|
|
| Line 1: |
Line 1: |
| In [[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. 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]], [http://janroman.dhis.org/finance/Interest%20Rates/3%20interest%20rates%20models.pdf] and is still widely used. [http://books.google.com/books?id=GnR3g9lvwfkC&pg=PP1&dq=Fixed+income+analysis+By+Frank+J.+Fabozzi,+Mark+Jonathan+Paul+Anson&ei=tpTVS7LjKILYNoPk7I8I&cd=1#v=snippet&q=Black-Derman-Toy&f=false][http://www.soa.org/library/professional-actuarial-specialty-guides/professional-actuarial-specialty-guides/2003/september/spg0308alm.pdf]
| | Oscar is what my spouse enjoys to call me and I completely dig that title. South Dakota is exactly where me and my spouse reside and my family members loves it. Since she was 18 she's been working as a meter reader but she's usually needed her own business. What I adore performing is doing ceramics but I haven't made a dime with it.<br><br>My page :: [http://www.karachicattleexpo.com/blog/56 std testing at home] |
| | |
| The model was introduced by [[Fischer Black]], [[Emanuel Derman]], and [[William Toy|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: Reflections on Physics and Finance|My Life as a Quant]]."[http://www.ederman.com/new/index.html]
| |
| | |
| 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). Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and [[interest rate derivative]]s. Calibration here means that: (1) we assume the probability of an up move = 50%; (2) for each input [[spot rate]], we: (a) [[iteration|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]]:[http://help.derivativepricing.com/2327.htm][http://janroman.dhis.org/finance/Interest%20Rates/3%20interest%20rates%20models.pdf]
| |
| | |
| :<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.[http://www.cfapubs.org/toc/rf/2001/2001/4] Relatedly, the model was originally described in [[algorithm]]ic language, and not using [[stochastic calculus]] or [[martingale (probability theory)|martingale]]s. [http://www.ederman.com/new/docs/fen-interview.html]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| {{refbegin}}
| |
| | |
| *{{cite journal|first=S.|last=Benninga|coauthors=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|coauthors=Derman, E. and 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}}
| |
| *{{cite journal|first=P.|last=Boyle|authorlink=Phelim Boyle|coauthors=Tan, K. and 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}}
| |
| *{{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''|coauthors=|url=http://www.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote23.pdf|authorlink=John C. Hull}}
| |
| *{{cite web|last=Klose|first=C.|year=2003|title=Implementation of the Black, Derman and Toy Model|publisher=Seminar Financial Engineering, University of Vienna|coauthors=Li C. Y.|url=http://www.lcy.net/files/BDT_Seminar_Paper.pdf}}
| |
| | |
| {{refend}}
| |
| | |
| ==External links==
| |
| *[http://lombok.demon.co.uk/financialTap/interestrates/bdtshortrates Online: Black-Derman-Toy short rate tree generator] Dr. Shing Hing Man, Thomson-Reuters' Risk Management
| |
| *[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
| |
| {{Bond market}}
| |
| {{Stochastic processes}}
| |
| | |
| {{DEFAULTSORT:Black-Derman-Toy Model}}
| |
| [[Category:Financial economics]]
| |
| [[Category:Mathematical finance]]
| |
| [[Category:Fixed income analysis]]
| |
| [[Category:Short-rate models]]
| |
Oscar is what my spouse enjoys to call me and I completely dig that title. South Dakota is exactly where me and my spouse reside and my family members loves it. Since she was 18 she's been working as a meter reader but she's usually needed her own business. What I adore performing is doing ceramics but I haven't made a dime with it.
My page :: std testing at home