Monte-Carlo simulation for Variance Gamma

reza · May 25th, 2003, 1:16 am

Say we want to simulate a VG processif I understand correctly this is like havingLn S(t_{k+1}) = Ln S(t_k) + w dt + \theta g(dt) + \sigma \sqrt{g(dt)} B_kfor w, \theta, \sigma given parametersB_k independent Gaussiansdt = t_{k+1}-t_kand g() a gamma process with a variance rate \nuhow would we proceed to get g(dt) ?I understand we can truncate the Levy density to get a Poisson like process ...?

mj · May 25th, 2003, 9:57 am

First you draw random time. To do this you get a uniform and then you just have to take an inverse cumulative gamma function. of the uniform. Taking the inverse cumulative gamma function can be done via Newton Raphson on the cumulative gamma function . (The cumulative gamma function has a good approximation in Numerical Recipes) You can speed things up using a table.Once the random time is known you are back in the Gaussian case,MJ

reza · May 25th, 2003, 6:30 pm

Thanks very much Mark,so calling G() the Gamma CDF for dt and H() its inversefirst I take dt*= H(u[0,1])and then use this dt* in Ln S(t_{k+1}) = Ln S(t_k) + w dt + \theta dt*+ \sigma \sqrt{dt*} B_kright?

mj · May 26th, 2003, 6:29 am

Anton · May 26th, 2003, 7:13 am

Reza, you might want to have a look at Nick Webber's homepage, he has a paper on fast Monte Carlo for the variance gamma processAnton

reza · May 26th, 2003, 12:30 pm

Thank you both !