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Finance

  

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

κ3.99

(1)

θ0.014

θ0.014

(2)

σ0.27

σ0.27

(3)

ρ0.79

ρ0.79

(4)

r0.0319

r0.0319

(5)

α0.1

α0.1

(6)

β0.15

β0.15

(7)

λ0.11

λ0.11

(8)

T5.0

T5.0

(9)

K100

K100

(10)

δ0.1

δ0.1

(11)

M100;N104

M100

N10000

(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: float[8]Storage: 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

κ0.

(15)

θ0.0

θ0.

(16)

λ1.0

λ1.0

(17)

σ0.0

σ0.

(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: float[8]Storage: 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]