I am trying to get a better understanding of Central Limit Theorem and how it can be used in life and in finance. From what I have read, the BSM model assumes the underlying asset's simple returns are normally distributed. It also assumes the underlying asset prices are lognormally distributed.
My first question is: Do we need to even assume the underlying simple returns are normal, given that the CLT exists?
Assume simple return, R, follows an unknown shaped distribution. S is the price of the underlying asset. We know:
S1 = S0 * (1 + R0)
S2 = S1 * (1 + R1) = S0 * (1 + R0) * (1 + R1)
Sn = S0 * (1 + R0) * (1 + R1) * ... * (1 + Rn)
ln(Sn) = ln(S0) + ln(1 + R0) + ln(1 + R1) + ... + ln(1 + Rn)
Per CLT, as n approaches infinity, the distribution of the sum of ln(1 + R0) + ... + ln(1 + Rn), should approach a normal distribution, implying that ln(Sn) is normal, implying that S itself is lognormally distributed. As a result, does it matter if we even make the assumption that simple returns are normal since CLT will make ln(S) approach normal regardless of the distribution of R?
My second question is: If it does not matter that we make the normal assumption, why in option pricing, or in broader finance , do we often talk about fat tails and non normality? Shouldn't CLT make the shape of the distribution irrelevant? I understand empirically there are fat tails, and I understand the normal assumption makes math easy.