|Short-rate tree calibration under BDT:
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 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 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 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.
For constant (time independent) 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 (the secant 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.
- "Impact of Different Interest Rate Models on Bond Value Measures, G, Buetow et al" (PDF). Archived from the original (PDF) on 2011-10-07. Retrieved 2011-07-21.
- Fixed Income Analysis, p. 410, at Google Books
- "Archived copy". Archived from the original on 2010-03-28. Retrieved 2010-04-26.CS1 maint: archived copy as title (link)
- "Archived copy". Archived from the original on 2016-03-03. Retrieved 2010-04-26.CS1 maint: archived copy as title (link)
- Benninga, S.; Wiener, Z. (1998). "Binomial Term Structure Models" (PDF). Mathematica in Education and Research: vol.7 No. 3.
- Black, F.; Derman, E.; Toy, W. (January–February 1990). "A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options" (PDF). Financial Analysts Journal: 24–32. Archived from the original (PDF) on 2008-09-10.
- Boyle, P.; Tan, K.; Tian, W. (2001). "Calibrating the Black–Derman–Toy model: some theoretical results" (PDF). Applied Mathematical Finance: 8, 27–48. Archived from the original (PDF) on 2012-04-22.
- Hull, J. (2008). "The Black, Derman, and Toy Model" (PDF). Technical Note No. 23, Options, Futures, and Other Derivatives. Archived from the original (PDF) on 2011-01-29. Retrieved 2011-04-08.
- Klose, C.; Li C. Y. (2003). "Implementation of the Black, Derman and Toy Model" (PDF). Seminar Financial Engineering, University of Vienna.
- Online: Black-Derman-Toy short rate tree generator Dr. Shing Hing Man, Thomson-Reuters' Risk Management
- Online: Pricing A Bond Using the BDT Model Dr. Shing Hing Man, Thomson-Reuters' Risk Management
- Calculator for BDT Model QuantCalc, Online Financial Math Calculator
- Excel BDT calculator and tree generator, Serkan Gur