# Do we need to assume underlying returns are normal in BSM model, given Central Limit Theorem?

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.

Thanks!

## 2 Answers

I will try to be as concise as possible. The sum of log returns over long time horizons for many assets tends to be less divergent from a normal than the distribution of log return at daily level, and the proximity gets better as your horizon increases (i.e the frequency of returns decreases). Indeed you can view low-frequency log returns as the sum of higher-frequency log returns, which is the beautiful thing about log returns. But when you price options in the real world, then you cannot make gifts to the market, so you cannot take an asymptotic distribution for long-term low-frequency returns as

• the option will have a certain discrete maturity
• in most case the maturity is also not so long-term (it will be a matter of days not tens of years)

Therefore the assumption of normal log returns is just a theoretical assumption, not to be used in real world for several applications. And of course it is a clear simplifications of the empirical real-world dynamic of asset prices. Remember that theory needs to make assumptions otherwise the problems would soon become intractable.

The “true” probability distribution of returns at a certain frequency is unknown. What we can do is to study the empirical evidence and try to approximate it with non-parametric techniques. And when you do that, you see the in the vast majority of cases, the assumption of normality tends to be rejected, due to both skewness and/or excess Kurthosis.. the latter may in part be mitigated when you fit some GARCH, so the assumption of normal conditional innovations in GARCH models may be empirically less wrong than the assumption of normal distribution of log returns. But in most cases also the latter hypothesis is just a proxy for a real-world complex phenomenon.

Regarding your second question, maybe you should also consider the general version of the CLT with no assumptions on second and higher moments. In those circumstances, the limiting distribution may be non-normal. They will be of alpha stable type, possessing deep connections with your questions on fat tails and so on.