CFA Level I Exam · Valuing a Derivative Using a One-Period Binomial Model

Binomial Model Arbitrage and Extensions to Other Derivatives

Updated 7 October 2026 · Fact-checked

In a one-period binomial model, an option's no-arbitrage value is the discounted risk-neutral expected payoff. If the market price differs, you have arbitrage: sell the overpriced option and buy the replicating portfolio, or do the reverse. For American options, compare exercise value with continuation value at each node and take the larger.

Understand Arbitrage and Extension to Other Derivatives

The binomial model says the underlying can move to only two prices after one period: an up price and a down price. Because there are only two outcomes, you can build a portfolio of the underlying and a risk-free bond that matches the option's payoff in both states. That portfolio is the replicating portfolio. Its cost today is the option's no-arbitrage value.

Arbitrage appears when the market price differs from that value. If the option is overpriced, sell it and buy the replicating portfolio (h shares, which is a short position when h is negative, as for puts, financed by borrowing or lending as needed). The payoffs cancel in both states, so you keep today's price gap with no risk. If the option is underpriced, buy it and short the replicating portfolio. In both cases the profit is the price gap, locked in today.

The same logic works for other derivatives. A forward on an asset with no income has no-arbitrage price F0 = S0 × (1 + r). Its payoff at expiry is S − F0, so the up and down payoffs can be replicated too. An interest rate option has a rate (or a bond price) as its underlying. You build a tree of the possible rates, compute the payoff in each state, take the risk-neutral expected payoff and discount it. The risk-neutral probabilities must be consistent with the forward rate or price, so exam questions usually give them or the tree inputs.

American-style options can be exercised early. In a one-period model, the American option can be exercised now (time 0) or held to expiry, where its value is the European value. So the American value is the larger of the immediate exercise value and the European (continuation) value. An American call on an asset with no income or other benefits has the same value as the European call, because early exercise is never better. An American put can be worth more than the European put when it is deep enough in the money and interest is positive.

Key formulas to remember

Up and down factors: u = S+ ÷ S0 ; d = S− ÷ S0; S+ and S− are the possible prices after one period.
Risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d); Use for an asset with no income. r is the risk-free rate for the period. The down probability is 1 − π.
Option value (risk-neutral): c0 = [π × c+ + (1 − π) × c−] ÷ (1 + r); Works for puts too: use the put payoffs p+ and p−.
Hedge ratio: Calls: h = (c+ − c−) ÷ (S+ − S−). Puts: h = (p+ − p−) ÷ (S+ − S−); h is the number of shares you hold to hedge one short option. A positive h means a long position in the shares. A negative h means a short position in the shares. For a short call, h is positive, so you hold h shares long. For a short put, the put payoff is larger in the down state, so h is negative, and the hedged position is a short put plus |h| shares sold short. If you are long the option, take the opposite share position.
Arbitrage profit: Profit today = |market price − model value|; Overpriced: sell the option, buy the replicating portfolio. Underpriced: buy the option, sell the replicating portfolio.
No-income forward price: F0 = S0 × (1 + r); A forward priced at this level has zero value at initiation.
American option value (one period): Value = max(immediate exercise value, European value); Call exercise value = max(0, S0 − X). Put exercise value = max(0, X − S0).

How to solve Arbitrage and Extension to Other Derivatives questions

Use the same sequence for any one-period binomial question, whether it asks for a value, an arbitrage profit or an American feature.

1Write down S0, the up and down prices, X, r and the market price of the option if given.
2Compute u and d, then the risk-neutral probability π = (1 + r − d) ÷ (u − d). Check it lies between 0 and 1.
3Compute the option payoff at the up node and the down node. Call: max(0, S − X). Put: max(0, X − S). For an interest rate option, use the rate payoff times the notional.
4Compute the European value: [π × up payoff + (1 − π) × down payoff] ÷ (1 + r).
5If the option is American, compute the immediate exercise value at time 0 and take the larger of the two values.
6Compare the model value with the market price. If the market price is higher, sell the option and buy the replicating portfolio. If it is lower, buy the option and sell the replicating portfolio.
7For the arbitrage profit, take the difference between the market price and the model value. Check it by confirming the hedge pays the same in both states.
8Match your answer to the three options. Numerical options run from smallest to largest.

Quickest way: Value first, then compare

When to use it: Use when you have about 90 seconds and the question asks for an option value, an arbitrage direction or an American value.

Get π in one line from (1 + r − d) ÷ (u − d).
Find the two payoffs and discount the weighted average. Keep the calculator memory for the intermediate result.
If the market price is above your value, the answer is sell the option. If it is below, buy it.
For American options, jump straight to max(exercise now, European value). Only check early exercise for puts, or for calls on assets that pay income.
The arbitrage profit equals the price gap, so do not rebuild the full hedge unless the question asks for the shares or the borrowing.

Common mistakes in Arbitrage and Extension to Other Derivatives

Using the real-world probability of an up move instead of π.
Questions sometimes give an 'actual' probability to tempt you, and the expected-payoff idea feels natural.
Fix: Ignore the real-world probability. Always compute π from u, d and r.
Buying the option when it is overpriced, or selling when it is underpriced.
Students mix up the direction of the comparison between market price and model value.
Fix: Say it aloud: market above value, so sell the option. Market below value, so buy it.
Forgetting to discount the expected payoff by (1 + r).
The weighted payoff looks like a finished answer.
Fix: Always divide by (1 + r) as the last step. Include this in your checklist.
Reporting the American value as the European value, or adding the two together.
Students compute the tree value and stop, or confuse exercise value with premium.
Fix: American value = max(exercise now, European value). The early-exercise premium is the difference.
Using the wrong sign for the hedge ratio on puts, or ignoring the time-0 exercise value of a put.
The call formula is memorised and applied mechanically.
Fix: For puts, payoffs are larger in the down state, so h is negative (short the underlying). Always compute X − S0 for an American put.
Treating an interest rate option like an equity option and discounting at the wrong rate or using the wrong payoff.
The underlying is a rate, so the strike, notional and payoff timing look unfamiliar.
Fix: Compute the payoff as the rate difference times the notional, apply the given risk-neutral probabilities, and discount at the rate stated for that period.

Worked examples

Example 1

A stock trades at $50. In one year it will be $60 or $40. The risk-free rate is 3%. A one-year European call with strike $50 trades at $6.50. What is the arbitrage profit per call today if you exploit the mispricing? A: $0.00 B: $0.92 C: $5.58

Show the solution

u = 60 ÷ 50 = 1.2 and d = 40 ÷ 50 = 0.8.
π = (1 + 0.03 − 0.8) ÷ (1.2 − 0.8) = 0.23 ÷ 0.40 = 0.575.
Call payoffs: up = 60 − 50 = 10, down = 0.
Model value = 0.575 × 10 ÷ 1.03 = 5.75 ÷ 1.03 = 5.5825.
The market price of $6.50 is above $5.58, so the call is overpriced. Sell the call.
Hedge ratio h = (10 − 0) ÷ (60 − 40) = 0.5. The hedge is long 0.5 shares and short 1 call. Buying 0.5 shares costs $25.00, and selling the call brings in $6.50, so the net cost today is 25.00 − 6.50 = $18.50.
Check: at expiry the hedge is worth 0.5 × 60 − 10 = $20 in the up state and 0.5 × 40 − 0 = $20 in the down state.
The sure $20 payoff is worth 20 ÷ 1.03 = $19.42 today (19.4175 unrounded). Since the hedge costs $18.50, the strategy earns 19.4175 − 18.50 = $0.9175, about $0.92 today. This equals 6.50 − 5.5825.

Answer: B: $0.92. Sell the overpriced call and buy 0.5 shares.

Example 2

A stock trades at $40 and will be $50 or $30 in one period. The risk-free rate is 5% for the period. An American put has strike $50. What is its value? A: $7.62 B: $10.00 C: $20.00

Show the solution

u = 50 ÷ 40 = 1.25 and d = 30 ÷ 40 = 0.75.
π = (1.05 − 0.75) ÷ (1.25 − 0.75) = 0.30 ÷ 0.50 = 0.6, so 1 − π = 0.4.
Put payoffs: up = max(0, 50 − 50) = 0, down = 50 − 30 = 20.
European value = (0.6 × 0 + 0.4 × 20) ÷ 1.05 = 8 ÷ 1.05 = 7.619.
Immediate exercise value = 50 − 40 = 10.
American value = max(10, 7.619) = 10.
Early-exercise premium = 10 − 7.619 = 2.38.

Answer: B: $10.00. The put is exercised immediately because exercise value exceeds the European value of $7.62.

Exam tips

Questions often hand you a market price and ask only whether the option is overpriced or underpriced. Compute the model value and compare. You rarely need the full hedge.
Eliminate options fast: a call value cannot exceed the stock price, and an American option cannot be worth less than its European equivalent or its exercise value.
For American puts, always compute X − S0 and compare it with the tree value. For calls on assets with no income, early exercise does not help.
For interest rate options and forwards, set up payoffs per node first, then use the given probabilities or the no-arbitrage forward price. Do not look for a new formula.
Keep the stem's interest rate straight: use the period rate in π and in discounting. No penalty for a wrong answer, so always pick one option.

Practice questions from Valuing a Derivative Using a One-Period Binomial Model

Arbitrage and Extension to Other Derivatives: frequently asked questions

How do I know if an option is mispriced in a binomial model?

Compute the model value using the risk-neutral probability and discount at the risk-free rate. If the market price is higher, the option is overpriced. If lower, it is underpriced.

How do I lock in the arbitrage profit?

If the option is overpriced, sell it and buy the replicating portfolio of h shares funded by borrowing or lending as needed. If it is underpriced, buy it and do the opposite. The payoffs cancel in both states, so you keep today's price gap.

What is the difference between European and American value in a one-period binomial model?

The European value is the discounted risk-neutral expected payoff at expiry. The American value is the larger of that and the payoff from exercising now. If exercising now is worth more, the option is exercised immediately.

How is an interest rate option priced in a binomial model?

Build the tree of possible rates, compute the payoff in each state as the rate difference times the notional, take the risk-neutral expected payoff and discount it. The probabilities must be consistent with no arbitrage, and exam questions usually give them.