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

where

$r$ is the risk-neutral drift,

$\mathrm{\theta }$ is the long-run mean of the variance process,

$\mathrm{\kappa }$ is the speed of mean reversion of the variance process,

$\mathrm{\sigma }$ is the volatility of the variance process,

$\mathrm{\delta }$ is the volatility jump size,

and

$W\left(t\right)$ is the two-dimensional Wiener process with instantaneous correlation $\mathrm{\rho }$,

$N\left(t\right)$ is a Poisson process, independent of $W\left(t\right)$, with constant intensity $\mathrm{\lambda }$,

$J$ is a lognormal random variable with mean $\mathrm{\alpha }$ and variance ${\mathrm{\beta }}^2$.

The parameters $\mathrm{\mu }$, $\mathrm{\alpha }$, and $\mathrm{\beta }$ are related by the following equation

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

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$Y≔\mathrm{SVJJProcess}\left(100\,0.008836\,r\,\mathrm{\theta }\,\mathrm{\kappa }\,\mathrm{\sigma }\,\mathrm{\rho }\,\mathrm{\lambda }\,\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\delta }\,t\right)\:$

$\mathrm{\kappa }≔3.99$

$\mathrm{\kappa }≔3.99$

$\mathrm{\theta }≔0.014$

$\mathrm{\theta }≔0.014$

$\mathrm{\sigma }≔0.27$

$\mathrm{\sigma }≔0.27$

$\mathrm{\rho }≔-0.79$

$\mathrm{\rho }≔\mathrm{-0.79}$

$r≔0.0319$

$r≔0.0319$

$\mathrm{\alpha }≔0.1$

$\mathrm{\alpha }≔0.1$

$\mathrm{\beta }≔0.15$

$\mathrm{\beta }≔0.15$

$\mathrm{\lambda }≔0.11$

$\mathrm{\lambda }≔0.11$

$T≔5.0$

$T≔5.0$

$K≔100$

$K≔100$

$\mathrm{\delta }≔0.1$

$\mathrm{\delta }≔0.1$

$M≔100\;$$N≔{10}^4$

$M≔100$

$N≔10000$

$A≔\mathrm{SamplePath}\left(Y\left(t\right)\,t=0..T\,\mathrm{timesteps}=30\,\mathrm{replications}=10\right)$

$\mathrm{PathPlot}\left(A\,1\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(A\,2\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\exp \left(-rT\right)\mathrm{ExpectedValue}\left(\max \left(Y\left(T\right)\left[1\right]-K\,0\right)\,\mathrm{timesteps}=M\,\mathrm{replications}=N\,\mathrm{output}=\mathrm{value}\right)$

$19.96717514$

$\mathrm{\kappa }≔0.$

$\mathrm{\kappa }≔0.$

$\mathrm{\theta }≔0.$

$\mathrm{\theta }≔0.$

$\mathrm{\lambda }≔1.0$

$\mathrm{\lambda }≔1.0$

$\mathrm{\sigma }≔0.$

$\mathrm{\sigma }≔0.$

$A≔\mathrm{SamplePath}\left(Y\left(t\right)\,t=0..T\,\mathrm{timesteps}=30\,\mathrm{replications}=10\right)$

$\mathrm{PathPlot}\left(A\,1\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(A\,2\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

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.