Definitions

# Short rate model

In the context of interest rate derivatives, a short rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate.

## The short rate

The short rate, usually written rt 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 entire yield curve. However no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of rt as a stochastic process under a risk-neutral measure Q then the price at time t of a zero-coupon bond maturing at time T is given by

$P\left(t,T\right) = mathbb\left\{E\right\}left\left[left. exp\left\{left\left(-int_t^T r_s, dsright\right) \right\} right| mathcal\left\{F\right\}_t right\right]$

where $mathcal\left\{F\right\}$ 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

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

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.

1. The Rendleman-Bartter model models the short rate as $dr_t = theta r_t, dt + sigma r_t, dW_t$
2. The Vasicek model models the short rate as $dr_t = a\left(b-r_t\right), dt + sigma , dW_t$
3. The Ho-Lee model models the short rate as $dr_t = theta_t, dt + sigma, dW_t$
4. The Hull-White model (also called the extended Vasicek model sometimes) posits $dr_t = \left(theta_t-alpha r_t\right),dt + sigma_t , dW_t$. In many presentations one or more of the parameters $theta, alpha$ and $sigma$ are not time-dependent. The process is called an Ornstein-Uhlenbeck process.
5. The Cox-Ingersoll-Ross model supposes $dr_t = \left(theta_t-alpha r_t\right),dt + sqrt\left\{r_t\right\},sigma_t, dW_t$
6. In the Black-Karasinski model a variable Xt is assumed to follow an Ornstein-Uhlenbeck process and rt is assumed to follow $r_t = exp\left\{X_t\right\}$.
7. The Black-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"):

1. The Longstaff-Schwartz model supposes the short rate dynamics is given by the following two equations: $dX_t = \left(theta_t-Y_t\right),dt + sqrt\left\{X_t\right\},sigma_t, dW_t$, 

d Y_t = (zeta_t-Y_t),dt + sqrt{Y_t},sigma_t, dW_t.

1. The Chen model models the short rate, also called stochastic mean and stochastic volatility of the short rate, is given by : $dr_t = \left(theta_t-alpha_t\right),dt + sqrt\left\{r_t\right\},sigma_t, dW_t$, 

d alpha_t = (zeta_t-alpha_t),dt + sqrt{alpha_t},sigma_t, dW_t, $d sigma_t = \left(beta_t-sigma_t\right),dt + sqrt\left\{sigma_t\right\},eta_t, dW_t$.

## 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 and market models, is often preferred for models of higher dimension.

## References

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