The Central Limit Theorem cannot be used to model market returns. It is a well-known fact in finance that market returns exhibit fat-tails. That is, the distribution of market returns is not normally distributed but rather has some distribution with excess kurtosis greater than zero. A consequence of this is that extreme deviations in market price occur more frequently than the normal distribution would predict. There are several reasons for this.
Non-Independence: Market returns are often not independent. Economic conditions, policy changes, global events, and investor sentiment can influence multiple periods, leading to autocorrelation where past returns influence future returns. The assumption of independence is crucial for the CLT, and its violation means the theorem may not apply as expected.
Non-Identical Distribution: Market returns do not always follow identical distributions across time. Market volatility can change due to various factors, leading to periods of low volatility (stable returns) and high volatility (variable returns). This variability challenges the assumption of identically distributed variables required by the CLT.
Limited Data: The CLT is most reliable when applied to a large number of observations. In the context of financial markets, while there may be many data points, the truly independent periods (considering economic cycles, for example) may be limited.