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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)
 > $S≔\mathrm{SampleValues}\left(G\left(2\right)-G\left(1.98\right),\mathrm{timesteps}={10}^{2},\mathrm{replications}={10}^{3}\right)$
  (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.