If one is a random variable, so is the other. It's a simple change of variables. What's your point?
> As for the quotes, I encourage you to strongly think about the meaning of the work of Sharpe, Black & Scholes and Markowitz applied to non normal distributions (both Nobel prizes, we understand each other).
Could you quote the part where they say that actual, real-world stock prices (or returns, whatever) are random?
Not sure I follow your reasoning. Prices are positive only, and non stationary. That is very much not the same for returns. Usually prices are log normal, leading to normal (log) returns.
> Could you quote the part where they say that actual, real-world stock prices (or returns, whatever) are random?
It is not said, but rather implied. Take the Sharpe ratio for instance, it is a measure that:
1) is used to compare different assets / portfolio returns
2) rely on the 2nd moment of the returns.
The standard deviation is less relevant the further away from the normal distribution you go, so since this is a comparison metric, it can only be reasonably applied to compare normally distributed returns.
If you believe the returns are not generally normal, then you reject the use of the Sharpe ratio as a relevant measure of comparison.
I don't have a B&S reference at hand, and I did not read it since 15 years, but I'm pretty sure it assumes lognormals prices as well.
Perhaps we should take it back to the beginning. What do you believe "random variable" means...?
> It is not said, but rather implied.
I see.
The market is incredibly efficient at pricing many things incorrectly and Markowtiz then BSM was no exception.
To be pedantic, stock returns, not prices.
As for the quotes, I encourage you to strongly think about the meaning of the work of Sharpe, Black & Scholes and Markowitz applied to non normal distributions (both Nobel prizes, we understand each other).
In particular, try to articulate the relevancy of sharpe ratios between two non normally distributed portfolios.