The synthetic duplicate

The binomial model values options by constructing a replicating portfolio of stock and risk-free bonds that matches the option payoff in every future state. If a stock currently at S0 can move either up to Su = u * S0 or down to Sd = d * S0, the option payoff in the up state is Cu and in the down state is Cd. The replicating portfolio holds Delta = (Cu - Cd) / (Su - Sd) shares of stock. Under zero interest rates (r = 0), no-arbitrage guarantees that the option fair price is C = q * Cu + (1 - q) * Cd, where q = (1 - d) / (u - d) = (S0 - Sd) / (Su - Sd) is the risk-neutral probability of an up move.

Words you need

Option: A contract that gives the right, but not the obligation, to buy or sell something at a fixed price by a set date.
Strike: The fixed price written into an option at which you may buy (call) or sell (put).
Underlying: The stock or asset the option is written on.
Payoff: What the option is worth at expiry, before subtracting the premium you paid.
Delta: How much an option's price changes when the underlying moves by 1.
Arbitrage: A set of trades that locks in a profit with no risk. Prices that allow it are mispriced.

Why it works

Because stock and cash replicate the option payoff exactly in both states, the option must trade at the replicating cost to prevent arbitrage.

Delta = (Cu - Cd) / (Su - Sd), and q = (S0 - Sd) / (Su - Sd) (at r = 0)

Spot it: Pricing options without continuous Black-Scholes mathematics.

Technique reference and sources

Think of the binomial tree as a two-equation system with two unknowns: shares of stock and dollars in cash. By matching the up payoff and down payoff simultaneously, you build a synthetic copy of the option. The price of the option must be the exact cost to build that copy today.

When this method fits

Pricing options without continuous Black-Scholes mathematics.
Deriving risk-neutral probabilities from no-arbitrage replication.
Computing binomial delta hedge ratios across discrete states.
Interview questions on one-step trees with zero interest rates.

Determine state prices. Find up stock price Su = u * S0 and down stock price Sd = d * S0.
Compute terminal payoffs. Calculate Cu = max(0, Su - K) and Cd = max(0, Sd - K).
Calculate replicating delta. Compute Delta = (Cu - Cd) / (Su - Sd).
Calculate risk-neutral probability. Under zero rates, q = (S0 - Sd) / (Su - Sd).
Discount expected payoff. Compute option price = q * Cu + (1 - q) * Cd.

Symmetric up and down percentage moves: When the percentage up move equals the percentage down move (such as +20% and -20%), the risk-neutral probability under zero rates is exactly q = 0.50.

S0 = 50, Su = 60, Sd = 40: q = (50 - 40) / (60 - 40) = 10 / 20 = 0.50
S0 = 100, Su = 120, Sd = 80: q = (100 - 80) / (120 - 80) = 20 / 40 = 0.50

At-the-money call price at q = 0.50: For an ATM call (K = S0) with Cd = 0 and q = 0.50, the option price is simply 0.50 * Cu.

S0 = 50, Su = 60, K = 50: Cu = 10, Cd = 0, Call = 0.50 * 10 = 5.00
S0 = 100, Su = 125, K = 100: Cu = 25, Cd = 0, Call = 0.50 * 25 = 12.50

Common traps

Using subjective real-world probabilities instead of risk-neutral probability q. Option valuation relies on no-arbitrage replication, not subjective market forecasts. The unique risk-neutral probability q is determined by S0, Su, Sd, and r.

Confusing the numerator and denominator in the delta formula. Delta is change in option price divided by change in stock price: Delta = (Cu - Cd) / (Su - Sd).

Forgetting to discount expected payoff when interest rates are strictly positive. When r > 0, the expected payoff under q must be discounted by dividing by 1 + r.

Another example: Binomial put option pricing

Up payoff: Pu = max(0, 70 - 100) = 0.00
Down payoff: Pd = max(0, 70 - 60) = 10.00
Risk-neutral probability: q = (80 - 60) / (100 - 60) = 20 / 40 = 0.50
Fair put price = q * Pu + (1 - q) * Pd = 0.50 * 0.00 + 0.50 * 10.00 = 5.00
Replicating delta = (Pu - Pd) / (Su - Sd) = (0 - 10) / (100 - 60) = -10 / 40 = -0.25 shares

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

John C. Hull, Options, Futures, and Other Derivatives (11th Edition)
Recommended textbook reference. Publisher overview and table of contents checked for option properties, binomial trees, Black-Scholes-Merton and Greeks; full derivations and exercises have not been audited against these lessons. Lesson examples and exercises are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.