Black–Derman–Toy model


Short-rate tree calibration under BDT:
Step 0. Set the risk-neutral probability of an up move, p, to 50%

Step 1. For each input spot rate, iteratively:
  • adjust the rate at the top-most node at the current time-step, i;
  • find all other rates in the time-step, where these are linked to the node immediately above via ;
  • discount recursively through the tree using the rate at each node, i.e. via "backwards induction", from the time-step in question to the first node in the tree ;
  • repeat until the discounted value at the first node in the tree equals the zero-price corresponding to the given spot interest rate for the i-th time-step.
Step 2. Once solved, retain these known short rates, and proceed to the next time-step, "growing" the tree until it incorporates the full input yield-curve.

In mathematical finance, the Black–Derman–Toy model is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see. It is a one-factor model; that is, a single stochastic factor—the short rate—determines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the log-normal distribution, and is still widely used.

History

The model was introduced by Fischer Black, Emanuel Derman, and William Toy. It was first developed for in-house use by Goldman Sachs in the 1980s and was published in the Financial Analysts Journal in 1990. A personal account of the development of the model is provided in Emanuel Derman's memoir My Life as a Quant.

Formulae

Under BDT, using a binomial lattice, one calibrates the model parameters to fit both the current term structure of interest rates, and the volatility structure for interest rate caps ; see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and interest rate derivatives.
Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous stochastic differential equation:
For constant short rate volatility,, the model is:
One reason that the model remains popular, is that the "standard" Root-finding algorithms—such as Newton's method or bisection—are very easily applied to the calibration. Relatedly, the model was originally described in algorithmic language, and not using stochastic calculus or martingales.