financial mathematics Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...

, the Black–Karasinski model is a

mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

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, t ...

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 s ...

. 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 zero- ...

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. Background Fischer Sheffey Black was born on January 11, 1938. He graduated from Harvard ...

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 u ...

): :

\, dt + \sigma_t\, dW_t

where ''dW''_''t'' is a standard

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

. 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 normally distributed. Thus, if the random variable is log-normally distributed, then has a norma ...

for the short rate and therefore the

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

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 In computer science, a binomial heap is a data structure that acts as a priority queue but also allows pairs of heaps to be merged. It is important as an implementation of the mergeable heap abstract data type (also called meldable heap), wh ...

with variable spacing, but a

trinomial tree The trinomial tree is a lattice-based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also ...

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 Exotic may refer to: Mathematics and physics * Exotic R4, a differentiable 4-manifold, homeomorphic but not diffeomorphic to the Euclidean space R4 *Exotic sphere, a differentiable ''n''-manifold, homeomorphic but not diffeomorphic to the ordinar ...

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 diff ...

s such as

American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, pe ...

and Bermudan bond options and

swaptions A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps. Types o ...

, once its parameters have been calibrated to the current term structure of interest rates and to the prices or implied volatilities of

caps Caps are flat headgear. Caps or CAPS may also refer to: Science and technology Computing * CESG Assisted Products Service, provided by the U.K. Government Communications Headquarters * Composite Application Platform Suite, by Java Caps, a Ja ...

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 In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the intensity that varies across the underlying mathematical space (often space or time ...

; the guaranteed positive rates are an important feature of the model here. Recent work o
Perturbation Methods in Credit Derivatives
has 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