Short-rate model
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A short-rate model, in the context of interest rate derivatives, 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, ...
that describes the future evolution 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, th ...
s by describing the future evolution of the short rate, usually written r_t \,.


The short rate

Under a short rate model, the
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
state variable A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
is taken to be the instantaneous
spot rate In finance, a spot contract, spot transaction, or simply spot, is a contract of buying or selling a commodity, security or currency for immediate settlement (payment and delivery) on the spot date, which is normally two business days after ...
. The short rate, r_t \,, then, is the ( continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time t. Specifying the current short rate does not specify the entire
yield curve In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...
. However, no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of r_t \, as a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
under a
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 ...
Q, then the price at time t of a
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- ...
maturing at time T with a payoff of 1 is given by : P(t,T) = \operatorname^Q\left \mathcal_t \right where \mathcal is the
natural filtration In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time. It ...
for the process. The interest rates implied by the zero coupon bonds form a yield curve, or more precisely, a zero curve. Thus, specifying a model for the short rate specifies future bond prices. This means that instantaneous
forward rate The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a ''forward rate''.. Forward rate calculation To extract the forward rate, we ne ...
s are also specified by the usual formula : f(t,T) = - \frac \ln(P(t,T)).


Particular short-rate models

Throughout this section W_t\, represents 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 ...
under a
risk-neutral In economics and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indif ...
probability measure and dW_t\, its differential. Where the model is
lognormal 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 ...
, a variable X_t is assumed to follow an
Ornstein–Uhlenbeck process In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
and r_t \, is assumed to follow r_t = \exp\,.


One-factor short-rate models

Following are the one-factor models, where a single
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
factor – the short rate – determines the future evolution of all interest rates. Other than Rendleman–Bartter and Ho–Lee, which do not capture the mean reversion of interest rates, these models can be thought of as specific cases of Ornstein–Uhlenbeck processes. The Vasicek, Rendleman–Bartter and CIR models have only a finite number of
free parameter A free parameter is a variable in a mathematical model which cannot be predicted precisely or constrained by the model and must be estimated experimentally or theoretically. A mathematical model, theory, or conjecture is more likely to be right ...
s and so it is not possible to specify these
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
values in such a way that the model coincides with observed market prices ("calibration"). This problem is overcome by allowing the parameters to vary deterministically with time. In this way, Ho-Lee and subsequent models can be calibrated to market data, meaning that these can exactly return the price of bonds comprising the yield curve. The implementation is usually via a (
binomial Binomial may refer to: In mathematics *Binomial (polynomial), a polynomial with two terms * Binomial coefficient, numbers appearing in the expansions of powers of binomials *Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition ...
) short rate tree Binomial Term Structure Models
''Mathematica in Education and Research'', Vol. 7 No. 3 1998. Simon Benninga and Zvi Wiener.
or simulation; see and
Monte Carlo methods for option pricing In mathematical finance, a Monte Carlo option model uses Monte Carlo methodsAlthough the term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s, some trace such methods to the 18th century French naturalist Buffon, and a question he as ...
. # Merton's model (1973) explains the short rate as r_t = r_+at+\sigma W^_: where W^_ is a one-dimensional Brownian motion under the spot
martingale 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 ...
. #The Vasicek model (1977) models the short rate as dr_t = (\theta-\alpha r_t)\,dt + \sigma \, dW_t; it is often written dr_t = a(b-r_t)\, dt + \sigma \, dW_t. #The Rendleman–Bartter model (1980) explains the short rate as dr_t = \theta r_t\, dt + \sigma r_t\, dW_t. #The
Cox–Ingersoll–Ross model In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" ( short-rate model) as it describes interest rate movements as driven by only one source of mark ...
(1985) supposes dr_t = (\theta-\alpha r_t)\,dt + \sqrt\,\sigma\, dW_t, it is often written dr_t = a(b-r_t)\, dt + \sqrt\,\sigma\, dW_t. The \sigma \sqrt factor precludes (generally) the possibility of negative interest rates. #The
Ho–Lee model In financial mathematics, the Ho–Lee model is a short-rate model widely used in the pricing of bond options, swaptions and other interest rate derivatives, and in modeling future interest rates. It was developed in 1986 by Thomas Ho and Sang B ...
(1986) models the short rate as dr_t = \theta_t\, dt + \sigma\, dW_t. #The
Hull–White model In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively str ...
(1990)—also called the extended Vasicek model—posits dr_t = (\theta_t-\alpha r_t)\,dt + \sigma_t \, dW_t. In many presentations one or more of the parameters \theta, \alpha and \sigma are not time-dependent. The model may also be applied as lognormal. Lattice-based implementation is usually
trinomial In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. Examples of trinomial expressions # 3x + 5y + 8z with x, y, z variables # 3t + 9s^2 + 3y^3 with t, s, y variables # 3ts + 9t + 5s with t, s variables # ...
. # The
Black–Derman–Toy model In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see . It is a one-factor model; that is, a single stochastic facto ...
(1990) has d\ln(r) = theta_t + \frac\ln(r)t + \sigma_t\, dW_t for time-dependent short rate volatility and d\ln(r) = \theta_t\, dt + \sigma \, dW_t otherwise; the model is lognormal. #The Black–Karasinski model (1991), which is lognormal, has d\ln(r) = theta_t-\phi_t \ln(r)\, dt + \sigma_t\, dW_t . The model may be seen as the lognormal application of Hull–White; its lattice-based implementation is similarly trinomial (binomial requiring varying time-steps). #The
Kalotay–Williams–Fabozzi 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 sho ...
(1993) has the short rate as d \ln(r_t) = \theta_t\, dt + \sigma\, dW_t, a lognormal analogue to the Ho–Lee model, and a special case of the Black–Derman–Toy model. This approach is effectively similar to “the original
Salomon Brothers Salomon Brothers, Inc., was an American multinational bulge bracket investment bank headquartered in New York. It was one of the five largest investment banking enterprises in the United States and the most profitable firm on Wall Street durin ...
model" (1987), also a lognormal variant on Ho-Lee.


Multi-factor short-rate models

Besides the above one-factor models, there are also multi-factor models of the short rate, among them the best known are the Longstaff and Schwartz two factor model and the Chen three factor model (also called "stochastic mean and stochastic volatility model"). Note that for the purposes of risk management, "to create realistic interest rate simulations", these multi-factor short-rate models are sometimes preferred over One-factor models, as they produce scenarios which are, in general, better "consistent with actual yield curve movements". * The
Longstaff–Schwartz 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 ...
(1992) supposes the short rate dynamics are given by :: \begin dX_t & = (a_t-b X_t)\,dt + \sqrt\,c_t\, dW_, \\ ptd Y_t & = (d_t-e Y_t)\,dt + \sqrt\,f_t\, dW_, \end : where the short rate is defined as :: dr_t = (\mu X + \theta Y)\,dt + \sigma_t \sqrt \,dW_. * The
Chen model In finance, the Chen model is a mathematical model describing the evolution of interest rates. It is a type of "three-factor model" ( short-rate model) as it describes interest rate movements as driven by three sources of market risk. It was the ...
(1996) which has a stochastic mean and volatility of the short rate, is given by :: \begin dr_t & = (\theta_t-\alpha_t)\,dt + \sqrt\,\sigma_t\, dW_t, \\ ptd\alpha_t & = (\zeta_t-\alpha_t)\,dt + \sqrt\,\sigma_t\, dW_t, \\ ptd\sigma_t & = (\beta_t-\sigma_t)\,dt + \sqrt\,\eta_t\, dW_t. \end


Other interest rate models

The other major framework for interest rate modelling is the
Heath–Jarrow–Morton framework The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates). When the volatility and drift of the in ...
(HJM). Unlike the short rate models described above, this class of models is generally non-Markovian. This makes general HJM models computationally intractable for most purposes. The great advantage of HJM models is that they give an analytical description of the entire yield curve, rather than just the short rate. For some purposes (e.g., valuation of mortgage backed securities), this can be a big simplification. The Cox–Ingersoll–Ross and Hull–White models in one or more dimensions can both be straightforwardly expressed in the HJM framework. Other short rate models do not have any simple dual HJM representation. The HJM framework with multiple sources of randomness, including as it does the
Brace–Gatarek–Musiela model The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to the names of some of the inventors) is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic derivat ...
and
market model Market is a term used to describe concepts such as: *Market (economics), system in which parties engage in transactions according to supply and demand *Market economy *Marketplace, a physical marketplace or public market Geography *Märket, an ...
s, is often preferred for models of higher dimension. Models based on
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 ...
's shadow rate are used when interest rates approach the
zero lower bound The Zero Lower Bound (''ZLB'') or Zero Nominal Lower Bound (''ZNLB'') is a macroeconomic problem that occurs when the short-term nominal interest rate is at or near zero, causing a liquidity trap and limiting the central bank's capacity to stimulate ...
.


See also

*
Fixed-income attribution Fixed-income attribution is the process of measuring returns generated by various sources of risk in a fixed income portfolio, particularly when multiple sources of return are active at the same time. For example, the risks affecting the return ...


References


Further reading

* * * * *Andrew J.G. Cairns (2004)
Interest-Rate Models
entry in * * * * Lane Hughston (2003)
The Past, Present and Future of Term Structure Modelling
entry in * * * * * * {{derivatives market Interest rates * Mathematical finance