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MertonJumpDiffusion

  

create new jump diffusion process

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

MertonJumpDiffusion(X, lambda, a, b)

MertonJumpDiffusion(S0, sigma, r, d, lambda, a, b, t, S)

Parameters

X

-

Black-Scholes process

lambda

-

intensity of the lognormal Poisson process

a

-

scale parameter of the lognormal Poisson process

b

-

shape parameter of the lognormal Poisson process

S0

-

non-negative constant; initial value

sigma

-

non-negative constant, procedure, or local volatility structure; volatility

r

-

non-negative constant, procedure, or yield term structure; risk-free rate

d

-

non-negative constant, procedure, or yield term structure; dividend yield

t

-

name; time variable

S

-

name; state variable

Description

• 

The MertonJumpDiffusion command creates a new jump diffusion process that is governed by the stochastic differential equation (SDE)

dStS`t-`=μtdt+σtdWt+dJt

  

where

– 

μt is the drift parameter

– 

σt is the volatility parameter

– 

Wt is the standard Wiener process

  

and

– 

Jt is a compound Poisson process of the form

Jt=j=1NtYj1

  

such that logYi is independent and lognormally distributed with mean a and standard deviation b.

• 

Both the drift parameter mu and the volatility parameter sigma can be either constant or time-dependent. In the second case they can be specified either as an algebraic expression containing one indeterminate, or as a procedure that accepts one parameter (the time) and returns the corresponding value of the drift (volatility).

• 

Similar to the drift and the volatility parameters, the intensity parameter lambda can be either constant or time-dependent. In the second case it can be specified either as an algebraic expression containing one indeterminate or as a procedure that accepts one parameter (the time).

• 

Both the scale parameter a and the shape parameter b of the underlying lognormal Poisson process must be real constants.

Examples

withFinance:

First consider two examples of jump diffusion with low volatility to observe the effect of jumps.

S0100

S0:=100

(1)

r0.05

r:=0.05

(2)

d0.01

d:=0.01

(3)

σ10.01

σ1:=0.01

(4)

a0.0

a:=0.

(5)

b0.5

b:=0.5

(6)

λ12.0

λ1:=2.0

(7)

λ20.2

λ2:=0.2

(8)

X1MertonJumpDiffusionS0,σ1,r,d,λ1,a,b:

PathPlotX[1]t,t=0..1,timesteps=100,replications=5,color=red..blue,thickness=3,axes=BOXED,gridlines=true

X2MertonJumpDiffusionS0,σ1,r,d,λ2,a,b:

PathPlotX[2]t,t=0..1,timesteps=100,replications=5,color=red..blue,thickness=3,axes=BOXED,gridlines=true

Now consider similar processes but with relatively high volatility.

σ20.5

σ2:=0.5

(9)

X3MertonJumpDiffusionS0,σ2,r,d,λ2,a,b:

PathPlotX[3]t,t=0..1,timesteps=100,replications=5,color=red..blue,thickness=3,axes=BOXED,gridlines=true

YBlackScholesProcessS0,σ2,r,d:

X4MertonJumpDiffusionY,λ1,0,b:

PathPlotX[4]t,t=0..1,timesteps=100,replications=5,color=red..blue,thickness=3,axes=BOXED,gridlines=true

ExpectedValuemaxX[4]190,0,timesteps=100,replications=105

value=65.32590075,standarderror=0.5535149183

(10)

S1SampleValuesY1,timesteps=100,replications=105

S1:= 1 .. 100000 ArrayData Type: float8Storage: rectangularOrder: C_order

(11)

S2SampleValuesX[4]1,timesteps=100,replications=105

S2:= 1 .. 100000 ArrayData Type: float8Storage: rectangularOrder: C_order

(12)

P1Statistics[FrequencyPlot]S1,range=0..300,thickness=3,color=red,bincount=50:

P2Statistics[FrequencyPlot]S2,range=0..300,thickness=3,color=blue,bincount=50:

plots[display]P1,P2,axes=BOXED,gridlines=true:

Here is another way to define the same jump diffusion process.

JPoissonProcessλ2,Normala,b

J:=_P

(13)

Zt→YtⅇJt

Z:=t→YtⅇJt

(14)

ExpectedValuemaxZ190,0,timesteps=100,replications=105

value=30.42046277,standarderror=0.1770232342

(15)

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.

  

Merton, R.C., On the pricing when underlying stock returns are discontinuous, Journal of Financial Economics, (3) 1976, pp. 125-144.

Compatibility

• 

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

• 

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

See Also

Finance[BlackScholesProcess]

Finance[BrownianMotion]

Finance[Diffusion]

Finance[Drift]

Finance[ExpectedValue]

Finance[ForwardCurve]

Finance[GeometricBrownianMotion]

Finance[ImpliedVolatility]

Finance[ItoProcess]

Finance[LocalVolatility]

Finance[LocalVolatilitySurface]

Finance[PathPlot]

Finance[SamplePath]

Finance[SampleValues]

Finance[StochasticProcesses]

Finance[SVJJProcess]

 


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