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Finance

 CoxIngersollRossModel
 define Cox-Ingersoll-Ross interest rate model

 Calling Sequence CoxIngersollRossModel(r, theta, kappa, sigma, ${x}_{0}$)

Parameters

 r - initial term structure theta - long term mean level kappa - speed of reversion sigma - volatility ${x}_{0}$ - initial value

Description

 • The CoxIngersollRossModel command creates a Cox-Ingersoll-Ross model with the specified parameters. Under this model the short rate process $r\left(t\right)$ has the following dynamics with respect to the risk-neutral measure

$\mathrm{dr}\left(t\right)=\mathrm{\kappa }\left(\mathrm{\theta }-r\left(t\right)\right)\mathrm{dt}+\mathrm{\sigma }\sqrt{r\left(t\right)}\mathrm{dW}\left(t\right)$

where $\mathrm{\theta }$, $\mathrm{\kappa }$, $\mathrm{\sigma }$, and ${x}_{0}$ are non-negative constants and W(t) is a Wiener process modeling the random market risk factor.

It is reasonable to require that ${\mathrm{\sigma }}^{2}<2\mathrm{\kappa }\mathrm{\theta }$.

Examples

 > $\mathrm{with}\left(\mathrm{Finance}\right):$

First define a Cox-Ingersoll-Ross model with parameters ${r}_{0}=0.03$, $\mathrm{\theta }=0.05$, $\mathrm{\kappa }=0.5$, $\mathrm{\sigma }=0.002$, and ${x}_{0}=0.1$.

 > $M≔\mathrm{CoxIngersollRossModel}\left(\mathrm{ZeroCurve}\left(0.03\right),0.05,0.5,0.002,0.1\right)$
 ${M}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (1)

Here is the corresponding short rate tree.

 > $T≔\mathrm{ShortRateTree}\left(M,5,40\right)$
 ${T}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (2)
 > $\mathrm{TreePlot}\left(T,\mathrm{axes}=\mathrm{BOXED},\mathrm{thickness}=2,\mathrm{gridlines}=\mathrm{true}\right)$ References

 Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice. New York: Springer-Verlag, 2001.
 Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.
 Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

Compatibility

 • The Finance[CoxIngersollRossModel] command was introduced in Maple 15.