Questions: Binomial Option Pricing and Replicating Portfolios

Question 1

Two analysts are pricing a call option using the binomial model. Analyst A believes the stock will rise with 80% probability; Analyst B believes 30%. Who will compute the higher option price?

• A. Analyst A — higher real-world probability of a rise increases expected payoff
• B. Analyst B — lower probability makes the option more of a hedge, increasing its value
• C. Neither — the real-world probability is not an input to the binomial option price
• D. It depends on the strike price relative to the current stock price

Answer: (C)

The real-world probability of the stock rising has no role in the binomial pricing formula. Option price is determined by the replicating portfolio — Δ shares + bond — which is derived entirely from the up/down factors (u and d), the current stock price, and the risk-free rate. The risk-neutral probability p* is a mathematical construct that replaces the real-world probability; it is calibrated to make expected stock return equal the risk-free rate, not to reflect anyone's beliefs. Both analysts will compute identical prices if they share the same S, u, d, K, and r.

Question 2

A call option has the following binomial payoffs: Cᵤ = $10 (up state), Cᵈ = $0 (down state). You construct a replicating portfolio of Δ shares and a bond. If this portfolio currently costs $4, what must the option's price be?

• A. More than $4, to compensate the seller for risk
• B. Less than $4, since the option only pays in one state
• C. Exactly $4, by the no-arbitrage principle
• D. Exactly $5, since the expected payoff is $5 under equal probabilities

Answer: (C)

By the no-arbitrage principle, two portfolios with identical payoffs in every state must have the same current price. The replicating portfolio is constructed to match the option's payoff exactly in both the up and down state. If the option priced above or below $4, you could buy the cheaper one and sell the more expensive one to lock in a riskless profit — a contradiction in an arbitrage-free market. Option pricing is thus a consequence of no-arbitrage, not of expected-value calculations.

Question 3

In the binomial model, a call option on a stock is worth more if investors collectively believe the stock is more likely to rise.

• T. True
• F. False

Answer: False

This is the most important misconception about option pricing. The binomial model prices options through replication — finding a portfolio of stock and bonds that perfectly matches the option's payoffs. The cost of that portfolio depends on the up/down factors (u and d), the current stock price, and the risk-free rate. Real-world beliefs about probabilities are irrelevant because the hedge ratio Δ and bond position B are uniquely determined by no-arbitrage, regardless of what investors believe. The risk-neutral probability p* is not a belief — it is a mathematical tool that produces the same answer as the replicating portfolio method.

Question 4

The cost of the replicating portfolio must equal the option's price; if it did not, a trader could construct a riskless profit.

• T. True
• F. False

Answer: True

This is the no-arbitrage foundation of the model. If the option traded above the replicating portfolio cost, you could sell the option and buy the replicating portfolio, locking in the difference risk-free (the replicating portfolio covers all obligations). If the option traded below, you could buy the option and short the replicating portfolio. In both cases, the position has zero net cash flow at expiration but positive cash flow today — a riskless profit. Competitive markets eliminate such opportunities, forcing the option price to equal the replicating portfolio cost exactly.

Question 5

Why doesn't the real-world probability of the stock rising affect the option price in the binomial model? What does determine the price instead?

Model answer: The option price is determined by the cost of the replicating portfolio — a combination of Δ shares and a bond position that exactly replicates the option's payoff in every state. This Δ and bond position are uniquely solved from the up/down payoffs and the up/down stock prices, with no reference to how likely each state is. The risk-free rate, current stock price, and up/down factors (u and d) fully determine the price. Real-world probabilities are irrelevant because any option price inconsistent with the replicating portfolio cost would permit riskless arbitrage, which the market eliminates.

This non-intuitive result — that beliefs don't matter — is the central insight of modern derivatives pricing. It follows from no-arbitrage: since the replicating portfolio costs what it costs regardless of probabilities, the option must too. The risk-neutral probability p* is a convenient reformulation of this same no-arbitrage condition, not a belief anyone holds.