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.

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

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

KEY: The risk-neutral probability $p^{\ast }$ 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.

$$C=e^{-rh}[p^{\ast }{C}_u+(1-p^{\ast })C_d]$$

Common mistakes

• Using the wrong $h$ in $p^{\ast }$. Two periods over one year means $h=0.5$, so $e^{rh}=e^{0.025}$, not $e^{0.05}$. Trap: $p^{\ast }$ 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 $p^{\ast }$ with the wrong formula. Numerator is $e^{rh}-d$, not $u-e^{rh}$. Reversing gives $1-p^{\ast }$. Trap: swapped up/down payoff weights.

Bottom line

• Risk-neutral probability: $p^{\ast }=(e^{rh}-d)/(u-d)$, valid only when $d<e^{rh}<u$ (no arbitrage).
• One-period price: $C=e^{-rh}[p^{\ast }{C}_u+(1-p^{\ast })C_d]$; actual probabilities are irrelevant.
• Replicating portfolio: $\Delta =(C_u-C_d)/[(u-d)S]$, $C=\Delta S+B$, with $\Delta >0$ for calls and $\Delta <0$ for puts.
• Multi-period: work backwards from terminal payoffs, with $p^{\ast }$ the same at every node.

Exam shortcut

Compute $p^{\ast }$ 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 1,813 words, 12 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 6b

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