create new jump diffusion process

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Finance[MertonJumpDiffusion] - create new jump diffusion process

	Calling Sequence
	MertonJumpDiffusion(X, lambda, a, b) MertonJumpDiffusion(, 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
	-	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)

where

–	is the drift parameter

–	is the volatility parameter

–	is the standard Wiener process

and

–	is a compound Poisson process of the form

J(t) = Sum(Y[j]-1, j = 1 .. N(t))

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

•

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.

Compatibility

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

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

Examples

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

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

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

PathPlot(X[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.

(9)

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

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

(10)

S[1] := RTABLE(18446790986282489606, float[8], Array, rectangular, C_order, [], 1, 1 .. 100000)

(11)

S[2] := RTABLE(18446790986282490326, float[8], Array, rectangular, C_order, [], 1, 1 .. 100000)

(12)

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

(13)

(14)

(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]