The ItoProcess command creates a new one- or multi-dimensional Ito process, which is a stochastic process $X\left(t\right)$ governed by the stochastic differential equation (SDE)

$\mathrm{\mu }\left(X\left(t\right)\,t\right)$ is the drift parameter

$\mathrm{\sigma }\left(X\left(t\right)\,t\right)$ is the diffusion parameter

$W\left(t\right)$ is the standard Wiener process.

The parameter $x_0$ defines the initial value of the underlying stochastic process. It must be a real constant.

The parameter mu is the drift. In the simplest case of a constant drift mu is real number (that is, any expression of type realcons). Time-dependent drift can be given either as an algebraic expression or as a Maple procedure. If mu is given as an algebraic expression, then the parameter t must be passed to specify which variable in mu should be used as a time variable. A Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift.

The parameter sigma is the diffusion. Similar to the drift parameter, the volatility can be constant or time-dependent.

One can use the ItoProcess command to construct a multi-dimensional Ito process with the given correlation structure. To be more precise, assume that $X$ is an $n$-dimensional vector whose components $X_1$, ..., $X_n$ are one-dimensional Ito processes. Let ${\mathrm{\mu }}_1$,...,${\mathrm{\mu }}_n$, and ${\mathrm{\sigma }}_1$,...,${\mathrm{\sigma }}_n$ be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an $n$-dimensional Ito process $Y$ such that

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

$Y≔\mathrm{ItoProcess}\left(1.0\,\mathrm{\mu }\,\mathrm{\sigma }\,x\,t\right)$

$Y≔\mathrm{\_X0}$

$\mathrm{Drift}\left(Y\left(t\right)\right)$

$\mathrm{\mu }$

$\mathrm{Diffusion}\left(Y\left(t\right)\right)$

$\mathrm{\sigma }$

$\mathrm{Drift}\left(\exp \left(Y\left(t\right)\right)\right)$

$\mathrm{\mu }{\ⅇ}^{\mathrm{\_X0}\left(t\right)}+\frac{{\mathrm{\sigma }}^2{\ⅇ}^{\mathrm{\_X0}\left(t\right)}}{2}$

$\mathrm{Diffusion}\left(\exp \left(Y\left(t\right)\right)\right)$

$\mathrm{\sigma }{\ⅇ}^{\mathrm{\_X0}\left(t\right)}$

$\mathrm{\mu }≔0.1$

$\mathrm{\mu }≔0.1$

$\mathrm{\sigma }≔0.5$

$\mathrm{\sigma }≔0.5$

$\mathrm{PathPlot}\left(\exp \left(Y\left(t\right)\right)\,t=0..3\,\mathrm{timesteps}=100\,\mathrm{replications}=10\right)$

$\mathrm{\mu }≔'\mathrm{\mu }'$

$\mathrm{\mu }≔\mathrm{\mu }$

$\mathrm{\sigma }≔'\mathrm{\sigma }'$

$\mathrm{\sigma }≔\mathrm{\sigma }$

$\mathrm{X0}≔⟨100.0\,0.⟩$

$\mathrm{X0}≔\left[\begin{array}{c}100.0\\ 0.\end{array}\right]$

$\mathrm{{\rm M}}≔⟨\mathrm{\mu }X\left[1\right]\,\mathrm{\kappa }\left(\mathrm{\theta }-X\left[2\right]\right)⟩$

$\mathrm{{\rm M}}≔\left[\begin{array}{c}\mathrm{\mu }{X}_1\\ \mathrm{\kappa }\left(\mathrm{\theta }-X_2\right)\end{array}\right]$

$\mathrm{\Sigma }≔⟨⟨\mathrm{sqrt}\left(X\left[2\right]\right)X\left[1\right]|0.⟩\,⟨0.|\mathrm{\sigma }X\left[2\right]⟩⟩$

$\mathrm{\Sigma }≔\left[\begin{array}{cc}\sqrt{X_2}{X}_1& 0.\\ 0.& \mathrm{\sigma }{X}_2\end{array}\right]$

$S≔\mathrm{ItoProcess}\left(\mathrm{X0}\,\mathrm{{\rm M}}\,\mathrm{\Sigma }\,X\,t\right)$

$S≔\mathrm{\_X2}$

$\mathrm{Drift}\left(S\left(t\right)\right)$

$\left[\begin{array}{c}\mathrm{\mu }{\mathrm{\_X2}\left(t\right)}_1\\ \mathrm{\kappa }\left(\mathrm{\theta }-{\mathrm{\_X2}\left(t\right)}_2\right)\end{array}\right]$

$\mathrm{Diffusion}\left(S\left(t\right)\right)$

$\left[\begin{array}{cc}\sqrt{{\mathrm{\_X2}\left(t\right)}_2}{\mathrm{\_X2}\left(t\right)}_1& 0\\ 0& \mathrm{\sigma }{\mathrm{\_X2}\left(t\right)}_2\end{array}\right]$

$\mathrm{\mu }≔0.1$

$\mathrm{\mu }≔0.1$

$\mathrm{\sigma }≔0.5$

$\mathrm{\sigma }≔0.5$

$\mathrm{\kappa }≔1.0$

$\mathrm{\kappa }≔1.0$

$\mathrm{\theta }≔0.4$

$\mathrm{\theta }≔0.4$

$A≔\mathrm{SamplePath}\left(S\left(t\right)\,t=0..1\,\mathrm{timesteps}=100\,\mathrm{replications}=10\right)$

$\mathrm{PathPlot}\left(A\,1\,\mathrm{thickness}=3\,\mathrm{markers}=\mathrm{false}\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(A\,2\,\mathrm{thickness}=3\,\mathrm{markers}=\mathrm{false}\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{ExpectedValue}\left(\max \left(S\left(1\right)\left[1\right]-100\,0\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^4\right)$

$\left[\mathrm{value}=21.41114565\,\mathrm{standarderror}=0.3390630872\right]$

$X≔\mathrm{GeometricBrownianMotion}\left(100.0\,0.05\,0.3\,t\right)$

$X≔\mathrm{\_X4}$

$Y≔\mathrm{GeometricBrownianMotion}\left(100.0\,0.07\,0.2\,t\right)$

$Y≔\mathrm{\_X5}$

$\mathrm{\Sigma }≔⟨⟨1|0.5⟩\,⟨0.5|1⟩⟩$

$\mathrm{\Sigma }≔\left[\begin{array}{cc}1& 0.5\\ 0.5& 1\end{array}\right]$

$Z≔\mathrm{ItoProcess}\left(⟨X\,Y⟩\,\mathrm{\Sigma }\right)$

$Z≔\mathrm{\_X6}$

$\mathrm{Drift}\left(Z\left(t\right)\right)$

$\left[\begin{array}{c}0.05{\mathrm{\_X6}\left(t\right)}_1\\ 0.07{\mathrm{\_X6}\left(t\right)}_2\end{array}\right]$

$\mathrm{Diffusion}\left(Z\left(t\right)\right)$

$\left[\begin{array}{cc}0.3{\mathrm{\_X6}\left(t\right)}_1& 0.15{\mathrm{\_X6}\left(t\right)}_1\\ 0.10{\mathrm{\_X6}\left(t\right)}_2& 0.2{\mathrm{\_X6}\left(t\right)}_2\end{array}\right]$

$\mathrm{ExpectedValue}\left(\max \left(X\left(1\right)-Y\left(1\right)\,0\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^4\right)$

$\left[\mathrm{value}=14.28360857\,\mathrm{standarderror}=0.2462050212\right]$

$\mathrm{ExpectedValue}\left(\max \left(Z\left(1\right)\left[1\right]-Z\left(1\right)\left[2\right]\,0\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^4\right)$

$\left[\mathrm{value}=8.073063829\,\mathrm{standarderror}=0.1512011974\right]$

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.

The Finance[ItoProcess] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.