create new SVJJ process
SVJJProcess(S0, V0, r, theta, kappa, sigma, rho, lambda, alpha, beta, delta, t)
real constant; initial value of the return process
non-negative constant; initial value of the variance
real constant; risk-neutral drift
non-negative constant, algebraic expression or procedure; long-run mean of the volatility
positive constant; speed of mean reversion
real constant; volatility of the variance process
non-negative constant; instantaneous correlation between the return process and the variance process
non-negative constant; jump intensity
non-negative constant; mean relative jump size
real constant; standard deviation of the relative jump size
real constant; jump size of the variance process
name; time variable
The SVJJProcess command creates a new stochastic volatility process with jumps (SVJJ). This is a process governed by the stochastic differential equation (SDE)
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,
W⁡t is the two-dimensional Wiener process with instantaneous correlation ρ,
N⁡t is a Poisson process, independent of W⁡t, 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
This process was introduced by A. Matytsin. Special cases of this process include
Bates SVJ process
Heston SV process
First construct an SVJJ process with variable parameters. You will assign numeric values to these parameters later.
Y ≔ SVJJProcess⁡100,0.008836,r,θ,κ,σ,ρ,λ,α,β,δ,t:
κ ≔ 3.99
θ ≔ 0.014
σ ≔ 0.27
ρ ≔ −0.79
r ≔ 0.0319
α ≔ 0.1
β ≔ 0.15
λ ≔ 0.11
T ≔ 5.0
K ≔ 100
δ ≔ 0.1
M ≔ 100;N ≔ 104
Generate 10 replications of the sample path and plot sample paths for the state variable and the variance process.
A ≔ SamplePath⁡Y⁡t,t=0..T,timesteps=30,replications=10
Consider different parameters.
κ ≔ 0.0
θ ≔ 0.0
λ ≔ 1.0
σ ≔ 0.0
Generate 10 replications of the sample path of the new process and plot sample paths for the state variable and the variance process.
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.
The Finance[SVJJProcess] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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