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

History

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.^[4]

Formulae

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.

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
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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

v t e Bond market
Bond Debenture Fixed income
Types of bonds by issuer	Agency bond Corporate bond Senior debt Subordinated debt Distressed debt Government bond Infrastructure bond Municipal bond Global bond
Types of bonds by payout	Accrual bond Auction rate security Callable bond Commercial paper Consol Contingent convertible bond Convertible bond Exchangeable bond Extendible bond Fixed rate bond Floating rate note High-yield debt Inflation-indexed bond Inverse floating rate note Lottery bond Perpetual bond Puttable bond Reverse convertible securities Zero-coupon bond
Bond valuation	Clean price Convexity Coupon Credit spread Current yield Dirty price Duration I-spread Mortgage yield Nominal yield Option-adjusted spread Risk-free bond Weighted-average life Yield curve Yield spread Yield to maturity Z-spread
Securitized products	Asset-backed security Collateralized debt obligation Collateralized mortgage obligation Commercial mortgage-backed security Mortgage-backed security
Bond options	Callable bond Convertible bond Embedded option Exchangeable bond Extendible bond Puttable bond
Institutions	Commercial Mortgage Securities Association (CMSA) International Capital Market Association (ICMA) Securities Industry and Financial Markets Association (SIFMA)

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category