Finance & Options
A Coin-Flip Tree That Becomes Black-Scholes
Paths that end at the same price fold into one node, so a 252-step tree prices in well under a millisecond, and as the steps shrink its price settles on the Black-Scholes number.
What to remember
- Any fixed u and d give a recombining tree: n + 1 final nodes where the path tree has 2ⁿ leaves.
- Backward induction: vᵢ(s) = e^(−rΔt)[q vᵢ₊₁(us) + (1 − q) vᵢ₊₁(ds)], one average per node, so the work grows like n².
- CRR sets u = e^(σ√Δt) and d = 1/u so that n steps add up to variance σ²T.
- The CRR price zig-zags onto Black-Scholes with error roughly proportional to 1/n; averaging n and n + 1 steps smooths it.
- A path-dependent payoff needs a bigger state, such as (price, running max) for a lookback.
Read the lessonA Coin-Flip Tree That Becomes Black-Scholes, with a checkpoint at the end.