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Black–Derman–Toy model

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Short-rate tree calibration under BDT:

Step 0. Set therisk-neutral probability of an up move, p, to 50%
Step 1. For each inputspot 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 (r_u; r_d being the node in question) via $\ln(r_{u}/r_{d})/2=\sigma _{i}{\sqrt {\Delta t}}$ (this node-spacing being consistent with p = 50%; Δt being the length of the time-step);
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 (i.e. i=0);
repeat until the discounted value at the first node in the tree equals thezero-price corresponding to the givenspot interest rate for the i-th time-step.

Step 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.

Inmathematical finance, theBlack–Derman–Toy model (BDT) is a popularshort-rate model used in the pricing ofbond options,swaptions and otherinterest rate derivatives; seeLattice model (finance) § Interest rate derivatives. It is a one-factor model; that is, a singlestochastic factor—the short rate—determines the future evolution of all interest rates. It was the first model to combine themean-reverting behaviour of the short rate with thelog-normal distribution,^[1] and is still widely used.^[2]^[3]

History

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The model was introduced byFischer Black,Emanuel Derman, and Bill Toy. It was first developed for in-house use byGoldman Sachs in the 1980s and was published in theFinancial Analysts Journal in 1990. A personal account of the development of the model is provided in Emanuel Derman'smemoirMy Life as a Quant.^[4]

Formulae

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Under BDT, using abinomial lattice, onecalibrates the model parameters to fit both the current term structure of interest rates (yield curve), and thevolatility structure forinterest rate caps (usuallyas implied by theBlack-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 andinterest rate derivatives.

Although initially developed for a lattice-based environment, the model has been shown to imply the following continuousstochastic differential equation:^[1]^[5]

d\ln(r)=\left[\theta _{t}+{\frac {\sigma '_{t}}{\sigma _{t}}}\ln(r)\right]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 standardBrownian motion under arisk-neutral probability measure;

dW_{t}\,

itsdifferential.

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 asNewton's method (thesecant method) orbisection—are very easily applied to the calibration.^[6] Relatedly, the model was originally described inalgorithmic language, and not usingstochastic calculus ormartingales.^[7]

References

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Notes

^^a ^b"Impact of Different Interest Rate Models on Bond Value Measures, G, Buetow et al"(PDF). Archived fromthe original(PDF) on 2011-10-07. Retrieved2011-07-21.
^Fixed Income Analysis, p. 410, atGoogle Books
^"Society of Actuaries Professional Actuarial Specialty Guide Asset-Liability Management"(PDF).soa.org. Retrieved19 March 2024.
^"My Life as a Quant: Reflections on Physics and Finance". Archived fromthe original on 2010-03-28. Retrieved2010-04-26.
^"Black-Derman-Toy (BDT)". Archived fromthe original on 2016-05-24. Retrieved2010-06-14.
^Phelim Boyle, Ken Seng Tan and Weidong Tian (2001).Calibrating the Black–Derman-Toy model: some theoretical results, Applied Mathematical Finance 8, 27– 48 (2001)
^"One on One Interview with Emanuel Derman (Financial Engineering News)". Retrieved2021-06-09.

Articles

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 fromthe 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 fromthe 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 fromthe original(PDF) on 2011-01-29. Retrieved2011-04-08.
Klose, C.; Li C. Y. (2003)."Implementation of the Black, Derman and Toy Model"(PDF). Seminar Financial Engineering, University of Vienna.

External links

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R function for computing the Black–Derman–Toy short rate tree, Andrea Ruberto
Excel BDT calculator and tree generator, Serkan Gur

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