## Glossary

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# Variance gamma process

The Variance gamma process (VG-process) can be seen as a Brownian motion with drift $\theta$ and volatity $\sigma$ timechanged with a Gamma process with shape paramter $1/\kappa$ and scale parameter $\kappa$ [Madan and Seneta, 1990] (See also Cont and Tankov (2004) p. 117) . The VG-process is a Lévy process with finite variation over finite intervals. A Variance Gamma (VG) process $Z(t)$ has the characteristic function $\phi_{Z(t)}=E[e^{iyZ(t)}]=e^{tK(y)},$ $K(y)=iy\mu-1/\kappa\log(1-(iy\theta\kappa-(y^2/2)\sigma^2\kappa)),~\mu,\theta\in {\mathbb R},~\sigma,\kappa>0$. If we let $\kappa$ tend zero the VG-process will converge in distribution to a Brownian motion with drift $\mu+\theta$ and volatility $\sigma$.
In financial applications we put $S(t)=S(0)\exp(Z(t))$ with $\mu=r+1/\kappa\log(1-\theta\kappa-\sigma^2\kappa/2)$ for the risk neutral case (Q-dynamics). If we let $\kappa$ tend to zero in the Q-dynamics the stock price model will converge to the Black-Scholes model.