- Short rate model
In the context of

interest rate derivatives, a**short rate model**is amathematical model that describes the future evolution ofinterest rate s by describing the future evolution of the**short rate**.**The short rate**The short rate, usually written "r"

_{"t"}is the (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 entireyield curve . 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 arisk-neutral measure "Q" then the price at time "t" of a zero-coupon bond maturing at time "T" is given by:$P(t,T)\; =\; mathbb\{E\}left\; [left.\; exp\{left(-int\_t^T\; r\_s,\; ds\; ight)\; \}\; ight|\; mathcal\{F\}\_t\; ight]$

where $mathcal\{F\}$ is the natural filtration for the process. 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

:$f(t,T)\; =\; -\; frac\{partial\}\{partial\; T\}\; ln(P(t,T)).$

And its third equivalent, the yields are given as well.

**Particular short-rate models**Throughout this section $W\_t$ represents a standard

Brownian motion and $dW\_t$ its differential.#The

Rendleman-Bartter model models the short rate as $dr\_t\; =\; heta\; r\_t,\; dt\; +\; sigma\; r\_t,\; dW\_t$

#TheVasicek model models the short rate as $dr\_t\; =\; a(b-r\_t),\; dt\; +\; sigma\; ,\; dW\_t$

#TheHo-Lee model models the short rate as $dr\_t\; =\; heta\_t,\; dt\; +\; sigma,\; dW\_t$

#TheHull-White model (also called the extended Vasicek model sometimes) posits $dr\_t\; =\; (\; heta\_t-alpha\; r\_t),dt\; +\; sigma\_t\; ,\; dW\_t$. In many presentations one or more of the parameters $heta,\; alpha$ and $sigma$ are not time-dependent. The process is called anOrnstein-Uhlenbeck process .

#TheCox-Ingersoll-Ross model supposes $dr\_t\; =\; (\; heta\_t-alpha\; r\_t),dt\; +\; sqrt\{r\_t\},sigma\_t,\; dW\_t$

#In theBlack-Karasinski model a variable "X"_{"t"}is assumed to follow an Ornstein-Uhlenbeck process and "r"_{"t"}is assumed to follow $r\_t\; =\; exp\{X\_t\}$.

# TheBlack-Derman-Toy model Besides the above one-factor models, there are also multi-factor models of the short rate, among them the best known are Longstaff and Schwartz two factor model and Chen three factor model (also called "stochastic mean and stochastic volatility model"):

#The

Longstaff-Schwartz model supposes the short rate dynamics is given by the following two equations: $dX\_t\; =\; (\; heta\_t-Y\_t),dt\; +\; sqrt\{X\_t\},sigma\_t,\; dW\_t$, $d\; Y\_t\; =\; (zeta\_t-Y\_t),dt\; +\; sqrt\{Y\_t\},sigma\_t,\; dW\_t$.

#TheChen model models the short rate, also called stochastic mean and stochastic volatility of the short rate, is given by : $dr\_t\; =\; (\; heta\_t-alpha\_t),dt\; +\; sqrt\{r\_t\},sigma\_t,\; dW\_t$, $d\; alpha\_t\; =\; (zeta\_t-alpha\_t),dt\; +\; sqrt\{alpha\_t\},sigma\_t,\; dW\_t$, $d\; sigma\_t\; =\; (eta\_t-sigma\_t),dt\; +\; sqrt\{sigma\_t\},eta\_t,\; dW\_t$.**Other interest rate models**The other major framework for interest rate modelling is the

Heath-Jarrow-Morton framework (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 andmarket model s, is often preferred for models of higher dimension.**References***

*

* cite book | author = Jessica James and Nick Webber | year = 2000 | title = Interest Rate Modelling

publisher = Wiley Finance | id = ISBN 0-471-97523-0*

*

*cite book | title = Interest Rate Models - An Introduction | author = Andrew J.G. Cairns | publisher =

Princeton University Press | year = 2004 | id = ISBN 0-691-11894-9

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