The log-normal distribution is often used to model the price of a stock. Though it is not a perfect model, it has remained at the core of many pricing algorithms for decades.

An investor wishes to invest $750. He can either invest $30 into a stock whose daily returns are log-normally distributed with mean 3% and standard deviation 1.5% for 25 days, or invest all $750 into a single stock whose daily returns are log-normally distributed with mean \(\mu\) and standard deviation \(\sigma\) for a single day. If these two options are equivalent, what is \(\sigma\)?

Here, daily returns is referring to \(\frac{S_1-S_0}{S_0},\) where \(S_0\) is the original stock price and \(S_1\) is the stock price on the following day.

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