Risk-neutral probability

The letter p in the binomial tree is the most misread symbol in derivatives, so it is worth stopping on it before it is used another forty times. p is not a forecast, and it is not anybody's probability of anything. It is the number that makes a particular equation balance, and the equation is the replication argument.

Where does the risk-neutral probability come from?

Take one step of the tree. The stock is at S, and one period from now it is either uS or dS. An option on it pays V_u in the up state and V_d in the down state. Build a portfolio of Δ shares and B rupees of borrowing, and choose Δ and B so that the portfolio pays exactly the option's payoff in both states. Two equations, two unknowns, and the solution is forced:

Δ = (V_u − V_d) / (uS − dS)          B = e^(−rΔt) · (Δ·dS − V_d)
Option value today = Δ·S − B

Nothing else is needed. The portfolio and the option have identical payoffs in every state the world can reach, so they must cost the same today, or one of them is free money. Notice what has not entered: no probability, no expected return on the stock, no risk premium, no view. Two investors who disagree violently about whether the stock is going up must still agree on the option's price, because the argument never asked either of them.

Now do the algebra one more step. Substitute Δ and B back into Δ·S − B and collect terms, and the answer rearranges into a shape that looks exactly like a discounted expectation:

Option value = e^(−rΔt) · [ p·V_u + (1 − p)·V_d ]      with   p = (e^((r−q)Δt) − d) / (u − d)

The expectation shape is an artefact of the algebra, not a new assumption. p is whatever number makes the stock itself satisfy the same relation. Equivalently, it is whatever number makes the stock's own discounted expected value equal its price today, which is to say whatever number makes the stock earn the risk-free rate in this fictitious world. That is why p is called the risk-neutral probability: not because anyone is neutral about risk, but because pricing under p gives the same answer as replication, and replication does not care what anyone thinks about risk.

Three consequences of risk-neutral pricing

Three consequences fall straight out, and each of them is a check you can run.

The real-world probability never appears anywhere in the argument. If a colleague says the tree is wrong because "the stock is much more likely to go up than p suggests", they have misunderstood the object. Their forecast belongs in a position-sizing decision (M9.01) and in an expected-return estimate; it does not belong in an option price. Every hedged replication argument in this branch has the same property.

A second consequence is that p must land strictly between 0 and 1, or the inputs themselves admit arbitrage. If e^((r−q)Δt) sits outside the interval [d, u], then the forward price of the stock is outside the range of prices the tree lets the stock reach, which means a riskless position with a certain profit exists in the model itself. A pricer that returns a p of −0.3 or 1.4 is not returning a bad price; it is telling you the inputs are inconsistent, most often because σ is far too small for the step length or r is far too large.

The third consequence is reach: the expectation under p prices any payoff written on the same tree. Digital options, barriers, anything whose value at each terminal node you can write down: same tree, same p, same backward recursion. That generality is why the tree survives as a workhorse long after the closed form exists, and it is also why the same machinery reappears in FI1.04's interest-rate tree, where the underlying is a short rate rather than a stock and the calibration is to a par curve rather than to σ alone.

The continuous-time version of this argument is the Black-Scholes-Merton partial differential equation, and its structure is identical: build an instantaneously riskless portfolio, insist it earns r, and read off the equation the option price must satisfy. The formula on the Black-Scholes model page is that equation's solution for a European payoff. Nothing new is assumed between here and there except that the steps become infinitely small.