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SVJJProcess

  

create new SVJJ process

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

SVJJProcess(S0, V0, r, theta, kappa, sigma, rho, lambda, alpha, beta, delta, t)

Parameters

S0

-

real constant; initial value of the return process

V0

-

non-negative constant; initial value of the variance

r

-

real constant; risk-neutral drift

theta

-

non-negative constant, algebraic expression or procedure; long-run mean of the volatility

kappa

-

positive constant; speed of mean reversion

sigma

-

real constant; volatility of the variance process

rho

-

non-negative constant; instantaneous correlation between the return process and the variance process

lambda

-

non-negative constant; jump intensity

alpha

-

non-negative constant; mean relative jump size

beta

-

real constant; standard deviation of the relative jump size

delta

-

real constant; jump size of the variance process

t

-

name; time variable

Description

• 

The SVJJProcess command creates a new stochastic volatility process with jumps (SVJJ). This is a process governed by the stochastic differential equation (SDE)

dStSt=λμ+rdt+VtdW1t+J1dNt

dVt=κθVtdt+σVtdW2t+δdNt

  

where

– 

r is the risk-neutral drift,

– 

θ is the long-run mean of the variance process,

– 

κ is the speed of mean reversion of the variance process,

– 

σ is the volatility of the variance process,

– 

δ is the volatility jump size,

  

and

– 

Wt is the two-dimensional Wiener process with instantaneous correlation ρ,

– 

Nt is a Poisson process, independent of Wt, with constant intensity λ,

– 

J is a lognormal random variable with mean α and variance β2.

• 

The parameters μ, α, and β are related by the following equation

ln1+μ=α+β22

• 

This process was introduced by A. Matytsin. Special cases of this process include

 

Bates SVJ process

δ=0

 

 

Heston SV process

λ=0

 

Examples

withFinance:

First construct an SVJJ process with variable parameters. You will assign numeric values to these parameters later.

YSVJJProcess100,0.008836,r,θ,κ,σ,ρ,λ,α,β,δ,t:

κ3.99

κ:=3.99

(1)

θ0.014

θ:=0.014

(2)

σ0.27

σ:=0.27

(3)

ρ0.79

ρ:=0.79

(4)

r0.0319

r:=0.0319

(5)

α0.1

α:=0.1

(6)

β0.15

β:=0.15

(7)

λ0.11

λ:=0.11

(8)

T5.0

T:=5.0

(9)

K100

K:=100

(10)

δ0.1

δ:=0.1

(11)

M100;N104

M:=100

N:=10000

(12)

Generate 10 replications of the sample path and plot sample paths for the state variable and the variance process.

ASamplePathYt,t=0..T,timesteps=30,replications=10

A:= 1..10 x 1..2 x 1..31 ArrayData Type: float8Storage: rectangularOrder: C_order

(13)

PathPlotA,1,thickness=3,color=red..blue,axes=BOXED,gridlines=true

PathPlotA,2,thickness=3,color=red..blue,axes=BOXED,gridlines=true

ⅇrTExpectedValuemaxYT1K,0,timesteps=M,replications=N,output=value

19.96717514

(14)

Consider different parameters.

κ0.0

κ:=0.

(15)

θ0.0

θ:=0.

(16)

λ1.0

λ:=1.0

(17)

σ0.0

σ:=0.

(18)

Generate 10 replications of the sample path of the new process and plot sample paths for the state variable and the variance process.

ASamplePathYt,t=0..T,timesteps=30,replications=10

A:= 1..10 x 1..2 x 1..31 ArrayData Type: float8Storage: rectangularOrder: C_order

(19)

PathPlotA,1,thickness=3,color=red..blue,axes=BOXED,gridlines=true

PathPlotA,2,thickness=3,color=red..blue,axes=BOXED,gridlines=true

References

  

Bates, D., Jumps and stochastic volatility: the exchange rate processes implicit in Deutsche Mark options, Review of Financial Studies, Volume 9, 69-107, 1996.

  

Duffie, D., Pan, J., and Singleton, K.J. Transform analysis and asset pricing for affine jump-diffusions. Econometrica, Volume 68, 1343-1376, 2000.

  

Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

  

Matytsin, A. Modelling volatility and volatility derivatives, Columbia Practitioners Conference on the Mathematics of Finance, 1999.

Compatibility

• 

The Finance[SVJJProcess] 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[GeometricBrownianMotion]

Finance[HestonProcess]

Finance[ItoProcess]

Finance[SamplePath]

Finance[SampleValues]

Finance[SquareRootDiffusion]

Finance[StochasticProcesses]

Finance[WienerProcess]

 


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