From my point of view BS pricing states that at any point during lifetime of the option the solution of the BSE provides risk free rates of return of the BS hedged portfolio. The keep hedging over the whole interval [ 0 , T ] we need additional stochastic cash flow. Nevertheless this is another problem, which can effect BS price to make non zero deviation between theoretical BS price and observed market prices but such fact does not make BSE wrong.Are you considering a one time static hedge at t=0, or a continuous dynamic re-heding at all points in time in the interval [0,T] ..like B&S assume?
In order to present BS price we choose arbitrary point t, t < T and define BS portfolio for future moments u ≥ t, ie call and stock are functions of a variable u and delta argument is a fixed parameter t. Following BS we arrive at BSE at any point of time during lifetime of the option. This present option price based on no arbitrage principle. It might be good or bad approximation for the market pricing which is basically represents price as a settlement price.I do not fully grasp your argument.
Where do you do the portfolio rebalancing step?
At each time step t, we have a value before and after rebalancing / rehedging... I miss this step in your last post, but maybe I didn't get what you are saying
BS hedged portfolio values are related to risk free rate and therefore option which is priced synthetically based on BS portfolio also depends on r. The stock's expected return is linked to mu and if risk free does not appeared in stock equation to tell that stock return formally relates to risk free rate somewhat incorrect.In BS the cashflows are values against a constant rate you can hedge with a bond. Just like the stock's expected return is linked to that same rate.
money in the bank, money invested in stock, borrowed money, .. al assumed having the same rate/expected yield in B&S
I interpret BS formula in other way.Your mu is set to r for stock options!