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Free Questions › FRM › Valuation and Risk Models
Free FRM Valuation and Risk Models Practice Questions & Answers
400 exam-style Valuation and Risk Models questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 181
A delta-hedged option position is exposed to which of the following risks?
Select an option first.
Correct answer: B — Gamma risk — large moves in the underlying cause the delta to change, leaving the position with residual P&L from the curvature of the option payoff.
Explanation: Delta hedging neutralises directional (first-order) price risk. It does not eliminate gamma (curvature) risk — for large moves, the position is no longer delta-neutral and incurs P&L. A perfectly gamma-hedged position additionally requires a second option to neutralise gamma. C and D are partial — both vega and theta remain after delta hedging, but gamma is the primary concern for large moves.
Question 182
The duration of a coupon-paying bond is always:
Select an option first.
Correct answer: C — Less than its time to maturity (because coupons are received before maturity).
Explanation: Macaulay duration < maturity for coupon-paying bonds: coupons received before maturity pull the weighted average time earlier. Only a zero-coupon bond has duration = maturity. D is the definition (correct in concept) but C correctly contrasts coupon bond duration vs zero-coupon, which is the tested point here.
Question 183
Negative convexity in a bond typically arises from:
Select an option first.
Correct answer: B — Embedded call options (e.g., callable bonds, MBS prepayment options) — when yields fall, the issuer/borrower prepays, capping price appreciation.
Explanation: Negative convexity: the bond's price rises less than a straight bond when yields fall, because the probability of early redemption/prepayment increases. Callable bonds and MBS exhibit negative convexity in low-rate environments. A is credit risk; C and D don't create negative convexity in the bond itself.
Question 184
Which of the following interest rate models is characterised by explicitly ensuring that interest rates cannot become negative (log-normal rates)?
Select an option first.
Correct answer: B — Cox-Ingersoll-Ross (CIR) model
Explanation: CIR model: dr = κ(θ − r)dt + σ√r dW. The √r term means the diffusion vanishes as r → 0, preventing negative rates (under parameter conditions). Vasicek (normal distribution) allows negative rates. Ho-Lee and Hull-White are also normal models. Under CIR, rates are always non-negative.
Question 185
The Vasicek interest rate model features:
Select an option first.
Correct answer: B — Mean reversion toward a long-run level θ, with normally distributed rate changes.
Explanation: Vasicek: dr = κ(θ − r)dt + σdW. κ is the speed of mean reversion; θ is the long-run mean; σ is volatility; dW is a Wiener process. Rate changes are normally distributed — rates can be negative. Mean reversion is explicit (A wrong); C is CIR; D has no randomness.
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Question 186
In a two-period binomial tree, a European put option value is computed by:
Select an option first.
Correct answer: B — Working backwards from expiry: compute put payoffs at terminal nodes, weight by risk-neutral probabilities, discount at the risk-free rate to find the value at each prior node.
Explanation: Binomial pricing (risk-neutral): (1) compute max(K−S,0) at each terminal node; (2) work backward: V_node = [p×V_up + (1-p)×V_down] × e^(−rΔt); (3) repeat until the initial node is reached. For European options, never take early exercise maximum (that applies to American options). A, C (American exercise logic), D are wrong.
Question 187
Which of the following best explains why implied volatility differs across strike prices (volatility smile/skew)?
Select an option first.
Correct answer: B — Actual return distributions have fat tails and skewness that BSM (which assumes log-normality) cannot capture — the smile reflects the market's corrections to BSM's limitations.
Explanation: The smile/skew: BSM assumes constant log-normal returns. Real distributions have: (1) negative skew (fat left tail in equities), (2) excess kurtosis (fat tails on both sides). OTM puts are priced with higher implied vol to reflect the market's higher probability of crash than BSM predicts. A is wrong (BSM has limitations); C is wrong; D is over-simplified.
Question 188
The relationship between futures price and spot price under the cost-of-carry model is:
Select an option first.
Correct answer: B — F = S × e^((r + u − y) × T), where r is risk-free rate, u is storage cost, y is convenience yield.
Explanation: Cost of carry: F₀ = S₀ × e^((r + u − y)T). For financial assets without storage costs or dividends: F₀ = S₀ × e^(rT). For commodities: add storage cost (u), subtract convenience yield (y). For stocks with dividends: F₀ = S₀ × e^((r−q)T). A subtracts; C uses discrete discounting; D is incorrect direction.
Question 189
An interest rate cap is best described as:
Select an option first.
Correct answer: B — A series of European call options on a floating interest rate — each caplet pays the difference between the floating rate and a strike rate if the floating rate exceeds the strike.
Explanation: An interest rate cap: portfolio of caplets. Each caplet is a call on the floating rate (e.g., 3-month SOFR) — pays max(floating − K, 0) × notional × day count fraction. A is an interest rate swap; C is a fixed-rate loan commitment; D is a bond put (floor-related concept).
Question 190
An interest rate floor pays the holder when:
Select an option first.
Correct answer: B — Interest rates fall below the floor rate (each floorlet pays max(K − floating rate, 0)).
Explanation: Floor = series of put options on a floating rate. Each floorlet pays max(K − floating, 0). Used by floating-rate investors to guarantee a minimum return. When rates fall below K, the floor compensates. A describes a cap; C is a bond feature; D is a yield curve scenario.
Question 191
Which of the following statements about the Greeks is correct?
Select an option first.
Correct answer: B — Delta, gamma, and vega are always positive for long calls; theta is always negative for long calls.
Explanation: Long call Greeks: Delta (+), Gamma (+), Vega (+), Theta (−), Rho (+). A is wrong — theta is negative. C is wrong — put delta is negative. D is wrong — gamma is positive for long options (positive convexity). Short options reverse the signs of all Greeks.
Question 192
For a non-dividend-paying stock, which of the following statements about American vs European call options is correct?
Select an option first.
Correct answer: B — For a non-dividend-paying stock, it is never optimal to exercise an American call early, so American and European calls have the same value.
Explanation: Classic options result: for non-dividend-paying stocks, an American call should never be exercised early because the time value is always positive. Exercising early loses the time value. Therefore, the American call equals the European call in value. This changes when dividends exist — early exercise may be rational just before an ex-dividend date.
Question 193
A bond's 'duration gap' in the context of ALM (Asset-Liability Management) is:
Select an option first.
Correct answer: B — The difference between the duration of assets and the (liability-weighted) duration of liabilities.
Explanation: Duration gap = D_assets − (L/A) × D_liabilities. If positive: assets are more rate-sensitive than liabilities; rising rates reduce equity value. If zero: immunised. If negative: liabilities more rate-sensitive; rising rates increase equity value. Banks and insurance companies actively manage duration gap to control interest rate risk.
Question 194
A callable bond has the same maturity and coupon as an otherwise identical straight bond. How does the call feature affect the bond's convexity?
Select an option first.
Correct answer: B — The callable bond has lower (potentially negative) convexity because the call cap limits price appreciation when yields fall.
Explanation: Callable bond = straight bond − call option value (to issuer). At low yields, the call option becomes valuable — the price appreciation of the callable bond is capped (the issuer is likely to call). This creates negative convexity at low yields. At high yields (unlikely call), the callable bond behaves like a straight bond with positive convexity. D is partially correct but B is more complete.
Question 195
In the Fundamental Review of the Trading Book (FRTB), the Internal Models Approach uses 'Expected Shortfall at 97.5%' calibrated to a stress period. Which of the following is an additional major change from the pre-FRTB framework?
Select an option first.
Correct answer: B — Liquidity horizons are assigned to risk factors (10-120 trading days) rather than using a uniform 10-day horizon, reflecting differences in asset liquidity.
Explanation: FRTB key changes: (1) ES at 97.5% replaces 99% VaR; (2) liquidity horizons of 10, 20, 40, 60, or 120 days depending on risk factor type (not a blanket 10-day); (3) P&L attribution test; (4) non-modellable risk factors (NMRFs); (5) desk-level approval. A is wrong (back-testing retained); C is wrong (IMA still exists); D is wrong.
Question 196
A portfolio manager holds a long bond position and wants to hedge both duration and convexity. To hedge convexity, the manager must:
Select an option first.
Correct answer: B — Use an option or option-like instrument, since convexity (gamma) in a bond portfolio can only be hedged with an instrument that has non-linear price behaviour.
Explanation: Duration (first-order) can be hedged with linear instruments (futures, other bonds). Convexity (second-order) requires an option or option-like instrument with positive gamma to offset the convexity. Adding another bond only changes the duration hedge ratio — it cannot independently control convexity. A and C are linear adjustments; D addresses only duration.
Question 197
Under the risk-neutral valuation framework, why can we discount at the risk-free rate rather than the asset's actual expected return?
Select an option first.
Correct answer: B — By changing to the risk-neutral probability measure, where all assets have an expected return equal to the risk-free rate, derivative values can be computed as discounted expected payoffs without knowing actual risk preferences.
Explanation: Risk-neutral pricing (Girsanov's theorem / equivalent martingale measure): we change probability measure from the real-world P to the risk-neutral Q, under which all assets earn the risk-free rate. In Q, expected payoffs discounted at r give the correct price. This is not because all assets earn r in the real world — it is a mathematical measure-change convenience. A is wrong; C is too narrow; D is partially true but not the correct explanation.
Question 198
A bank holds a portfolio of MBS (Mortgage-Backed Securities) with negative convexity. When interest rates fall significantly, which of the following best describes the impact on the portfolio?
Select an option first.
Correct answer: B — The portfolio gains less than a comparable straight bond portfolio because prepayments accelerate, effectively calling the securities at par and preventing full price appreciation.
Explanation: MBS negative convexity: when rates fall, mortgage borrowers refinance (prepay) — the MBS investor receives face value back at a time when reinvesting proceeds yields lower rates. This caps the price appreciation below what a straight bond would achieve. Duration also shortens as prepayments accelerate, reducing sensitivity further. A and C have the direction wrong; D is incorrect.
Question 199
The sensitivity of an option's delta to changes in the underlying (gamma) is highest when:
Select an option first.
Correct answer: B — The option is at the money and near expiry.
Explanation: Gamma peaks when: (1) the option is at the money (delta transition from ~0 to ~1 is fastest around ATM), AND (2) near expiry (the transition from 0 to 1 happens over an ever-shorter time window). Deep ITM/OTM options have low gamma (delta is already near 1 or 0, not changing much). Long time to expiry → lower gamma (gradual delta change). This makes near-expiry ATM options most expensive to gamma-hedge.
Question 200
A bond portfolio manager uses PCA to decompose yield curve risk. The portfolio has large exposure to PC2 (slope factor) but minimal exposure to PC1 (level factor). Which of the following hedging instruments is most appropriate to reduce the PC2 (slope) exposure?
Select an option first.
Correct answer: B — A steepener/flattener trade — simultaneously shorting short-duration bonds and going long long-duration bonds (or vice versa) to hedge the slope sensitivity.
Explanation: PC2 (slope): short-term yields move opposite to long-term yields. To hedge slope risk, you need instruments with offsetting short-end and long-end exposures — a steepener or flattener trade (e.g., short 2-year treasury + long 10-year treasury). A hedges PC1 (level); C hedges volatility; D only hedges PC1 (level).
Question 201
When computing the value of an American put option using a binomial tree, at each node you must take:
Select an option first.
Correct answer: B — The maximum of the intrinsic value (immediate exercise) and the discounted expected continuation value.
Explanation: American put binomial: V = max(K−S, [p×V_up + (1-p)×V_down]×e^(−rΔt)). At each node, compare: (1) exercise now (intrinsic value K−S) vs (2) hold (continuation value from risk-neutral pricing). Take the maximum. D is wrong — early exercise is sometimes optimal for deep ITM puts. C takes the minimum (incorrect).
Question 202
Which of the following explains why an at-the-money straddle (long call + long put at the same strike) profits from high realised volatility?
Select an option first.
Correct answer: B — The straddle buyer profits if the underlying makes a large move in either direction — the combined payoff exceeds the total premium paid when the magnitude of the move is large enough.
Explanation: Long straddle = long call + long put (same strike, same expiry). Payoff profile: V-shaped — profits when S moves far from K in either direction. Breakeven: S_T < K − Premium or S_T > K + Premium. The buyer is long volatility (positive vega) — high realised volatility benefits the position. A reverses buyer/seller; C eliminates risk is wrong; D is wrong (profits from both sides).
Question 203
The 'Fundamental Theorem of Asset Pricing' relates to the absence of arbitrage and states:
Select an option first.
Correct answer: B — The absence of arbitrage is equivalent to the existence of at least one risk-neutral (equivalent martingale) probability measure under which all discounted asset prices are martingales.
Explanation: First fundamental theorem: no arbitrage ↔ existence of at least one equivalent martingale measure (EMM). Second fundamental theorem: the market is complete (all contingent claims replicable) ↔ uniqueness of the EMM. This is the mathematical foundation of derivatives pricing. A, C, D are not the theorem.
Question 204
A bond portfolio manager is concerned about 'basis risk' when hedging with Treasury futures. What is the primary source of this basis risk?
Select an option first.
Correct answer: B — Imperfect correlation between the yield movements of the bonds being hedged and the Treasury futures' cheapest-to-deliver bond, leaving a residual exposure after hedging.
Explanation: Basis risk in bond futures hedging: the futures price tracks the CTD (cheapest-to-deliver) bond. If the hedged bond's yield changes differently from the CTD bond's yield (different spread dynamics, sector, credit quality), the hedge is imperfect. This residual is called basis risk. A affects the hedge ratio calculation; C creates convexity risk but not basis risk per se; D is a hedge ratio mismatch, addressable.
Question 205
A risk manager wants to delta-gamma hedge an options portfolio. After delta-hedging with the underlying, the portfolio still has a significant gamma of −500. Which of the following actions would reduce gamma exposure?
Select an option first.
Correct answer: B — Buying an option with a positive gamma (e.g., a long call or put) with an appropriate notional.
Explanation: Gamma can only be adjusted using options (which have non-linear payoffs and thus non-zero gamma). The underlying has zero gamma. To neutralise a gamma of −500: buy options with total gamma = +500. After delta-adjusting for the new option position, delta-hedging with the underlying restores delta neutrality. A, C have zero gamma; D addresses vega, not gamma.
Question 206
Under BSM, the price of a European put option on a non-dividend-paying stock increases when which of the following occurs (all else equal)?
Select an option first.
Correct answer: C — The strike price increases.
Explanation: Put price increases with: (1) higher K (deeper ITM for put), (2) lower S (put ITM), (3) higher σ (more chance of large downward move), (4) lower r (PV of strike K×e^(−rT) is higher when r is lower), (5) longer T (usually). A (higher S) reduces put value; B (shorter T) generally reduces put value; D (higher r) reduces put value by reducing PV(K).
Question 207
The 'cheapest-to-deliver' (CTD) bond concept is relevant in Treasury futures contracts because:
Select an option first.
Correct answer: B — The seller of a Treasury futures contract has the option to deliver any qualifying bond from a basket — they will deliver the bond that is least expensive to acquire relative to the futures price (maximising profit or minimising cost).
Explanation: CTD: the futures seller has a delivery option — they can deliver any bond from the qualifying basket. They choose the bond where: CTD = argmin(Bond price − Futures price × Conversion factor). The conversion factor adjusts for coupon differences. This delivery option has value and is embedded in the futures price. A, C, D are wrong.
Question 208
A portfolio manager uses a 'barbell' strategy (holding bonds concentrated at short and long maturities) versus a 'bullet' strategy (concentrated at intermediate maturities). How do these strategies compare on yield curve curvature risk?
Select an option first.
Correct answer: B — The barbell has more positive convexity than the bullet, but the barbell is more exposed to curvature (butterfly) shifts where the middle of the curve rises relative to the wings.
Explanation: Barbell vs bullet: for the same modified duration, the barbell typically has higher convexity (more spread around the average duration → more curvature benefit). However, the barbell is short the intermediate maturity → it is hurt by a butterfly shift where intermediate yields rise (barbell outperforms if mid-curve falls relative to wings). Duration matching doesn't immunise against non-parallel shifts. C is wrong; D is partially related but doesn't capture curvature.
Question 209
The Greeks relationship in Black-Scholes satisfies which of the following (for a non-dividend-paying call)?
Select an option first.
Correct answer: A — Delta + Theta + ½σ²S²Gamma = rC (the BSM PDE: Theta + rSΔ + ½σ²S²Γ = rC)
Explanation: The BSM PDE: ∂V/∂t + rS(∂V/∂S) + ½σ²S²(∂²V/∂S²) = rV, i.e., Theta + rS×Delta + ½σ²S²×Gamma = r×V. This is the fundamental partial differential equation that all derivative prices satisfying BSM assumptions must obey. B has no mathematical basis; C is not always true; D is incorrect.
Question 210
A risk manager calculates that the portfolio's 'gamma P&L' (from a delta-neutral options book) over a period where the underlying moved $5 is approximately:
Select an option first.
Correct answer: B — Gamma P&L ≈ ½ × Gamma × (ΔS)²
Explanation: Taylor expansion of option price: ΔV ≈ Δ×ΔS + ½Γ×(ΔS)² + Θ×Δt + ν×Δσ. For a delta-neutral portfolio (Δ=0 at hedge initiation), the dominant P&L source from underlying moves is ½Γ×(ΔS)². A is the delta P&L; C is vega P&L; D is theta P&L.
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