Binomial Option Pricing Model

The stock can only go up or down by a known factor next period. It sounds too simple, yet this binomial framework is the backbone of option pricing. Every multi-period tree and even Black-Scholes emerge as limiting cases. The key insight: price through replication, not prediction. The actual probability of the stock going up never appears in the formula.

Stock at $S$ moves to $uS$ (up) or $dS$ (down) over period $h$. No-arbitrage requires $d < e^{rh} < u$.

HIGH-FREQUENCY: The risk-neutral probability and one-period formula appear on most FAM sittings.

$$p^* = \frac{e^{rh} - d}{u - d}$$

This is not the true probability. It makes the expected stock return equal the risk-free rate.

KEY: The risk-neutral probability $p^*$ is NOT the actual probability of the stock going up. It is a pricing construct that makes the expected return equal the risk-free rate.

$$C = e^{-rh}[p^* C_u + (1-p^*) C_d]$$

Common mistakes

• Using the wrong $h$ in $p^*$. Two periods over one year means $h = 0.5$, so $e^{rh} = e^{0.025}$, not $e^{0.05}$. Trap: $p^*$ computed with total time instead of per-period time.
• Forgetting to discount at each step. Each backward step requires $e^{-rh}$. Skipping one gives an answer that is $e^{rh}$ too large. Trap: an option price about 2-3% too high.
• Computing $p^*$ with the wrong formula. Numerator is $e^{rh} - d$, not $u - e^{rh}$. Reversing gives $1 - p^*$. Trap: swapped up/down payoff weights.

Bottom line

• Risk-neutral probability: $p^* = (e^{rh} - d)/(u - d)$, valid only when $d < e^{rh} < u$ (no arbitrage).
• One-period price: $C = e^{-rh}[p^* C_u + (1-p^*) C_d]$; actual probabilities are irrelevant.
• Replicating portfolio: $\Delta=(C_u-C_d)/[(u-d)S]$, $C=\Delta S+B$; call delta is nonnegative and put delta nonpositive, including zero when both payoffs vanish.
• Multi-period: work backwards from terminal payoffs, with $p^*$ the same at every node.

Exam shortcut

Compute $p^*$ first (you use it at every node. For two-period problems, draw the full tree on scratch paper: stock prices at every node, then option values right to left. Remember: "p-star = (growth minus down) over (up minus down)") how far the risk-free growth exceeds the down move, relative to the full range. "Work backwards", start at terminal payoffs, always.

The full lesson (about 2,209 words, 15 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 6b

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