Implied Binomial Trees

A standard binomial tree for option pricing assumes the underlying moves up or down by a fixed factor at every step with a single constant probability, which implicitly assumes one constant volatility — exactly the assumption that fails once you notice real option markets price different strikes at different implied volatilities (the volatility smile). An implied binomial tree instead builds each node's probability and terminal price so that, when you price all the liquid listed options against the tree, it reproduces their actual observed market prices rather than a theoretical Black-Scholes price.

The construction, introduced by Rubinstein, works backward from the smile: it takes the risk-neutral probability distribution of the underlying's price at expiry implied by the full set of traded option prices, and builds a tree whose terminal nodes match that distribution exactly, then fills in the earlier nodes to be consistent with no-arbitrage. The resulting tree can then price other, less liquid derivatives — like exotic or American-style options — consistently with the market's smile, something a constant-volatility tree cannot do.

If the market prices out-of-the-money puts richer than out-of-the-money calls (a typical equity index skew), the implied tree's terminal nodes will show more probability mass, and larger down-moves, on the downside than a symmetric constant-volatility tree would — directly encoding the market's crash-risk pricing into the node structure itself, rather than layering it on afterward.