With the tree calibrated, pricing is a two-pass procedure: build the lattice of stock prices forward, then roll option values backward from the payoff, one discounted expectation per node. This note states the algorithm precisely (the version an exam asks you to write in full), works a complete numerical example, analyses convergence to the Black–Scholes price — including CRR's oscillatory error and the tilted-tree fix — and shows how delta, gamma and theta drop out of the first two tree layers for free.
Learning objectives. By the end of this note you should be able to:
- write the European and American binomial algorithms in full pseudocode, with cost and memory counts;
- execute backward induction by hand on a small tree;
- explain why CRR convergence is linear but oscillatory, what the tilted tree does, and when Richardson extrapolation is safe;
- estimate , , from tree nodes and explain why vega and rho need re-runs.
In this lesson
- Backward induction
- The basic algorithm
- Convergence and how to improve it
- Sensitivities from trees
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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