One-Period Binomial Model
Value a European call or put when the underlying can move only up or down over one period, by building a portfolio of shares and a bond that copies the option, and by risk-neutral pricing.
Learning focus: an option that can be copied with shares and a bond must cost what the copy costs, so its value doesn't depend on how likely the up move really is.
Leave blank to skip.
Each node shows the underlying's price (S) and the option's value (V). At expiry the value is the payoff; today it is the cost of the portfolio that copies those payoffs.
The portfolio of 0.5556 shares and a bond of −42.3280 pays exactly what the option pays in both states, so it must cost the same as the option, whichever state is more likely. The investor's view of the up move's chance, and their attitude to risk, are already in today's share price. The risk-neutral probability (0.5556) is just the weighting that makes the share earn the risk-free rate on average; using it with risk-free discounting gives the same answer as replication.