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A short-rate model, in the context of interest rate derivatives, is a mathematical model 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, ...
s by describing the future evolution of the short rate, usually written r_t \,.


The short rate

Under a short rate model, the stochastic
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. 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 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 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 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 rates 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, 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 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 parameters 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) 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. # 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 (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 (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 (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 (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 (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 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.


See also

* Fixed-income attribution


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