Binomial option pricing
The Cox-Ross-Rubinstein lattice, drawn live. Set the parameters, step the tree up and down, switch between European and American exercise, and watch the discrete price close in on Black-Scholes as the steps multiply.
Convergence to Black-Scholes
The tree price oscillates around the continuous-time value and settles as the steps shrink. With 60 steps the tree already agrees with Black-Scholes to a few pence.
How the tree works
Time to expiry is chopped into equal steps. Each step, the stock either rises by a factor u or falls by d = 1/u, chosen so the tree matches the volatility:
q is the risk-neutral probability of an up move: not the real-world chance, but the one that makes the expected discounted stock price equal today's price. Option values fill in backwards from expiry: each node is the discounted q-weighted average of its two children. For an American option, each node also checks whether exercising immediately beats holding on, and takes the larger.
Things worth trying
- American put, deep in the money — set a put with the strike well above spot and switch to American: gold nodes appear where early exercise is optimal, and the tree price rises above the European value.
- American call, no dividends — switch a call to American and notice nothing changes. Without dividends an American call is never exercised early, a classic exam result.
- The convergence wobble — the price oscillates as steps increase because the discrete strike keeps landing on different sides of a node. By 60 steps it has all but died out.
Studying CM2 or the CFA? Binomial trees, risk-neutral valuation and Black-Scholes are core syllabus. Memori is a flashcard app built by actuarial students, with a ready-made CM2 set covering exactly these results. Join the beta or try the option payoff tool.
For education only: no dividends, constant volatility and rates, and not investment advice.