Black–Derman–Toy model

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In 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. 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 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). Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and interest rate derivatives. Calibration here means that: (1) we assume the probability of an up move = 50%; (2) for each input spot rate, we: (a) iteratively adjust the rate at the top-most node at the current time-step, i; (b) find all other nodes in the time-step, where these are linked to the node immediately above via 0.5 ln (ru/rd) = σi sqrt(Δt); (c) discount recursively through the tree, from to the time-step in question to the first node in the tree; (d) repeat this until the calculated spot-rate (i.e. the discount factor at the first node in the tree) equals the assumed spot-rate; (3) Once solved, we 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.

Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous stochastic differential equation:

$d\ln(r)=[\theta _{t}+{\frac {\sigma '_{t}}{\sigma _{t}}}\ln(r)]dt+\sigma _{t}\,dW_{t}$

where,

$r\,$ = the instantaneous short rate at time t

$\theta _{t}\,$ = value of the underlying asset at option expiry

$\sigma _{t}\,$ = instant short rate volatility

$W_{t}\,$ = a standard Brownian motion under a Risk-neutral probability measure; $dW_{t}\,$ its differential.

For constant (time independent) short rate volatility, $\sigma \,$ , the model is:

$d\ln(r)=\theta _{t}\,dt+\sigma \,dW_{t}$

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.

References

Benninga, S.; Wiener, Z. (1998). "Binomial Term Structure Models". Mathematica in Education and Research: vol.7 No. 3. Cite uses deprecated parameters (help)
Black, F.; Derman, E. and Toy, W. (January–February 1990). "A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options". Financial Analysts Journal: 24–32. Cite uses deprecated parameters (help)
Boyle, P.; Tan, K. and Tian, W. (2001). "Calibrating the Black–Derman–Toy model: some theoretical results". Applied Mathematical Finance: 8, 27–48. Cite uses deprecated parameters (help)
Hull, J. (2008). "The Black, Derman, and Toy Model". Technical Note No. 23, Options, Futures, and Other Derivatives.
Klose, C.; Li C. Y. (2003). "Implementation of the Black, Derman and Toy Model". Seminar Financial Engineering, University of Vienna. Cite uses deprecated parameters (help)

External links

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

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