GammaProcess - Maple Help

Finance

 GammaProcess
 create new Gamma process

 Calling Sequence GammaProcess(mu, sigma)

Parameters

 mu - real constant; mean parameter sigma - real constant; variance parameter

Description

 • The GammaProcess command creates a Gamma process with the specified parameters. The Gamma process $G\left(t\right)$ with mean parameter mu and variance parameter sigma is a continuous-time process with stationary, independent gamma increments such that for any $0, $G\left(t+h\right)-G\left(t\right)$ has a Gamma distribution with shape parameter $\frac{{\mathrm{\mu }}^{2}h}{\mathrm{\sigma }}$ and scale parameter $\frac{\mathrm{\sigma }}{\mathrm{\mu }}$.
 • The parameter mu is the mean. The parameter sigma is the variance.

Examples

 > $\mathrm{with}\left(\mathrm{Finance}\right):$
 > $\mathrm{\mu }≔1:$$\mathrm{\sigma }≔3:$
 > $G≔\mathrm{GammaProcess}\left(\mathrm{\mu },\mathrm{\sigma }\right):$
 > $\mathrm{PathPlot}\left(G\left(t\right),t=0..3,\mathrm{timesteps}=100,\mathrm{replications}=10,\mathrm{thickness}=3,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)$
 > $\mathrm{ExpectedValue}\left(G\left(3\right),\mathrm{replications}={10}^{4}\right)$
 $\left[{\mathrm{value}}{=}{2.376349494}{,}{\mathrm{standarderror}}{=}{0.01792180394}\right]$ (1)
 > $S≔\mathrm{SampleValues}\left(G\left(2\right)-G\left(1.98\right),\mathrm{timesteps}={10}^{2},\mathrm{replications}={10}^{3}\right)$
 ${S}{≔}\left[{0.}{,}{0.}{,}{7.35991702799366}{×}{{10}}^{{-6}}{,}{0.}{,}{0.}{,}{7.35602354562381}{×}{{10}}^{{-6}}{,}{2.22044604925031}{×}{{10}}^{{-16}}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{2.68673971959288}{×}{{10}}^{{-14}}{,}{0.}{,}{0.}{,}{1.35483735341779}{×}{{10}}^{{-10}}{,}{1.40403509477430}{×}{{10}}^{{-6}}{,}{0.000394374611912163}{,}{0.}{,}{0.0154969225911339}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{7.51650208741239}{×}{{10}}^{{-9}}{,}{0.}{,}{0.}{,}{2.63580268722308}{×}{{10}}^{{-11}}{,}{0.}{,}{0.0971434323130915}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.0413351186423117}{,}{0.}{,}{0.0000276411228146145}{,}{0.}{,}{0.}{,}{0.}{,}{5.41153927224869}{×}{{10}}^{{-13}}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.0198811462516445}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.000210839334942481}{,}{0.}{,}{1.81071369009800}{×}{{10}}^{{-9}}{,}{5.21804821573824}{×}{{10}}^{{-14}}{,}{0.}{,}{0.}{,}{0.195217033877362}{,}{1.61692881306408}{×}{{10}}^{{-12}}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{1.01821726494755}{×}{{10}}^{{-7}}{,}{0.}{,}{0.}{,}{0.}{,}{0.}{,}{0.00408745142293243}{,}{1.31116352972340}{,}{5.16159894009549}{×}{{10}}^{{-6}}{,}{0.}{,}{\dots }{,}{\text{⋯ 900 Array entries not shown}}\right]$ (2)

The variance gamma process, introduced by Madan and Seneta, is the difference of two independent gamma processes representing the up and down movements of the underlying asset.

 > $\mathrm{Xu}≔\mathrm{GammaProcess}\left(1,3\right):$
 > $\mathrm{Xd}≔\mathrm{GammaProcess}\left(0.9,3\right):$
 > $X≔t↦\mathrm{Xu}\left(t\right)-\mathrm{Xd}\left(t\right)$
 ${X}{≔}{t}{↦}{\mathrm{Xu}}{}\left({t}\right){-}{\mathrm{Xd}}{}\left({t}\right)$ (3)
 > $\mathrm{PathPlot}\left(X\left(t\right),t=0..3,\mathrm{timesteps}=20,\mathrm{replications}=5,\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)$

References

 Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

Compatibility

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