Thank you for your thoughtful critique. You raise a crucial point regarding the use of the normal distribution in financial modeling, which merits careful consideration.

Firstly, it's important to clarify that while the normal distribution does maximize entropy under the constraints of a fixed mean and variance, this property assumes that the underlying data conforms to the assumptions of normality, including having finite variance. This assumption of maximum entropy implies minimal additional structure and is often appealing for its mathematical simplicity and elegance.

However, as you correctly pointed out, financial time series frequently exhibit properties (like fat tails and volatility clustering) that deviate significantly from the normal distribution. These characteristics include higher kurtosis and, in some cases, an undefined variance as suggested by Mandelbrot. The assumption of normality in these cases can indeed lead to underestimating the probability of extreme events, thus posing substantial risks in financial risk management.

The normal distribution's utility in entropy maximization is context-dependent. In fields where the data reasonably approximates normality, it offers a robust model due to its entropy characteristics. However, in finance, where the data often violates these assumptions, alternative models that capture fat tails and extreme events, such as the Cauchy distribution or other Levy stable distributions, should be considered.

While the normal distribution is a powerful tool in many statistical applications, its use in finance must be approached with caution. It serves as a reminder of the "no free lunch" theorem in statistical modeling—no one model is universally optimal across all scenarios. The choice of model should always be guided by the specific characteristics of the data and the practical implications of the modeling assumptions.