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Finance[CEVProcess] - create new constant elasticity of variance (CEV) process

Calling Sequence

CEVProcess(x0, mu, sigma, beta, opts)

Parameters

x0

-

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

Description

• 

The CEVProcess command creates new constant elasticity of variance (CEV) process St, which is governed by the stochastic differential equation (SDE)

dSt=μStdt+σStβ2dWt

where

– 

μ is the drift

– 

σ is the volatility

– 

β is the elasticity

and

– 

Wt is the standard Wiener process.

• 

The parameter x0 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 β=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

withFinance:

X:=CEVProcessx0,μ,σ,β

X:=_X

(1)

DriftXt

μ_Xt

(2)

DiffusionXt

σ_Xt12β

(3)

simplifyDriftXt2β

μ_Xt2β2β+12σ2β23β+2

(4)

simplifyDiffusionXt2β

σ_Xt12β+12β

(5)

x0:=1.0

x0:=1.0

(6)

μ:=0.05

μ:=0.05

(7)

σ:=0.3

σ:=0.3

(8)

β:=1.4

β:=1.4

(9)

T:=2.0

T:=2.0

(10)

S:=SamplePathXt,t=0..T,timesteps=100,replications=104:

The following set of examples estimates the distribution of max0,_X21 for different values of the elasticity parameter 1.4.

ExpectedValuemaxXT1,0,timesteps=100,replications=104

value=0.2296864556,standarderror=0.003433128717

(11)

β:=2.0

β:=2.0

(12)

S1:=SampleValuesXT,timesteps=100,replications=104

S1:= 1 .. 10000 ArrayData Type: float8Storage: rectangularOrder: C_order

(13)

β:=5

β:=5

(14)

S2:=SampleValuesXT,timesteps=100,replications=104

S2:= 1 .. 10000 ArrayData Type: float8Storage: rectangularOrder: C_order

(15)

β:=0.1

β:=0.1

(16)

S3:=SampleValuesXT,timesteps=100,replications=104

S3:= 1 .. 10000 ArrayData Type: float8Storage: rectangularOrder: C_order

(17)

P1:=Statistics[FrequencyPlot]S1,range=0..3,bincount=10,averageshifted=5,thickness=3,color=red:

P2:=Statistics[FrequencyPlot]S2,range=0..3,bincount=10,averageshifted=5,thickness=3,color=blue:

P3:=Statistics[FrequencyPlot]S3,range=0..3,bincount=10,averageshifted=5,thickness=3,color=green:

plots[display]P1,P2,P3,gridlines=true,tickmarks=10,10

See Also

Finance[BlackScholesProcess], Finance[BrownianMotion], Finance[Diffusion], Finance[Drift], Finance[ExpectedValue], Finance[GeometricBrownianMotion], Finance[ItoProcess], Finance[SamplePath], Finance[SampleValues], Finance[StochasticProcesses], Finance[WienerProcess]

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.


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