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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.
Copyright (C) Magnus Wiktorsson 2011 Error

References

Cont, R. and Tankov, P. (2004). Finanial Modelling with Jump Processes, Chapman and Hall.
Madan, D. B., Seneta, E., 1990. The Variance Gamma (VG) Model for Share Market returns. Journal of Business 63, 511-524. Article

 

Questions: Magnus Wiktorsson
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