made after the end of the period and therefore the payment is discounted back to the node by discounting it by Rt(I) and then using the Green function to discount it to day 0. The market value of the caplet is the sum of all the node values at time t, of the above generated tree. Using the same theory a series of caplets one at each period t can be priced.
* These prices are then compared to the prices of caplets currently trading in the market for the same strike and over similar time periods. The difference between the predicted and actual price should be zero at each t. This is then enforced by using the solver and constraining each of the differences (the ones referred to earlier and the ones generated here) to zero by changing At and Bt variables simultaneously.
The solution will generate a short rate tree which conforms to our two constraints and can be used to price any cap, floor, or swaption using methodology described in the second section. This is a fully functional application of the BDT although limited to a few time periods. The model itself can be expanded to a full scale version by using a more powerful solver program as the solver in excel is only powerful enough to build a 5-6 period tree.
This week's Learning Curve was written by Manish Gupta a consultant with the Mitchell Madison Group.
