Black–Derman–Toy model

0. Set the Risk-neutral probability of an up move, p, = 50%
1. For each input spot rate, iteratively:

2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve.

In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) #Interest rate derivatives. It is a one-factor model; that is, a single 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 lognormal distribution, and is still widely used.

The model was introduced by Fischer Black, Emanuel Derman, and Bill 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 one of the chapters in Emanuel Derman's memoir "My Life as a Quant."

Under BDT, using a binomial lattice, one calibrates the model parameters to fit both the current term structure of interest rates (yield curve), and the volatility structure for interest rate caps (usually as implied by the Black-76-prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and interest rate derivatives.

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