Black–Derman–Toy model

Short-rate tree calibration under BDT: Step 0. Set the risk-neutral probability of an up move, p, to 50% Step 1. For each input spot 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 the zero-price corresponding to the given spot 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.

In

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 log-normal distribution,^[1] and is still widely used.^[2]^[3]

History

The model was introduced by

My Life as a Quant.^[4]

Formulae

Under BDT, using a

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

.

Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous stochastic 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 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.^[6] Relatedly, the model was originally described in algorithmic language, and not using stochastic calculus or martingales.^[7]

References

Notes

^ ^a ^b "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
^ "Society of Actuaries Professional Actuarial Specialty Guide Asset-Liability Management" (PDF). soa.org. Retrieved 19 March 2024.
^ "My Life as a Quant: Reflections on Physics and Finance". Archived from the original on 2010-03-28. Retrieved 2010-04-26.
^ "Black-Derman-Toy (BDT)". Archived from the original on 2016-05-24. Retrieved 2010-06-14.
^ 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)". Retrieved 2021-06-09.

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 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.

External links

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

[Buetow-1] "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.

[2] Fixed Income Analysis, p. 410, at Google Books

[3] "Society of Actuaries Professional Actuarial Specialty Guide Asset-Liability Management" (PDF). soa.org. Retrieved 19 March 2024.

[4] "My Life as a Quant: Reflections on Physics and Finance". Archived from the original on 2010-03-28. Retrieved 2010-04-26.

[5] "Black-Derman-Toy (BDT)". Archived from the original on 2016-05-24. Retrieved 2010-06-14.

[6] 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)

[7] "One on One Interview with Emanuel Derman (Financial Engineering News)". Retrieved 2021-06-09.

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