I want to do a stock price simulation. First of all, I used the GBM. To simulate the values, I didn't use the closed form solution for the GBM given by: $$ S_t=S_0\exp[(μ−σ^2)t+σWt] $$ but the discrete version instead, so I that can "see" a realization every day: $$ S_{i+1}=μΔt\cdot S_i+σφΔt*S_i+S_i $$ Now I wanted to do the same with the variance gamma distribution model given by: $$ S_T=S_0 \cdot \exp((r-q)T+w+z) $$ but the problem is, that with this formula, I can only observe the final realizations on time point $T$. Not the values between. I need a discrete version. Can you tell me which formula I have to use? Or how I can solve this problem?
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