Why pricing as if investors were risk neutral works

Explain why valuing derivatives with risk-neutral expectations gives the correct price.

Because the option can be hedged perfectly, its price cannot depend on investors' risk preferences, so we may assume any preferences, including risk neutrality. Delta hedging removes the stock's drift from the pricing equation; the resulting value is therefore the same as in a world where every asset grows at the risk-free rate, in which price equals the discounted expected payoff. The risk-neutral probabilities are a calculation device, not real-world forecasts.