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In [[financial mathematics]], the '''Black–Karasinski model''' is a [[mathematical model]] of the [[term structure]] of [[interest rate]]s; see [[short rate model]]. It is a one-factor model as it describes interest rate movements as driven by a single source of randomness.
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It belongs to the class of no-arbitrage models, i.e. it can fit today's [[zero-coupon bond]] prices, and in its most general form, today's prices for a set of caps, floors or European [[swaption]]s. The model was introduced by [[Fischer Black]] and [[Piotr Karasinski]] in 1991.
 
==Model==
 
The main state variable of the model is the short rate, which is assumed to follow the stochastic differential equation (under the [[risk-neutral measure]]):
 
: <math> d\ln(r) = [\theta_t-\phi_t \ln(r)] \, dt + \sigma_t\, dW_t </math>
 
where ''dW''<sub>''t''</sub> is a standard [[Wiener process|Brownian motion]]. The model implies a [[log-normal distribution]] for the short rate and therefore the [[expected value]] of the money-market account is infinite for any maturity.
 
In the original article by Fischer Black and Piotr Karasinski the model was implemented using a [[Binomial options pricing model|binomial tree]] with variable spacing, but a [[trinomial tree]] implementation is more common in practice, typically a lognormal application of the [[Hull%E2%80%93White_model#Trees_and_lattices|Hull-White Lattice]].
 
==Applications==
The model is used mainly for the pricing of [[exotic option|exotic]] [[interest rate derivative]]s such as [[American option|American]] and [[Bermudan option|Bermudan]] [[bond option]]s and [[swaptions]], once its parameters have been calibrated to the current term structure of interest rates and to the prices or [[implied volatility|implied volatilities]] of [[Interest rate cap|caps]], [[Interest rate floor|floors]] or [[European option|European]] swaptions. [[Numerical method]]s (usually trees) are used in the calibration stage as well as for pricing.
 
==References==
{{reflist}}
{{refbegin}}
*{{cite journal|first=F.|last=Black|coauthors=Karasinski, P.|title=Bond and Option pricing when Short rates are Lognormal|date=July–August 1991|pages=52–59|journal=Financial Analysts Journal}}
*{{cite book | title = Interest Rate Models – Theory and Practice with Smile, Inflation and Credit| author = Damiano Brigo, Fabio Mercurio | publisher = Springer Verlag | year = 2001 | edition = 2nd ed. 2006 | isbn = 978-3-540-22149-4}}
{{refend}}
 
==External links==
*Simon Benninga and Zvi Wiener (1998). [http://simonbenninga.com/wiener/MiER73.pdf Binomial Term Structure Models], ''Mathematica in Education and Research'', Vol. 7 No. 3 1998
 
{{Stochastic processes}}
 
{{DEFAULTSORT:Black-Karasinski Model}}
[[Category:Mathematical modeling]]
[[Category:Short-rate models]]

Latest revision as of 16:37, 2 October 2014



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