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Finance

 CEVProcess
 create new constant elasticity of variance (CEV) process

 Calling Sequence CEVProcess(${x}_{0}$, mu, sigma, beta, opts)

Parameters

 ${x}_{0}$ - algebraic expression; initial value mu - algebraic expression; drift parameter sigma - algebraic expression; volatility parameter beta - algebraic expression; elasticity parameter opts - (optional) equation(s) of the form option = value where option is scheme; specify options for the CEVProcess command

Options

 • scheme = unbiased or Euler -- This option specifies which discretization scheme should be used for simulating this process.

Description

 • The CEVProcess command creates new constant elasticity of variance (CEV) process $S\left(t\right)$, which is governed by the stochastic differential equation (SDE)

$\mathrm{dS}\left(t\right)=\mathrm{\mu }S\left(t\right)\mathrm{dt}+\mathrm{\sigma }{S\left(t\right)}^{\frac{\mathrm{\beta }}{2}}\mathrm{dW}\left(t\right)$

where

 – $\mathrm{\mu }$ is the drift
 – $\mathrm{\sigma }$ is the volatility
 – $\mathrm{\beta }$ is the elasticity

and

 – $W\left(t\right)$ is the standard Wiener process.
 • The parameter ${x}_{0}$ is the initial value of the process.
 • The parameters mu, sigma and beta can be any algebraic expressions but must be constant if the process is to be simulated.
 • The constant elasticity of variance (CEV) process provides an alternative to the lognormal model for equity prices. This model includes the geometric Brownian motion as a special case $\mathrm{\beta }=2$. The main advantage of such a model is that the volatility of the stock price is no more constant but it is a function of the underlying asset price. In particular, in the CEV model the variations in the underlying asset price are negative correlated with the variations in the volatility level which helps to reduce the well-known volatility smile effect of the lognormal model.

Examples

 > $\mathrm{with}\left(\mathrm{Finance}\right):$
 > $X≔\mathrm{CEVProcess}\left(\mathrm{x0},\mathrm{μ},\mathrm{σ},\mathrm{β}\right)$
 ${X}{≔}{\mathrm{_X}}$ (1)
 > $\mathrm{Drift}\left(X\left(t\right)\right)$
 ${\mathrm{μ}}{}{\mathrm{_X}}{}\left({t}\right)$ (2)
 > $\mathrm{Diffusion}\left(X\left(t\right)\right)$
 ${\mathrm{σ}}{}{{\mathrm{_X}}{}\left({t}\right)}^{\frac{{1}}{{2}}{}{\mathrm{β}}}$ (3)
 > $\mathrm{simplify}\left(\mathrm{Drift}\left({X\left(t\right)}^{2-\mathrm{β}}\right)\right)$
 ${\mathrm{μ}}{}{{\mathrm{_X}}{}\left({t}\right)}^{{2}{-}{\mathrm{β}}}{}\left({2}{-}{\mathrm{β}}\right){+}\frac{{1}}{{2}}{}{{\mathrm{σ}}}^{{2}}{}\left({{\mathrm{β}}}^{{2}}{-}{3}{}{\mathrm{β}}{+}{2}\right)$ (4)
 > $\mathrm{simplify}\left(\mathrm{Diffusion}\left({X\left(t\right)}^{2-\mathrm{β}}\right)\right)$
 ${\mathrm{σ}}{}{{\mathrm{_X}}{}\left({t}\right)}^{{-}\frac{{1}}{{2}}{}{\mathrm{β}}{+}{1}}{}\left({2}{-}{\mathrm{β}}\right)$ (5)
 > $\mathrm{x0}≔1.0$
 ${\mathrm{x0}}{≔}{1.0}$ (6)
 > $\mathrm{μ}≔0.05$
 ${\mathrm{μ}}{≔}{0.05}$ (7)
 > $\mathrm{σ}≔0.3$
 ${\mathrm{σ}}{≔}{0.3}$ (8)
 > $\mathrm{β}≔1.4$
 ${\mathrm{β}}{≔}{1.4}$ (9)
 > $T≔2.0$
 ${T}{≔}{2.0}$ (10)
 > $S≔\mathrm{SamplePath}\left(X\left(t\right),t=0..T,\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right):$

The following set of examples estimates the distribution of $\mathrm{max}\left(0,\mathrm{_X}\left(2\right)-1\right)$ for different values of the elasticity parameter $1.4$.

 > $\mathrm{ExpectedValue}\left(\mathrm{max}\left(X\left(T\right)-1,0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)$
 $\left[{\mathrm{value}}{=}{0.2296864556}{,}{\mathrm{standarderror}}{=}{0.003433128717}\right]$ (11)
 > $\mathrm{β}≔2.0$
 ${\mathrm{β}}{≔}{2.0}$ (12)
 > $\mathrm{S1}≔\mathrm{SampleValues}\left(X\left(T\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)$
 ${\mathrm{S1}}{≔}\left[\begin{array}{c}{\mathrm{1 .. 10000}}{\mathrm{Array}}\\ {\mathrm{Data Type:}}{\mathrm{float}}{[}{8}{]}\\ {\mathrm{Storage:}}{\mathrm{rectangular}}\\ {\mathrm{Order:}}{\mathrm{C_order}}\end{array}\right]$ (13)
 > $\mathrm{β}≔5$
 ${\mathrm{β}}{≔}{5}$ (14)
 > $\mathrm{S2}≔\mathrm{SampleValues}\left(X\left(T\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)$
 ${\mathrm{S2}}{≔}\left[\begin{array}{c}{\mathrm{1 .. 10000}}{\mathrm{Array}}\\ {\mathrm{Data Type:}}{\mathrm{float}}{[}{8}{]}\\ {\mathrm{Storage:}}{\mathrm{rectangular}}\\ {\mathrm{Order:}}{\mathrm{C_order}}\end{array}\right]$ (15)
 > $\mathrm{β}≔0.1$
 ${\mathrm{β}}{≔}{0.1}$ (16)
 > $\mathrm{S3}≔\mathrm{SampleValues}\left(X\left(T\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)$
 ${\mathrm{S3}}{≔}\left[\begin{array}{c}{\mathrm{1 .. 10000}}{\mathrm{Array}}\\ {\mathrm{Data Type:}}{\mathrm{float}}{[}{8}{]}\\ {\mathrm{Storage:}}{\mathrm{rectangular}}\\ {\mathrm{Order:}}{\mathrm{C_order}}\end{array}\right]$ (17)
 > $\mathrm{P1}≔\mathrm{Statistics}[\mathrm{FrequencyPlot}]\left(\mathrm{S1},\mathrm{range}=0..3,\mathrm{bincount}=10,\mathrm{averageshifted}=5,\mathrm{thickness}=3,\mathrm{color}=\mathrm{red}\right):$
 > $\mathrm{P2}≔\mathrm{Statistics}[\mathrm{FrequencyPlot}]\left(\mathrm{S2},\mathrm{range}=0..3,\mathrm{bincount}=10,\mathrm{averageshifted}=5,\mathrm{thickness}=3,\mathrm{color}=\mathrm{blue}\right):$
 > $\mathrm{P3}≔\mathrm{Statistics}[\mathrm{FrequencyPlot}]\left(\mathrm{S3},\mathrm{range}=0..3,\mathrm{bincount}=10,\mathrm{averageshifted}=5,\mathrm{thickness}=3,\mathrm{color}=\mathrm{green}\right):$
 > $\mathrm{plots}[\mathrm{display}]\left(\mathrm{P1},\mathrm{P2},\mathrm{P3},\mathrm{gridlines}=\mathrm{true},\mathrm{tickmarks}=\left[10,10\right]\right)$

References

 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[CEVProcess] command was introduced in Maple 15.