Hello all, I have a bit of a problem at the moment solving this task. I done half of it already, just can solve it for the other half. Hope u can help! Thanks =)

Let $\displaystyle S_n = e^{\sum Y_i}$ where $\displaystyle Y_i, i = 1, 2,...,n$ all are independent with $\displaystyle N (\mu, \sigma) $ distribution, find a relation such that r can be expressed in terms of $\displaystyle \mu, \sigma^2$

Here's my working out so far:

Def of a martingale: $\displaystyle E(X_{n+1} | X_1,...,X_n) = X_n$

Hence: $\displaystyle E (S_{n+1} e^{-r(n+1)} | S_1,,,,,S_n) = S_ne^{-rn}$

Now because $\displaystyle S_n = e^{\sum Y_i}$ where $\displaystyle Y_i, i = 1, 2,...,n$

That becomes: $\displaystyle E(e^{Y_1 + ....+ Y_{n+1}} e^{-r(n+1)} |S_n)$

From that point im not too sure what to do. If anyone can help me or tell me what to do would be much appreciated. Thanks.