The binomial option tree

A share costs 100 today. Each step it is multiplied by u (up) or d (down). Slide u, d, the interest rate and the strike, and read the tree: at every node the share price in black, the option's value in blue, and underneath, the replicating portfolio (Δ shares plus a cash amount B) that is worth exactly the option one step later whichever way the share goes. The value today is just Δ × 100 + B. Then slide the real probability p of an up move. Nothing in the tree moves. Try to say why before reading on.

share price Soption value Vreplicating portfolio: Δ shares, B cashpayoff at expiry

What's going on

Work from the right. At expiry the option is worth its payoff, max(S − K, 0) for a call. One step earlier, at a node with share price S, suppose you hold Δ shares and B in cash (negative B means you borrowed). After an up move you have ΔSu + B(1 + r); after a down move, ΔSd + B(1 + r). Choose Δ and B so that these equal the option's two possible values Vu and Vd: two equations, two unknowns, giving Δ = (Vu − Vd) / (S(u − d)) and B = (Vu − ΔSu) / (1 + r). This portfolio is the option, in every future, so the option must cost what the portfolio costs: V = ΔS + B. If it cost more, sell the option and buy the portfolio; if less, do the reverse. Either way you pocket the difference with no risk, a free lunch, and markets with free lunches do not stay that way.

Rearranging V = ΔS + B gives V = [qVu + (1 − q)Vd] / (1 + r), with q = (1 + r − d)/(u − d). It looks like a discounted expected value, and q looks like a probability, but q was built from u, d and r alone. The real probability p that the share goes up never entered: the two equations for Δ and B did not mention it. That is why sliding p changes nothing. People who disagree about whether the share will rise can still agree on the option's price, because the price comes from copying, not from forecasting. The number q is called the risk-neutral probability, and the last line of the readout shows how far the "obvious" answer, the real-world expected payoff, is from the true price.

Repeat the one-step argument at every node and you have priced a three-step option. Make the steps smaller and more numerous and the tree becomes the Black–Scholes formula. Note what q needs: d < 1 + r < u. If the share were sure to beat the bank (d > 1 + r) or sure to lose to it, there would be a free lunch in the share alone and no sensible price for anything.

Sheet: What an option is worth. Related demos: Galton board (the same tree, counting paths) and Pascal's triangle.