The price of any claim is a discounted fair bet — and the stock's real expected return does not enter it. Price a claim two ways: by replication (a stock-plus-bond hedge that copies its payoff) and by the discounted risk-neutral expectation e−rT·EQ[payoff], where under the measure Q every asset drifts at the risk-free rate. They agree — exactly. The blue team builds and defends; the red team tries to break it.
source Harrison, J.M. & Kreps, D.M., Martingales and Arbitrage in Multiperiod Securities Markets (1979), J. Economic Theory 20(3): 381–408 — doi.org/10.1016/0022-0531(79)90043-7. Rendered, not quoted. These are models with named assumptions, not investment advice.
Replication. To copy a claim over one step, hold Δ shares and B in the risk-free bond so the portfolio pays the claim in both states: Δ = (fu−fd)/(S(u−d)), B = e−rΔt(fu−ΔSu). Its cost today is the price — any other price is an arbitrage.
Expectation. Solve the same two equations and the cost equals e−rΔt(q·fu+(1−q)·fd) with q = (erΔt−d)/(u−d). The hedge forces the measure Q.
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Harrison & Kreps made the link exact: no arbitrage ⇔ a risk-neutral measure Q exists under which discounted prices are martingales — the fundamental theorem of asset pricing. Pricing stops being a forecast and becomes an expectation under Q.
This is why the-black-scholes formula never mentions the stock's expected return μ: the hedge cancels it. Each sphere is the next one's premise — the binomial tree here is that formula's discrete skeleton.
The blue team's live check: re-derive the price by the hedge and by the Q-expectation and confirm they still coincide, that discounted S is a martingale, and that no-arbitrage tracks q∈(0,1). If red tampers, this badge is where it shows.
A binomial market: one stock at S₀ that over each step Δt goes up by factor u=eσ√Δt or down by d=1/u, plus a bond earning the risk-free rate r. A claim pays a known function of the terminal price (call, put, forward, digital). Real-world drift μ is fed in only to prove it is ignored.
Feed these into the engine below. Nothing here is looked up — u, d, q and every price are computed on the spot.
Change μ and re-read: the price does not move — μ never enters Q. The struck real-world number shows what a naive forecaster would (wrongly) quote.
What the machine produces, proven: the replication cost and the discounted Q-expectation agree to machine precision on one-step and multi-step trees, discounted S is a Q-martingale, and refining the tree converges to the Black–Scholes closed form. The current claim's price is above; the identity is the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make them disagree.
Real markets have all of the frictions this model deletes. The binomial tree is the first proof that a price can be pinned by a hedge, not the last word on what anything is worth — and none of it is investment advice.
"Risk-neutral means investors are risk-neutral." Cut. It means the price can be written as if they were — a change of measure, not a claim about anyone's preferences.
"q is the real probability of an up move." Cut. q depends on r, not on μ; the real probability depends on μ. They are equal only in the knife-edge case μ=r.
"A higher expected return makes the option worth more." Cut. The hedge cancels μ; two stocks with the same σ but different drifts price the same option identically.
The red team's move: price with the real-world probability and drift μ instead of Q. The number changes — and it no longer equals the replication cost. The blue team's witness (window 7) is watching.
Swap Q for the physical measure and the quote drifts off the hedge cost — the witness recomputes, the two prices disagree, and it turns red. Nothing is faked; the attack is real and it is caught.