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Math for Quantitative Finance

# Log-normal Distribution

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.

Which of the following are reasons that the log-normal distribution is used to model stock returns, rather than the normal distribution?

A stock currently sells for $30 per share, and the percentage daily returns are normally distributed with a mean of 3% and a standard deviation of 1.5%. An investor will exercise a call option on this stock if its price rises above$31. What is the approximate probability this happens the next day?

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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