In
financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the Finance#Quantitative_finance, financial field.
In general, there exist two separate ...
, the Black–Karasinski model is a
mathematical model
A mathematical model is an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical m ...
of the
term structure of
interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, ...
s; see
short-rate model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.
The short rate
Under a sh ...
. It is a one-factor model as it describes interest rate movements as driven by a single source of randomness. It belongs to the class of no-arbitrage models, i.e. it can fit today's
zero-coupon bond
A zero-coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. Unlike regular bonds, it does not make periodic interest payments or have so-called coupons, hence the term zer ...
prices, and in its most general form, today's prices for a set of caps, floors or European
swaptions. The model was introduced by
Fischer Black
Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist, best known as one of the authors of the Black–Scholes equation. Working variously at the University of Chicago, the Massachusetts Institute of Technology, ...
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
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or '' equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price un ...
):
:
where ''dW''
''t'' is a standard
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
. The model implies a
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
for the short rate and therefore the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
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 tree with variable spacing, but a
trinomial tree
The trinomial tree is a Lattice model (finance), lattice-based computational model used in financial mathematics to price option (finance), options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, ...
implementation is more common in practice, typically a log-normal application of the
Hull–White lattice.
Applications
The model is used mainly for the pricing of
exotic interest rate derivative
In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of dif ...
s such as
American and
Bermudan bond option
In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.
*A European bond option is an option to buy or sell a bond at a certain date in fu ...
s and
swaptions, once its parameters have been calibrated to the current term structure of interest rates and to the prices or
implied volatilities of
caps,
floors
A floor is the bottom surface of a room or vehicle. Floors vary from simple dirt in a cave to many layered surfaces made with modern technology. Floors may be stone, wood, bamboo, metal or any other material that can support the expected load ...
or
European swaptions.
Numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
s (
usually trees) are used in the calibration stage as well as for pricing. It can also be used in modeling
credit default risk, where the Black–Karasinski short rate expresses the (stochastic) intensity of default events driven by a
Cox process; the guaranteed positive rates are an important feature of the model here. Recent work o
Perturbation Methods in Credit Derivativeshas shown how analytic prices can be conveniently deduced in many such circumstances, as well as for interest rate options.
References
*
*
External links
* Simon Benninga and Zvi Wiener (1998)
Binomial Term Structure Models ''Mathematica in Education and Research'', Vol. 7 No. 3 1998
* Blanka Horvath, Antoine Jacquier and Colin Turfus (2017)
Analytic Option Prices for the Black–Karasinski Short Rate Model* Colin Turfus (2018)
Analytic Swaption Pricing in the Black–Karasinski Model* Colin Turfus (2018)
Exact Arrow-Debreu Pricing for the Black–Karasinski Short Rate Model* Colin Turfus (2019)
Perturbation Expansion for Arrow–Debreu Pricing with Hull-White Interest Rates and Black–Karasinski Credit Intensity* Colin Turfus and Piotr Karasinski (2021)
The Black-Karasinski Model: Thirty Years On
{{DEFAULTSORT:Black-Karasinski Model
Short-rate models
Financial models