How can we use real probabilities for pricing derivatives

Question

How can we use real probabilities for pricing derivatives?

Answer 1

Yes and no. There are many reasons why risk-neutral pricing does not work perfectly in practice, since markets are incomplete and dynamic hedging is not possible. If you cannot continuously dynamically hedge so you cannot remove risk and so risk neutrality is not too relevant. You might be tempted to try to price by using real probabilities in its place. It is fine, and there are plenty of theories on such topic, generally with some element of utility theory regarding them. For illustration, some theories exercise concepts from Modern Portfolio Theory and look at real averages and real standard deviations.