Free FRM Quantitative Analysis Practice Questions & Answers
400 exam-style Quantitative Analysis questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
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Question 151
Using Black-Scholes, the price of a European call option increases when:
Select an option first.
Correct answer: C — Volatility increases
Explanation: C
Question 152
The forward price of a non-dividend paying stock for delivery in T years is:
Select an option first.
Correct answer: B — S_0 × e^{rT} (continuous compounding) or S_0 × (1+r)^T (discrete)
Explanation: B
Question 153
A long futures position profits when:
Select an option first.
Correct answer: B — The futures price rises
Explanation: B
Question 154
The delta of an at-the-money call option is approximately:
Select an option first.
Correct answer: B — 0.5
Explanation: B
Question 155
Theta of an option represents:
Select an option first.
Correct answer: B — The rate of decay of option value with the passage of time, all else equal
Explanation: B
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The Greeks for a put option: which of the following is true?
Select an option first.
Correct answer: B — Delta is negative (ranging from -1 to 0), Gamma is positive (same as calls), Theta is typically negative, Vega is positive
Explanation: B
Question 157
The put-call parity for European options requires that:
Select an option first.
Correct answer: B — P - C = K×e^{-rT} - S (rearranged: C - P = S - K×e^{-rT})
Explanation: B
Question 158
A repo (repurchase agreement) is effectively:
Select an option first.
Correct answer: B — A short-term collateralised borrowing where the seller sells securities and agrees to repurchase them at a higher price on a later date, with the price difference representing the financing cost
Explanation: B
Question 159
The payoff of a binary (digital) cash-or-nothing call option is:
Select an option first.
Correct answer: B — A fixed payment Q if S_T > K, and zero otherwise
Explanation: B
Question 160
The payoff of a down-and-out barrier call option is:
Select an option first.
Correct answer: B — max(S_T - K, 0) if the underlying price has never fallen to or below a barrier level B during the option's life; zero if the barrier is breached
Explanation: B
Question 161
The binomial tree model for option pricing becomes the Black-Scholes model when:
Select an option first.
Correct answer: B — The number of time steps approaches infinity and step size approaches zero (continuous limit)
Explanation: B
Question 162
A bear spread using put options involves:
Select an option first.
Correct answer: B — Buying a high-strike put and selling a low-strike put
Explanation: B
Question 163
The Black model (Black-76) is used to price:
Select an option first.
Correct answer: B — Options on futures contracts, where the futures price replaces the spot price in the Black-Scholes formula, discounted at the risk-free rate
Correct answer: B — Delta changes as the underlying price, volatility, and time change (gamma and vanna effects), so a previously delta-neutral portfolio becomes non-neutral as these factors change
Explanation: B
Question 165
Rho of a call option is positive because:
Select an option first.
Correct answer: B — A higher risk-free rate reduces the present value of the strike price paid at expiry, effectively lowering the cost of the call, increasing its value
Explanation: B
Question 166
A butterfly spread using calls is constructed as:
Select an option first.
Correct answer: B — Long 1 call at low strike, short 2 calls at middle strike, long 1 call at high strike — creates a position that profits if the underlying stays near the middle strike at expiry
Explanation: B
Question 167
A calendar spread (time spread) in options involves:
Select an option first.
Correct answer: B — Buying a longer-dated option and selling a shorter-dated option at the same strike, profiting from time decay differences between the two expirations
Explanation: B
Question 168
The payoff of a long forward at maturity for a buyer who agreed to buy at forward price F₀ is:
Select an option first.
Correct answer: B — S_T - F₀ — the difference between the spot price at delivery and the agreed forward price, which can be positive or negative
Explanation: B
Question 169
The key difference between a CDS and a bond purchase from a risk perspective is:
Select an option first.
Correct answer: B — A bond purchase creates both credit risk and funding risk (must finance the purchase); a CDS creates credit risk synthetically with no funding requirement (except margin). CDS can also be short credit without the difficulty of shorting bonds
Explanation: B
Question 170
The value of a call option is higher when:
Select an option first.
Correct answer: C — The risk-free rate is higher (and underlying does not pay dividends)
Explanation: C
Question 171
The convexity of a bond is always positive for a straight (option-free) bond because:
Select an option first.
Correct answer: B — The price-yield relationship is always curved upward (convex), meaning price appreciation from yield declines exceeds price depreciation from equal yield increases — a mathematical property of discounting
Explanation: B
Question 172
A knock-in barrier option:
Select an option first.
Correct answer: B — Comes into existence only if the underlying price touches the barrier level during the option's life — it is worthless before activation
Explanation: B
Question 173
The total P&L of a delta-hedged long option position over a short period is approximately:
Select an option first.
Correct answer: B — (1/2) × Γ × (ΔS)² + Θ × Δt — the sum of gamma P&L (positive, from large moves) and theta P&L (negative, from time decay)
Explanation: B
Question 174
The relationship between the price of a European call and a European put with the same strike and expiry (put-call parity) breaks down for:
Select an option first.
Correct answer: B — American-style options — American options include early exercise rights that the European put-call parity relationship does not accommodate, resulting only in put-call inequalities rather than equalities
Explanation: B
Question 175
Options on dividend-paying stocks are adjusted in the Black-Scholes model by:
Select an option first.
Correct answer: B — Reducing the current stock price by the present value of expected dividends: S* = S - PV(dividends), or equivalently treating the dividend yield q as a continuous yield: S → S×e^{-qT}
Explanation: B
Question 176
A delta-hedged portfolio is gamma-neutral when:
Select an option first.
Correct answer: C — A position in options is added such that the total gamma equals zero, eliminating sensitivity to large price moves
Explanation: C
Question 177
The no-arbitrage condition for a currency forward rate is given by covered interest rate parity:
Select an option first.
Correct answer: B — F = S × (1 + r_d)^T / (1 + r_f)^T in discrete compounding or S × e^{(r_d - r_f)T} in continuous
Explanation: B
Question 178
The gamma exposure of a delta-hedged short option position is:
Select an option first.
Correct answer: B — Negative, losing money when the underlying makes large moves in either direction
Explanation: B
Question 179
The futures basis is defined as:
Select an option first.
Correct answer: B — Spot price minus futures price (or futures minus spot, depending on convention). Basis convergence — the basis approaches zero at delivery — is central to futures pricing.
Explanation: B
Question 180
Vanna measures:
Select an option first.
Correct answer: B — The sensitivity of delta to changes in implied volatility (∂Δ/∂σ = ∂Vega/∂S)
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