Binomial Option Pricing
Replicate an option with shares and a bond, and its price falls out — with no forecast of the stock.
The idea
The binomial model prices an option in the smallest market that carries any uncertainty.
Definition (Call option).
A call option with strike $K$ is the right, but not the obligation, to buy a share for $K$ on a fixed future date. If the share is worth $S_{1}$ on that date, the right is worth $\max(S_{1} - K, 0)$.
Over one period a stock priced $S$ moves either up to $uS$ or down to $dS$, and a risk-free bond turns one dollar into $1 + r$. In that market every payoff can be copied: to match a claim worth $C_{u}$ in the up state and $C_{d}$ in the down state, hold $\Delta$ shares and $B$ dollars of bond satisfying
$\Delta\,uS + B(1+r) = C_{u}, \qquad \Delta\,dS + B(1+r) = C_{d}.$
The two equations determine $\Delta$ and $B$. The portfolio pays what the claim pays in both states, so by the law of one price the claim must cost what the portfolio costs, $\Delta S + B$.
Substituting the solution and collecting terms rewrites that cost as a discounted expectation.
Theorem (One-period binomial pricing).
In the one-period market where a stock moves from $S$ to $uS$ or $dS$ and the bond grows by $1 + r$, with $d < 1 + r < u$, the arbitrage-free price of a claim paying $C_{u}$ in the up state and $C_{d}$ in the down state is $\frac{1}{1+r}\bigl(q\,C_{u} + (1-q)\,C_{d}\bigr), \qquad q = \frac{(1+r) - d}{u - d}.$
The form is a discounted expected payoff with $q$ in the role of the up-probability, yet $q$ is built from $u$, $d$ and $r$ alone. The real chance of an up move never enters, because replication matches the claim state by state and never asks how likely a state is.
Ways to work on it
- Walkthrough. Replicate a call in a one-period tree, then rewrite the cost as a risk-neutral expectation.
- Proof. See why — no arbitrage forces the replication price and pins down the risk-neutral weight.
- Practice. Compute risk-neutral weights, hedge ratios, and prices on one-period trees.
- Hardest. Price a call on a two-step tree.
Not sure where to start? Take the ten-question placement test.