Short rate model
From Freepedia
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.
Contents |
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
- <math> P(t,T) = \mathbb{E}\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right] </math>
where <math>\mathcal{F}</math> is the natural filtration for the process. Thus specifying a model for the short rate specifies future bond prices. This means that future instantaneous forward rates are also specified by the usual formula
- <math> f(t,T) = - \frac{\partial}{\partial T} P(t,T). </math>
Particular short-rate models
Throughout this section <math>W_t</math> represents a standard Brownian motion and <math>dW_t</math> its differential.
- The Ho-Lee model models the short rate as <math>dr_t = \theta_t\, dt + \sigma\, dW_t</math>
- The Hull-White model (also called the Vasicek model almost interchangeably) posits <math>dr_t = (\theta_t-\alpha r_t)\,dt + \sigma_t \, dW_t</math>. In many presentations one or more of the parameters <math>\theta, \alpha</math> and <math>\sigma</math> are not time-dependent. The process is called a Ornstein-Uhlenbeck or OU process.
- The Cox-Ingersoll-Ross model supposes <math>dr_t = (\theta_t-\alpha r_t)\,dt + \sqrt{r_t}\,\sigma_t\, dW_t</math>
In the Black-Karasinski model a variable Xt is assumed to follow an OU process and rt is assumed to follow <math>r_t = \exp{X_t}</math>.
Other interest rate models
The other major framework for interest rate modelling is the Heath-Jarrow-Morton framework. Whilst the two frameworks are actually equivalent in scope for modelling interest rates with one source of uncertainty (one driving Brownian motion), the latter, including as it does the Brace-Gatarek-Musiela model and market models, are often preferred for models of higher dimension.
References
- {{{Author|}}}{{|{{{3}}}}}}|show1| (1996)}}{{{{{Year|}}}}}}|show1|.}} {{|{{{3}}}}}}|show1|[{{{URL}}}}} Financial Calculus{{|{{{3}}}}}}|show1|]}}{{|{{{3}}}}}}|show1|, {{{Pages}}}}}{{|{{{3}}}}}}|Show1|, Cambridge University Press}}. {{{ID|}}}
- {{{Author|}}}{{|{{{3}}}}}}|show1| (2002)}}{{{{{Year|}}}}}}|show1|.}} {{|{{{3}}}}}}|show1|[{{{URL}}}}} Modern Pricing of Interest-Rate Derivatives{{|{{{3}}}}}}|show1|]}}{{|{{{3}}}}}}|show1|, {{{Pages}}}}}{{|{{{3}}}}}}|Show1|, Princeton University Press}}. {{{ID|}}}
