View source for Black–Derman–Toy model

{| class="wikitable floatright" | width="300"
|-
|'''Short-rate tree calibration under BDT:'''
0. Set the [[Risk-neutral probability]] of an up move, p, = 50%<br />
1. For each input [[spot rate]], [[iteration|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{{sub|u}}; r{{sub|d}} being the node in question) via 0.5×[[natural logarithm|ln]](r{{sub|u}}/r{{sub|d}}) = σ{{sub|i}}×{{radic|Δt}} (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-coupon bond|zero-price]] corresponding to the given [[Spot contract|spot interest rate]] for the ith time-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 option]]s, [[swaptions]] and other [[interest rate derivative]]s; 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 reversion (finance)|mean-reverting]] behaviour of the short rate with the [[lognormal distribution]],<ref name="Buetow">{{cite web|url=http://janroman.dhis.org/finance/Interest%20Rates/3%20interest%20rates%20models.pdf |title=Impact of Different Interest Rate Models on Bond Value Measures, G, Buetow et al.|accessdate=2011-07-21 |url-status=dead |archiveurl=https://web.archive.org/web/20111007091116/http://janroman.dhis.org/finance/Interest%20Rates/3%20interest%20rates%20models.pdf |archivedate=2011-10-07 }}</ref> and is still widely used.<ref>{{Google Books |id=GnR3g9lvwfkC |page=410 |title=Fixed Income Analysis }}</ref><ref>http://www.soa.org/library/professional-actuarial-specialty-guides/professional-actuarial-specialty-guides/2003/september/spg0308alm.pdf</ref>

==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: Reflections on Physics and Finance|My Life as a Quant]]".<ref>{{Cite web |url=http://www.ederman.com/new/index.html |title=Archived copy |access-date=2010-04-26 |archive-url=https://web.archive.org/web/20100328013307/http://www.ederman.com/new/index.html |archive-date=2010-03-28 |url-status=dead }}</ref>

==Formulae==
Under BDT, using a [[Binomial options pricing model|binomial lattice]], one [[root finding|calibrates]] the model parameters to fit both the current term structure of interest rates ([[yield curve]]), and the [[Volatility (finance)#Volatility terminology|volatility structure]] for [[interest rate cap]]s (usually [[Interest rate cap#Implied Volatilities|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 derivative]]s.

Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous [[stochastic differential equation]]:<ref name="Buetow"/><ref>http://help.derivativepricing.com/2327.htm</ref>

:<math> d\ln(r) = [\theta_t + \frac{\sigma'_t}{\sigma_t}\ln(r)]dt + \sigma_t\, dW_t </math>

::where,
::<math> r\,</math> = the instantaneous short rate at time t
::<math>\theta_t\,</math> = value of the underlying asset at option expiry
::<math>\sigma_t\,</math> = instant short rate volatility
::<math>W_t\,</math> = a standard [[Brownian motion]] under a [[Risk-neutral measure|risk-neutral]] probability measure; <math>dW_t\,</math> its [[differential (mathematics)|differential]].

For constant (time independent) short rate volatility, <math>\sigma\,</math>, the model is:

:<math>d\ln(r) = \theta_t\, dt + \sigma \, dW_t </math>

One reason that the model remains popular, is that the "standard" [[Root-finding algorithm]]s—such as [[Newton's method]] (the [[secant method]]) or [[bisection method|bisection]]—are very easily applied to the calibration.<ref>http://www.cfapubs.org/toc/rf/2001/2001/4</ref> Relatedly, the model was originally described in [[algorithm]]ic language, and not using [[stochastic calculus]] or [[martingale (probability theory)|martingale]]s.<ref>{{Cite web |url=http://www.ederman.com/new/docs/fen-interview.html |title=Archived copy |access-date=2010-04-26 |archive-url=https://web.archive.org/web/20160303192221/http://www.ederman.com/new/docs/fen-interview.html |archive-date=2016-03-03 |url-status=dead }}</ref>

==References==
'''Notes'''
{{reflist}}
{{refbegin}}

'''Articles'''
*{{cite journal|first=S.|last=Benninga|author2=Wiener, Z.|title=Binomial Term Structure Models|year= 1998|pages=vol.7 No. 3|journal=Mathematica in Education and Research|url=http://pluto.mscc.huji.ac.il/~mswiener/research/Benninga73.pdf}}
*{{cite journal|first=F.|last=Black|author2=Derman, E.|author3=Toy, W.|title=A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options|date=January–February 1990|pages=24–32|journal=Financial Analysts Journal|url=http://savage.wharton.upenn.edu/FNCE-934/syllabus/papers/Black_Derman_Toy_FAJ_90.pdf|url-status=dead|archiveurl=https://web.archive.org/web/20080910041743/http://savage.wharton.upenn.edu/FNCE-934/syllabus/papers/Black_Derman_Toy_FAJ_90.pdf|archivedate=2008-09-10}}
*{{cite journal|first=P.|last=Boyle|authorlink=Phelim Boyle|author2=Tan, K.|author3=Tian, W.|title=Calibrating the Black–Derman–Toy model: some theoretical results|year=2001|pages=8, 27–48|journal=Applied Mathematical Finance|url=http://belkcollegeofbusiness.uncc.edu/wtian1/bdt.pdf|url-status=dead|archiveurl=https://web.archive.org/web/20120422023305/http://belkcollegeofbusiness.uncc.edu/wtian1/bdt.pdf|archivedate=2012-04-22}}
*{{cite web|last=Hull|first=J.|year=2008|title=The Black, Derman, and Toy Model|publisher=Technical Note No. 23, Options, Futures, and Other  Derivatives|url=http://www.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote23.pdf|authorlink=John C. Hull|access-date=2011-04-08|archive-url=https://web.archive.org/web/20110129071012/http://www.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote23.pdf|archive-date=2011-01-29|url-status=dead}}
*{{cite web|last=Klose|first=C.|year=2003|title=Implementation of the Black, Derman and Toy Model|publisher=Seminar Financial Engineering, University of Vienna|author2=Li C. Y.|url=http://www.lcy.net/files/BDT_Seminar_Paper.pdf}}

{{refend}}

==External links==
*[https://web.archive.org/web/20160305002641/http://lombok.demon.co.uk/financialTap/interestrates/bdtshortrates Online: Black-Derman-Toy short rate tree generator] Dr. Shing Hing Man, Thomson-Reuters' Risk Management
*[https://web.archive.org/web/20110807174135/http://lombok.demon.co.uk/financialTap/interestrates/bdtbond Online: Pricing A Bond Using the BDT Model] Dr. Shing Hing Man, Thomson-Reuters' Risk Management
*[http://quantcalc.net/BDTBond.html Calculator for BDT Model] QuantCalc, Online Financial Math Calculator
*[http://www.serkangur.freeservers.com/ Excel BDT calculator and tree generator], Serkan Gur
{{Bond market}}
{{Stochastic processes}}

{{DEFAULTSORT:Black-Derman-Toy Model}}
[[Category:Fixed income analysis]]
[[Category:Short-rate models]]
[[Category:Financial models]]
[[Category:Options (finance)]]