option pricing - Risk Neutral Probability

I read that an option prices is the expected value of the payout under the risk neutral probability. Intuitively why is the expectation taken with respect to risk neutral as opposed to the actual probabilty.

Intuitively why would risk neutral probability differ from actual probability?

Answer

The following is a standard exercise that will help you answer your own question.

Consider a one-period binomial lattice for a stock with a constant risk-free rate.

Determine the initial cost of a portfolio that perfectly hedges a contingent claim with payoff $uX$ in the upstate and $dX$ in the downstate (you can do this so long as the up and down price are different in your lattice).

Assuming there exists no portfolio that yields a profit without downside risk (assume no arbitrage) and that your economy is frictionless and competitive, show that any other price for the contingent claim, other than the initial cost of the replicating portfolio you found, would lead to the existence of a portfolio that yields a profit without downside risk. Pause and reflect on the fact that you have determined the price of any contingent claim without any mention of probability. However, don't forget what you assumed! What did you actually need to do what you just did?

Now that you know that the price of the initial portfolio is the "arbitrage free" price of the contingent claim, find the number $q$ such that you can express that price of the contingent claim as the discounted payoff in the up state times a number $q$ plus the discounted payoff in the downstate times the number $1-q$. Solve for the number $q$. Interpret the number $q$ as a probability and compute the expected value of the discounted stock with this probability. This should be the same as the initial price of the stock. Pause and reflect on the fact that you have determined the unique number $q$ between $0$ and $1$ such that the expected value (using $q$) of the discounted stock is the initial price and that you can compute the price of any contingent claim by computing its expected (using $q$) discounted payoff.

It is clear from what you have just done that if you chose any other number $p$ between $0$ and $1$ other than the $q$ and computed the expected (using $p$) discount payoff, then you would not recover the arbitrage free price (remember you have shown that any other price than the one you found leads to an arbitrage portfolio). This means that if you had a real world probability $p$ for your initial lattice, it is not the correct probability to use when computing the price.