For variance of stock returns to equal the expected value of the square of returns, how large does the sample size have to be?Here's what I am doing.Var(X) = E[(X - E[X])^2]= E[X^2 - 2XE[X] + (E[X])^2]= E[X^2] -2E[X]E[E[X]] + E[(E[X])^2]= E[X^2] - 2E[X]E[X] + E[(E[X])^2]= E[X^2] - 2(E[X])^2 + (E[X])^2= E[X^2] - (E[X])^2= E[X^2] - (E[X])^2We assume that for stock equities, E[X] is ZERO. What are the pitfalls surrounding this assumption?Would greatly appreciate your thoughts.Thanks in advance.

So to rephrase my question, what are the assumptions underlying the statement that the expected return of an equity stock is ZERO?

It is not a question of sample size, but of time interval (sampling interval).Let us look at some realistic numbers.The typical stock index moves by 1% (0.01) per day and the typical stock by 0.02 per day. A reasonable estimate of expected return is 0.1 per year. Arguably that is not negligible.In dealing with daily data the expected return is 0.1/250 = 0.0004, that is quite small especially since your formula calls for the square of this. It is therefore common to set it to zero when calculating variances from daily data.At an hourly or minute time scale, the expected return on stocks is even smaller and thus even more negligible when squared (compared to the leading term in your formula), it would be absurd to incorporate it in the calculation.

Thank you! Makes much sense.

Isnt there a mathematical argumentation for estimating stock price return zero when evaluating e.g. var swaps? Something like Girsanovs theorem and saying change of measure has influence on expected value but not on variance?