Why Risk-Neutral Pricing Uses Different Probabilities

Use risk-neutral probabilities because discounted expected payoffs under that measure reproduce the cost of a replicating portfolio. Physical probabilities describe how often outcomes are expected to occur. To use them for pricing, you would also need a state-dependent discount factor that accounts for how the market values cash in each outcome.

Solve for the weight that prices the stock

A non-dividend-paying stock trades at $S_0$ and will be worth $S_u$ or $S_d$ after one period. Let $R$ be the gross risk-free return. With $S_d<RS_0<S_u$ , the unique risk-neutral up probability is

q=\frac{RS_0-S_d}{S_u-S_d}.

It follows by requiring

S_0=R^{-1}[qS_u+(1-q)S_d]

. The same

q

prices every replicated payoff

H_u,H_d

R^{-1}[qH_u+(1-q)H_d]

. To see this, solve

\Delta S_u+RB=H_u

and

\Delta S_d+RB=H_d

. Multiply the first equation by

q

, the second by

1-q

, add, and divide by

R

. The left side becomes the initial cost

\Delta S_0+B

If the physical up probability is $p$ and $0<p<1$ , a discount factor consistent with those same prices takes the values $m_u=q/(Rp)$ and $m_d=(1-q)/(R(1-p))$ . Then $\mathbb{E}^{\mathbb{P}}[mH]=R^{-1}\mathbb{E}^{\mathbb{Q}}[H]$ . The two expressions package the adjustment differently.

Start with a payoff you can already price, such as one share of stock. Its discounted weighted terminal value must equal

S_0

, so solving that equation fixes the weight before you look at any option. The cash account supplies the second traded payoff. In a two-state model these two payoffs span the two states, which is why the same

q

prices every claim in that tree. In an incomplete model there can be multiple pricing measures. The two-state argument gives uniqueness because of the spanning assumption.

Let $S_0=100$ , $S_u=120$ , $S_d=90$ , and $R=1.05$ . Then $q=(105-90)/30=0.50$ . A call struck at $100$ pays $20$ in the up state and zero in the down state. It costs $10/1.05=9.52$ . The replicating hedge holds $20/30=2/3$ of a share and borrows $60/1.05$ . It costs $66.67-57.14\approx9.52$ .

Suppose a forecaster believes the stock has a

70\%

chance of rising. Discounting that forecasted payoff at the risk-free rate gives

14/1.05=13.33

, which exceeds the cost of the payoff you can build using stock and cash. With these model assumptions, the

13.33

price would allow someone to sell the call and buy the same payoff more cheaply.

Suppose your physical forecast assigns a 70% chance to the up state of the numerical tree. Discounting the call's physical average at the cash rate would give $0.7(20)/1.05=13.33$ . You can still build its cash flows for about 9.52. Under the model assumptions, the discrepancy could be traded by selling the more expensive claim and buying the hedge. The problem is the choice of discounting for a risky payoff under the physical forecast. The risk-neutral weight adjusts the up and down states so that risk-free discounting reproduces the cost of the hedge.

Why Risk-Neutral Pricing Uses Different Probabilities

Solve for the weight that prices the stock

Watch a binomial tree approach the Black–Scholes price

References and next steps