I'm studying the risk-neutral derivation of Black-Scholes formula and feel confused about the requirement for the volatility of the underlying asset and the risk-free rate to be constant. It seems that in all stages of the derivation:
- Change probability measure to make discounted stock price a martingale
- Show that stock price is a Geometric Brownian Motion with drift equal to the risk-free rate
- Prove that the discounted wealth of any replicating portfolio consisting of stock and bond is a martingale
- Derive the formula using the above conclusion
It doesn't really matter weather the risk-free rate or volatility are constants or change with time. I wonder if the assumptions are removed, what stages of the derivation will not hold?