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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

μ:=0.1

(6)

σ0.5

σ:=0.5

(7)

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

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

μ'μ'

μ:=μ

(8)

σ'σ'

σ:=σ

(9)

X0100.0,0.0

X0:=100.00.

(10)

ΜμX1,κθX2

Μ:=μX1κθX2

(11)

ΣX2X1|0.0,0.0|σX2

Σ:=X2X10.0.σX2

(12)

SItoProcessX0,Μ,Σ,X,t

S:=_X1

(13)

DriftSt

μ_X1t1κθ_X1t2

(14)

DiffusionSt

_X1t2_X1t100σ_X1t2

(15)

μ0.1

μ:=0.1

(16)

σ0.5

σ:=0.5

(17)

κ1.0

κ:=1.0

(18)

θ0.4

θ:=0.4

(19)

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

A:= 1..10 x 1..2 x 1..101 ArrayData Type: float8Storage: 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

Σ:=10.50.51

(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]

 


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