Using the Risk-Neutral Probability Method in Option Valuation is a bit counterintuitive. Why doesn’t the option price depend on the expected movement of the underlying stock — its probability of going up or down? Why does it not matter if the probability of the stock going up is 95% or 15%? Why do calculations done in a risk-neutral world work even in the real world of risk?
The point is that we are not valuing the option in absolute terms; we are valuing it as a function of the current price of the underlying stock. The stock price itself already incorporates risk and expectations (Hull, Options, Futures & Other Derivatives, Chapter 12, Binomial Trees).
Substantially, I understand it this way: In a simple model (Risk-Free World), we have a stock price that can either increase by X% or decrease by Y%. Accordingly, a call option has two possible outcomes: positive payoff if the stock increases, or zero if the stock decreases. To calculate the PV of an investment in such an option, we need a discount rate, which is unknown, even though we know the range of volatility. We know X and Y, but we don’t know the real probability that would help weight them.
However, we know that discounting risky cash flows at the appropriate risky rate and discounting riskless cash flows at the risk-free rate will give the same PV. Therefore, we can adjust the probabilities so that, when weighting X% and Y%, we use the risk-free rate. In other words, we know what the “up” and “down” probabilities would be under risk-neutral cash flows. Then, discounting the expected option payoff using these probabilities at the risk-free rate gives the fair value of the option.
Interestingly, the fair value obtained using this method matches the fair value obtained from the asset replication method (Δ Stock + Bond, Portfolio Replication Method).
P.S.
As ChatGPT showed me, these two methods (often referred to as separate methods in academic literature) are in fact algebraically connected.
