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What is the Central Limit Theorem?
The Central Limit Theorem (CLT) is a fundamental concept in probability theory and statistics that has important implications in quantitative finance. It states that, under certain conditions, the distribution of the sum or average of many independent and identically distributed (i.i.d.) random variables approaches a normal distribution, regardless of the shape of the original distribution.
In the context of quantitative finance, the Central Limit Theorem has several key implications:
Market Returns: The Central Limit Theorem suggests that the distribution of market returns, which can be seen as the sum or average of numerous individual price changes, tends to be approximately normal. This assumption of normality is often made in quantitative models and statistical analysis.
Portfolio Returns: The CLT has important implications for portfolio returns. If the returns of individual assets in a portfolio are i.i.d., the portfolio's overall return distribution will tend to become more normal as the number of assets in the portfolio increases. This allows for the application of various statistical techniques and portfolio optimization methods that assume normality.
Hypothesis Testing: The Central Limit Theorem is frequently used in hypothesis testing in quantitative finance. Many statistical tests, such as t-tests and z-tests, rely on the assumption of normality. The CLT provides a justification for using these tests when sample sizes are sufficiently large.
Option Pricing: The Central Limit Theorem is relevant in option pricing models. For example, in the Black-Scholes model, which assumes that stock price changes follow a log-normal distribution, the CLT helps establish the assumption of continuous trading and the normality of log returns.
It's important to note that the Central Limit Theorem relies on specific assumptions, including the i.i.d. nature of the random variables and the existence of finite moments. Additionally, while the CLT provides a useful approximation in many situations, it may not hold precisely for all financial data, especially when dealing with extreme events or heavy-tailed distributions.
CLT is covered in more detail in module 1 of the CQF program.