Black–Derman–Toy model
Shortrate tree calibration under BDT:
0. Set the Riskneutral probability of an up move, p, = 50%
2. Once solved, retain these known short rates, and proceed to the next timestep (i.e. input spotrate), "growing" the tree until it incorporates the full input yieldcurve. 
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 onefactor 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 meanreverting behaviour of the short rate with the lognormal distribution, [1] and is still widely used. [2][3]
The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for inhouse 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."[4]
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 Black76prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interestrate sensitive securities and interest rate derivatives.
Although initially developed for a latticebased environment, the model has been shown to imply the following continuous stochastic differential equation:[5][6]

 where,
 = the instantaneous short rate at time t
 = value of the underlying asset at option expiry
 = instant short rate volatility
 = a standard Brownian motion under a riskneutral probability measure; its differential.
For constant (time independent) short rate volatility, , the model is:
One reason that the model remains popular, is that the "standard" Rootfinding algorithms – such as Newton's method (the secant method) or bisection – are very easily applied to the calibration.[7] Relatedly, the model was originally described in algorithmic language, and not using stochastic calculus or martingales. [8]
References
 Benninga, S.; Wiener, Z. (1998). "Binomial Term Structure Models" (PDF). Mathematica in Education and Research: vol.7 No. 3.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Black, F.; Derman, E.; Toy, W. (January–February 1990). "A OneFactor Model of Interest Rates and Its Application to Treasury Bond Options" (PDF). Financial Analysts Journal: 24–32.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Boyle, P.; Tan, K.; Tian, W. (2001). "Calibrating the Black–Derman–Toy model: some theoretical results" (PDF). Applied Mathematical Finance: 8, 27–48.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Hull, J. (2008). "The Black, Derman, and Toy Model" (PDF). Technical Note No. 23, Options, Futures, and Other Derivatives.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Klose, C.; Li C. Y. (2003). "Implementation of the Black, Derman and Toy Model" (PDF). Seminar Financial Engineering, University of Vienna.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
External links
 Online: BlackDermanToy short rate tree generator Dr. Shing Hing Man, ThomsonReuters' Risk Management
 Online: Pricing A Bond Using the BDT Model Dr. Shing Hing Man, ThomsonReuters' Risk Management
 Calculator for BDT Model QuantCalc, Online Financial Math Calculator
 Excel BDT calculator and tree generator, Serkan Gur