Valuing a Derivative Using a One-Period Binomial Model

A stock at $100 will be either $120 or $90 next period. You hold a call struck at $100. What is the call worth today? The binomial model gives one defensible answer, and it ignores your view on probabilities entirely. The sections below explain how to value that call two equivalent ways, by replication and by a risk-neutral expectation.

A stock priced $S_0$ today moves to either $S^{+}=S_0\cdot u$ (up state) or $S^{-}=S_0\cdot d$ (down state) at time T. The factors are gross returns: u = 1.20 means a 20% gain, d = 0.90 means a 10% loss. A risk-free rate r covers the period.

The derivative pays $c^{+}$ in the up state and $c^{-}$ in the down state. For a call struck at K, $c^{+}=\max (S^{+}-K,0)$ and $c^{-}=\max (S^{-}-K,0)$. For a put, flip to $\max (K-S,0)$.

KEY: The binomial model prices the derivative from the stock and the bond, not from your view of which state is likely. Historical probabilities play no role.

Common mistakes

• Using real-world probabilities in the pricing formula. The exam plants an "estimated" up probability in the prompt to bait you. The answer uses π, not the estimate. Trap: weighting payoffs by 0.70 instead of 0.48.
• Forgetting to discount. The π-weighted expected payoff is the time-T value. Divide by (1 + r) to bring it to today. Trap: reporting $4.80 instead of $4.615 in Example 1.
• Swapping u and d in the π formula. π = (1 + r − d) / (u − d). The numerator subtracts d, the denominator is u minus d. Trap: writing (1 + r − u) / (d − u), which yields a negative number.

Bottom line

• π = (1 + r − d) / (u − d) is the risk-neutral probability, a pricing weight, NOT a real probability.
• Derivative value today equals the π-weighted expected payoff discounted at the risk-free rate r.
• The actual probability of an up move is irrelevant. Investor risk preferences cancel through the replicating portfolio.
• u and d are gross return factors. Always confirm d < 1 + r < u, otherwise arbitrage exists.

Exam shortcut

For π recall, say aloud: "One plus r minus d, over u minus d." If a question lists a real-world probability, ignore it, it is a distractor planted to mislead. For put pricing, run the same π formula but flip payoffs to max(K − S, 0), and expect a negative hedge ratio sign.

The full lesson (about 1,975 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• one-period binomial model

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