Binomial and Black–Scholes pricing

Valuation compares future rights in today's money

A European call lets its holder buy the underlying at strike K on the expiry date. It has a positive payoff above K and may expire unused otherwise. Valuation asks what this uncertain future right is worth today, rather than the eventual profit after a premium has been paid. Compare this with the expiry payoff, which subtracts the premium paid.

Even when today's stock price equals the strike, future upside can make the right valuable. Models describe possible prices, assign pricing weights and discount future amounts. All numerical examples here are hypothetical teaching inputs, not market quotes.

Start with a two-state, one-step tree

Assume stock and strike are both 100, with a one-year rise to 120 or fall to 80, and a zero risk-free rate. Call payoffs are 20 and 0. Half a share financed by borrowing 40 costs 50 − 40 = 10 today and produces exactly 20 or 0 at expiry. The no-arbitrage call value is therefore 10.

This replicating portfolio explains why valuation need not start with a forecast probability of an increase. Equal payoffs in every model state should have equal prices; otherwise buying the cheaper payoff and selling the dearer one creates an arbitrage.

Risk-neutral probability is a pricing weight

For the same tree at zero rates, the risk-neutral up weight is (1 − 0.8)/(1.2 − 0.8) = 0.5, yielding 0.5 × 20 + 0.5 × 0 = 10. With a 5% effective annual rate, the growth factor is 1.05, the weight becomes 0.625 and value is 12.5/1.05 = 11.9048.

This weight gives the stock risk-free expected growth in the pricing world; it does not claim a 62.5% actual chance of rising. A formula using continuous rate r requires ln(1.05) for a 5% effective annual rate, not an unqualified 0.05. The down and up growth factors must straddle risk-free growth to produce a valid weight between zero and one.

What the interaction compares with Black–Scholes

The interaction does not use the fixed ±20% tree above. It sets u = exp(σ√T) and d = exp(−σ√T). With stock and strike 100, T = 1, continuous rate 5% and volatility 20%, the one-step value is about 12.1623 and Black–Scholes about 10.4506. Their terminal price distributions differ, so their answers need not match.

Black–Scholes treats a continuous range of prices under idealized assumptions including constant volatility and continuous trading. A one-step tree retains only two terminal states. More steps allow convergence analysis, but this page does not perform that calculation; a one-step difference alone does not establish a model error.

Change volatility to understand asymmetric rights

Increase volatility and observe both terminal tree prices and the two valuations. The call holder may abandon adverse outcomes while retaining favorable upside. With other inputs fixed in these models, greater uncertainty generally increases call value because its payoff is convex; that is not a prediction that the stock will rise.

This page covers non-dividend-paying European calls with constant parameters, no transaction costs and idealized risk-free financing. Actual volatility changes, prices can jump, and spreads and hedging constraints matter. A model value is not a guaranteed transaction price and does not directly value American early exercise or dividend-paying contracts.

Scope and limitations

European calls only, no dividends or transaction costs, constant parameters. A one-step tree is a teaching approximation; it is not a converged multi-step valuation or an American-option model.