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Finance[MertonJumpDiffusion] - create new jump diffusion process

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.

S0:=100

S0:=100

(1)

r:=0.05

r:=0.05

(2)

d:=0.01

d:=0.01

(3)

σ1:=0.01

σ1:=0.01

(4)

a:=0.0

a:=0.

(5)

b:=0.5

b:=0.5

(6)

λ1:=2.0

λ1:=2.0

(7)

λ2:=0.2

λ2:=0.2

(8)

X1:=MertonJumpDiffusionS0,σ1,r,d,λ1,a,b:

PathPlotX1t,t=0..1,timesteps=100,replications=5,color=red..blue,thickness=3,axes=BOXED,gridlines=true

X2:=MertonJumpDiffusionS0,σ1,r,d,λ2,a,b:

PathPlotX2t,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.

σ2:=0.5

σ2:=0.5

(9)

X3:=MertonJumpDiffusionS0,σ2,r,d,λ2,a,b:

PathPlotX3t,t=0..1,timesteps=100,replications=5,color=red..blue,thickness=3,axes=BOXED,gridlines=true

Y:=BlackScholesProcessS0,σ2,r,d:

X4:=MertonJumpDiffusionY,λ1,0,b:

PathPlotX4t,t=0..1,timesteps=100,replications=5,color=red..blue,thickness=3,axes=BOXED,gridlines=true

ExpectedValuemaxX4190,0,timesteps=100,replications=105

value=65.32590075,standarderror=0.5535149183

(10)

S1:=SampleValuesY1,timesteps=100,replications=105

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

(11)

S2:=SampleValuesX41,timesteps=100,replications=105

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

(12)

P1:=Statistics[FrequencyPlot]S1,range=0..300,thickness=3,color=red,bincount=50:

P2:=Statistics[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.

J:=PoissonProcessλ2,Normala,b

J:=_P

(13)

Z:=t→YtⅇJt

Z:=t→YtⅇJt

(14)

ExpectedValuemaxZ190,0,timesteps=100,replications=105

value=30.42046277,standarderror=0.1770232342

(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]

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.


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