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Finance

  

ItoProcess

  

create new Ito process

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

ItoProcess(x0, mu, sigma)

ItoProcess(x0, mu, sigma, x, t)

ItoProcess(X, Sigma)

Parameters

x0

-

the initial value

mu

-

the drift parameter

sigma

-

volatility parameter

X

-

Vector of one-dimensional Ito processes

Sigma

-

matrix

Description

• 

The ItoProcess command creates a new one- or multi-dimensional Ito process, which is a stochastic process Xt governed by the stochastic differential equation (SDE)

dXt=μXt,tdt+σXt,tdWt

where

– 

μXt,t is the drift parameter

– 

σXt,t is the diffusion parameter

and

– 

Wt is the standard Wiener process.

• 

The parameter x0 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 X1, ..., Xn are one-dimensional Ito processes. Let μ1,...,μn, and σ1,...,σn be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an n-dimensional Ito process Y such that

dYti=μiYti,t+σiYti,tdWti

where Wt is an n-dimensional Wiener process whose covariance matrix is Sigma. Note that the matrix Sigma must have numeric coefficients.

Examples

withFinance:

YItoProcess1.0,μ,σ,x,t

Y_X

(1)

DriftYt

μ

(2)

DiffusionYt

σ

(3)

DriftⅇYt

μⅇ_Xt+12σ2ⅇ_Xt

(4)

DiffusionⅇYt

σⅇ_Xt

(5)

You can generate sample paths for this stochastic process (in order to do this, we must assign numeric values to mu and sigma).

μ0.1

μ0.1

(6)

σ0.5

σ0.5

(7)

PathPlotⅇYt,t=0..3,timesteps=100,replications=10

Here is an example of a multi-dimensional Ito process.

μ'μ'

μμ

(8)

σ'σ'

σσ

(9)

X0100.0,0.0

X0100.00.

(10)

ΜμX1,κθX2

ΜμX1κθX2

(11)

ΣX2X1|0.0,0.0|σX2

ΣX2X10.0.σX2

(12)

SItoProcessX0,Μ,Σ,X,t

S_X1

(13)

DriftSt

μ_X1t1κθ_X1t2

(14)

DiffusionSt

_X1t2_X1t100σ_X1t2

(15)

μ0.1

μ0.1

(16)

σ0.5

σ0.5

(17)

κ1.0

κ1.0

(18)

θ0.4

θ0.4

(19)

ASamplePathSt,t=0..1,timesteps=100,replications=10

A 1..10 x 1..2 x 1..101 ArrayData Type: float[8]Storage: rectangularOrder: C_order

(20)

PathPlotA,1,thickness=3,markers=false,color=red..blue,axes=BOXED,gridlines=true

PathPlotA,2,thickness=3,markers=false,color=red..blue,axes=BOXED,gridlines=true

ExpectedValuemaxS11100,0,timesteps=100,replications=104

value=21.41114565,standarderror=0.3390630872

(21)

In this example, construct a two-dimensional Ito process using two one-dimensional projections and a given covariance matrix.

XGeometricBrownianMotion100.0,0.05,0.3,t

X_X3

(22)

YGeometricBrownianMotion100.0,0.07,0.2,t

Y_X4

(23)

Σ1|0.5,0.5|1

Σ10.50.51

(24)

ZItoProcessX,Y,Σ

Z_X5

(25)

DriftZt

0.05_X5t10.07_X5t2

(26)

DiffusionZt

0.3_X5t10.15_X5t10.10_X5t20.2_X5t2

(27)

ExpectedValuemaxX1Y1,0,timesteps=100,replications=104

value=14.32896059,standarderror=0.2447103632

(28)

ExpectedValuemaxZ11Z12,0,timesteps=100,replications=104

value=8.103315185,standarderror=0.1520913055

(29)

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[ItoProcess] command was introduced in Maple 15.

• 

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

See Also

Finance[BlackScholesProcess]

Finance[CEVProcess]

Finance[Diffusion]

Finance[Drift]

Finance[ExpectedValue]

Finance[GeometricBrownianMotion]

Finance[ItoProcess]

Finance[PathPlot]

Finance[SamplePath]

Finance[SampleValues]

Finance[StochasticProcesses]

Finance[WienerProcess]