Binomial Trees

Discrete-time option pricing that converges to Black–Scholes.

A binomial tree models the underlying as moving up by factor $u$ or down by factor $d$ over each time step, with risk-neutral up-probability

q = \frac{e^{r\Delta t} - d}{u - d}

The option price at each node is the discounted expectation of its children:

V = e^{-r\Delta t}\, [q\, V_u + (1-q)\, V_d]

You build the tree forward (asset prices), then evaluate the payoff at each leaf and roll back to the root.

Trees are flexible: handle American exercise (compare immediate payoff to roll-back value at each node), discrete dividends, and step-dependent dynamics. As $\Delta t \to 0$ with $u = e^{\sigma\sqrt{\Delta t}}$ , $d = 1/u$ , the tree converges to Black-Scholes.

Use trees when closed forms don't exist: American puts, dividend-paying underlyings, or when you want a transparent computational scheme that's easier to debug than Monte Carlo.