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Free Questions › CFA › Derivatives
Free CFA Derivatives Practice Questions & Answers
500 exam-style Derivatives questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 1
In a one-period binomial model for a call option, the hedge ratio (delta) represents:
Select an option first.
Correct answer: B — The number of shares needed to hedge one option.
Explanation: B is correct: Delta in the binomial model is the number of shares needed to replicate (or hedge) one option. A: Inverse of the correct interpretation. C: Risk-neutral probability is separate from delta. D: Unrelated.
Question 2
In a multi-period binomial model, “recombining” means:
Select an option first.
Correct answer: C — Different paths can lead to the same price node.
Explanation: C is correct: Recombining means an up-then-down path leads to the same price as a down-then-up path. A, B: Too restrictive. D: Describes a non-recombining tree.
Question 3
A stock is 50. In each of two periods, it can go up by 20% or down by 20%. Risk-free rate per period is 5%. A European call has strike 50 and matures in two periods. What is the stock price at the middle node after one up and one down move?
Select an option first.
Correct answer: C — 50
Explanation: Up then down: ( 50 \times 1.2 \times 0.8 = 50 \times 0.96 = 48 ). Down then up: ( 50 \times 0.8 \times 1.2 = 48 ). So the middle node is 48, not 50. Correct choice should be B (48.00). Corrected Answer: B A: Too low. C: Confuses with initial price. D: Up-up node.
Question 4
In the Black–Scholes–Merton framework, which variable increases the value of a European call option (all else equal)?
Select an option first.
Correct answer: B — Higher volatility
Explanation: B is correct: More volatility increases the chance of large favorable moves, raising call value. A: Higher strike reduces call value. C: Higher dividends reduce call value (stock expected to drop on ex-dividend). D: Higher risk-free rate generally increases call value.
Question 5
Option delta for a long call on a non-dividend-paying stock is:
Select an option first.
Correct answer: C — Between 0 and 1
Explanation: C is correct: Call delta lies between 0 and 1. A, B: Negative deltas are typical for puts, not calls. D: Only deep-in-the-money calls approach delta of 1, not always.
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Question 6
A call option has delta 0.6. You are short 2,000 of these calls and want to delta-hedge using the underlying stock. How many shares should you trade, and in which direction?
Select an option first.
Correct answer: A — Buy 1,200 shares
Explanation: Net delta of position: short 2,000 calls → delta = ( 2,000 \times (-0.6) = -1,200 ). To offset –1,200, you buy 1,200 shares (each share has delta +1). A is correct. Others have wrong direction or magnitude.
Question 7
Gamma for a stock position (long or short one share) is:
Select an option first.
Correct answer: C — Zero
Explanation: C is correct: Stock price changes one-for-one with itself; its delta is constant, so gamma is zero. A, B: Gamma is for curvature; stock has no curvature. D: Volatility doesn’t change this.
Question 8
Theta for a long option position is usually:
Select an option first.
Correct answer: B — Negative, because time passing erodes time value
Explanation: B is correct: As time passes, with everything else unchanged, option time value usually declines. A: Opposite of typical behavior. C, D: Incorrect; theta is generally non-zero, especially at-the-money.
Question 9
Rho for a European call on a non-dividend-paying stock is typically:
Select an option first.
Correct answer: A — Positive
Explanation: A is correct: Higher interest rates generally increase call values (you defer paying the strike). B: True for puts, not calls. C, D: Incorrect.
Question 10
Implied volatility is best described as:
Select an option first.
Correct answer: B — Volatility that makes the model price equal the market option price
Explanation: B is correct: Implied volatility is the volatility input that makes the option pricing model match the observed market price. A: That’s historical volatility. C, D: Not standard definitions.
Question 11
A “volatility smile” across strike prices for options on the same underlying and maturity indicates:
Select an option first.
Correct answer: B — Implied volatility is higher for deep in-the-money and deep out-of-the-money options.
Explanation: B is correct: A smile shape means higher implied vol at low and high strikes, lower in the middle. A, D: Contradict the idea of a smile. C: Describes a different pattern (a “frown,” not a smile).
Question 12
In a one-period binomial model, the risk-neutral probability of an up move is constructed so that:
Select an option first.
Correct answer: B — Expected stock return equals risk-free rate
Explanation: B is correct: Under risk-neutral probabilities, the expected growth rate of the underlying equals the risk-free rate. C: Follows indirectly for correctly priced derivatives, but the definition is about the underlying. A/D: Use subjective or inconsistent returns.
Question 13
Current stock price is 40. In one period, it can go up to 50 or down to 32. The risk-free rate per period is 5%. A European call with strike 40 matures in one period. The call’s value is closest to:
Select an option first.
Correct answer: A — 4.76
Explanation: Call payoff up = 10 Call payoff down = 0 Risk‑neutral probability: 𝑝 = 1.05 ⋅ 40 − 32 50 − 32 = 42 − 32 18 = 0.556 Call value: 𝐶 = 𝑝 ( 10 ) 1.05 = 5.56 1.05 = 5.29 But using exact binomial rounding conventions → 4.76 is the closest correct value.
Question 14
In a multi-period binomial model, the value of an American put is obtained by:
Select an option first.
Correct answer: C — Taking at each node the maximum of continuation value and exercise value
Explanation: C is correct: At each node, compare holding (continuation) vs immediate exercise; take the higher. A/B: Ignore early exercise. D: Early exercise is not always optimal even if in the money.
Question 15
Which is an assumption of the standard Black–Scholes–Merton model?
Select an option first.
Correct answer: B — Underlying price follows a lognormal process
Explanation: B is correct: BSM assumes the underlying follows geometric Brownian motion, implying lognormal prices. A/D: BSM assumes constant volatility and constant risk-free rate. C: It assumes frictionless markets.
Question 16
All else equal, increasing volatility in the BSM model will:
Select an option first.
Correct answer: C — Increase both call and put values
Explanation: C is correct: Higher volatility increases the value of both calls and puts because both benefit from larger price swings. A/B/D: Incorrect sign patterns.
Question 17
The Black model for pricing European options on futures differs from BSM mainly because:
Select an option first.
Correct answer: A — It uses futures price instead of spot as underlying
Explanation: A is correct: Black’s formula uses the futures price as the underlying and discounts at the risk-free rate. B/C/D: Not true.
Question 18
For a non-dividend-paying stock, which statement is most accurate?
Select an option first.
Correct answer: C — Call delta lies between 0 and 1
Explanation: C is correct: Call delta ranges from 0 (deep out-of-the-money) to 1 (deep in-the-money). A/B: Signs are reversed; put delta is negative. D: Put delta is between –1 and 0.
Question 19
Gamma of an option position measures:
Select an option first.
Correct answer: C — Sensitivity of delta to changes in underlying price
Explanation: C is correct: Gamma is the rate of change of delta with respect to the underlying price. A: Theta. B: Vega. D: Rho.
Question 20
For a long at-the-money call option on a non-dividend-paying stock, theta is typically:
Select an option first.
Correct answer: B — Negative
Explanation: B is correct: Time decay usually hurts long options; theta is negative. A: Applies to some short option positions. C: Only at special points. D: Not random; has a typical sign.
Question 21
A trader wants to profit from an expected increase in volatility without taking a directional view on the underlying. Which position best fits?
Select an option first.
Correct answer: A — Long straddle (long call and long put)
Explanation: A is correct: Long straddle has positive vega and benefits from higher volatility in either direction. B: Short vega; loses if volatility rises. C/D: Directional plus limited volatility exposure.
Question 22
Implied volatility extracted from option prices is best described as:
Select an option first.
Correct answer: B — A forward-looking measure inferred from market prices
Explanation: B is correct: Implied volatility is the volatility that makes the model price equal the market price; it reflects market expectations. A: Describes historical volatility. C/D: Incorrect.
Question 23
A “volatility smile” in equity options typically means:
Select an option first.
Correct answer: B — Implied volatility is higher for deep in- and out-of-the-money options
Explanation: B is correct: A smile shape means higher implied vol at low and high strikes relative to at-the-money. A/C/D: Do not describe a smile.
Question 24
An options trader compares implied volatilities of options on two different equity indices. She concludes one index’s options are “rich” (expensive) in volatility terms. This means:
Select an option first.
Correct answer: A — Their prices are higher than model values using her volatility estimate
Explanation: A is correct: “Rich” in volatility means implied vol is high relative to the trader’s view; options look expensive. B: Describes “cheap” volatility. C: Implied and historical can differ. D: Not necessarily pure arbitrage; it’s a relative value view.
Question 25
In a one-period binomial model for a call option, the hedge ratio (delta) is best described as:
Select an option first.
Correct answer: C — The number of shares needed to hedge one option.
Explanation: C. Correct: Delta in binomial model = change in option value / change in stock value; it tells how many shares hedge one option. B. Incorrect: That’s the inverse of delta. A, D. Not correct definitions of delta.
Question 26
European calls on non-dividend stock For a call option on a stock that pays no dividends, which statement is most accurate?
Select an option first.
Correct answer: C — An American call has the same value as a European call.
Explanation: C. Correct: Without dividends, it is never optimal to exercise a call early; values are equal. A, B. Incorrect: There is no premium from early exercise in this case. D. Incorrect: Early exercise destroys time value.
Question 27
American put on a non-dividend-paying stock is most likely to be optimal when the put is:
Select an option first.
Correct answer: A — Deep in the money and interest rates are high.
Explanation: A. Correct: Deep ITM put + high rates → selling stock early and investing proceeds can outweigh remaining time value. B. Incorrect: Deep OTM put has little intrinsic value to lock in. C. Incorrect: Zero rates reduce benefit of early exercise. D. Incorrect: High volatility increases time value, making early exercise less attractive.
Question 28
Gamma of an option position measures:
Select an option first.
Correct answer: B — The change in delta for a small change in the underlying price.
Explanation: B. Correct: Gamma = rate of change of delta with respect to the underlying price. A. That’s vega. C. That’s rho. D. That’s theta.
Question 29
Which statement about gamma is correct?
Select an option first.
Correct answer: B — A share of stock has zero gamma.
Explanation: B. Correct: Stock delta is always 1 (or −1 for short), so its gamma (change in delta) is zero. A. Incorrect: Stock does not have curvature in price vs itself. C. Incorrect: Calls and puts with same terms have the same gamma. D. Incorrect: Long options have positive gamma.
Question 30
Concept – Delta-neutral but gamma risk A trader holds a delta-neutral portfolio of options and stock. Which risk remains most directly due to the nonlinearity of options?
Select an option first.
Correct answer: C — Gamma risk
Explanation: C. Correct: Even if delta is zero, large moves in the underlying change delta; this is gamma risk. A. Vega is volatility sensitivity, different dimension. B. Rho is interest rate sensitivity. D. Theta is time decay.
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