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The Binomial-Tree (Option-Pricing)

The binomial model prices an option by building a tree of possible prices and working backwards. It is less elegant than a closed-form formula and far more instructive, because the mechanism is visible.

One step, worked

Stock at 100. In one period it goes to 110 or 90. A call struck at 100 is worth 10 or 0. Risk-free rate 0 for simplicity.

The trick is replication. Build a portfolio of Δ\Delta shares and BB cash that matches the option in both states:

110Δ+B=1090Δ+B=0110\Delta + B = 10 \qquad 90\Delta + B = 0

Subtracting: 20Δ=1020\Delta = 10, so Δ=0.5\Delta = 0.5 and B=−45B = -45. Half a share funded with 45 borrowed matches the option exactly.

If two things have identical payoffs in every state, they must have the same price today:

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