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Math for Quantitative Finance

# Normal Distribution in Quant Finance

The normal distribution is a famous statistical distribution that has been used to model random variables, such as human heights or weights.

In quantitative finance, it can be used to model stocks or other assets. While it could be used to model stock price (as is done sometimes in this quiz), it is more typically used to model the distribution of log-returns of the price of a stock. Though it is not a perfect model, it has remained at the core of many pricing algorithms for decades.

A stock currently sells for $30 per share, and the historical daily returns are known to be 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 that this occurs the next day?

An investor wishes to invest $750. He can either invest all$750 into a single stock that has a mean return of 3% and a standard deviation of $$x\%,$$ or he can invest $30 into each of 25 separate stocks that have a mean return of 3% and a standard deviation of 1.5%. For what value of $$x$$ are these two options equivalent? An investor is choosing between two stocks that are both currently trading at the same share price. The daily returns of the first stock are historically normally distributed with a mean of 3% and a standard deviation of 1.5%. The daily returns of the second stock are historically normally distributed with a mean of 4% and a standard deviation of 2%. What is the approximate probability that, after one day, the second stock outperforms the first stock? An investor wishes to invest$700.

There are two independent stocks the investor can choose to invest in, both of which are currently trading at the same share price. The daily returns of the first stock are historically normally distributed with a mean of 3% and a standard deviation of 1.5%. The daily returns of the second stock are historically normally distributed with a mean of 4% and a standard deviation of 2%.

How much should the investor choose to invest (in dollars) in the first stock to maximize his probability of having a positive profit over the course of a day?

An investor buys an equal number of shares in two stocks whose returns are both normally distributed with mean 3% and standard deviation 1.5%. What is the approximate probability that the investor makes a profit?

Assume that the stock prices are equal, and that the stocks have independent return distributions.

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