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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 241
A 10-year bond with modified duration 7.5 and convexity 85. For a 200bp yield rise, approximate price change including convexity:
Select an option first.
Correct answer: C — −13.30%
Explanation: ΔP/P ≈ −7.5×0.02 + ½×85×(0.02)² = −0.15 + 0.5×85×0.0004 = −0.15 + 0.017 = −0.133 = −13.3%. A ignores convexity; B has a computation error; D has wrong convexity sign. The convexity term is always positive — it reduces the price decline for yield rises.
Question 242
An interest rate floorlet with notional $5m, floor rate 3%, current SOFR=1.5%. The floorlet payoff:
Select an option first.
Correct answer: B — $75,000
Explanation: Floorlet payoff = max(K−SOFR,0) × Notional = max(3%−1.5%,0) × $5m = 1.5% × $5m = $75,000. A incorrectly states no payment — SOFR below the floor rate DOES trigger a floorlet payment to the buyer. C halves; D doubles.
Question 243
The CIR model prevents negative interest rates through:
Select an option first.
Correct answer: B — The √r diffusion term that vanishes as r→0, preventing negative rates under the Feller condition.
Explanation: CIR: dr = κ(θ−r)dt + σ√r dW. At r=0, the diffusion term σ√r = 0, and the drift κθ is positive — pulling rates back up. Under Feller condition (2κθ > σ²), rates remain strictly positive. Vasicek, Ho-Lee, Hull-White all have constant diffusion (no √r), allowing negative rates.
Question 244
A steepener trade (long short-dated bonds, short long-dated bonds, DV01-neutral) profits when:
Select an option first.
Correct answer: B — The yield curve steepens — long-term yields rise more than short-term yields.
Explanation: Steepener: long 2-year + short 10-year (DV01-neutral). Profit when: (1) 10-year yield rises → short 10-year gains, OR (2) 2-year yield falls → long 2-year profits, OR both (steepening). A parallel shift is DV01-neutral → approximately zero P&L. C is a credit trade; D typically flattens the curve (lower inflation expectations reduce long-end inflation premium).
Question 245
Rho of a European call option:
Select an option first.
Correct answer: B — Is positive — higher interest rates increase call value by reducing PV of the strike.
Explanation: Rho (ρ) = ∂C/∂r = K×T×e^(−rT)×N(d₂) > 0. Higher r → lower PV(K) = Ke^(−rT) → strike costs less in PV terms → call more valuable. For puts: rho < 0 (higher r reduces put value). A is wrong; C is wrong (rho is small but non-zero for short-dated); D describes dividend sensitivity (sometimes called psi or phi).
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Question 246
Each caplet in an interest rate cap resembles:
Select an option first.
Correct answer: B — A European call on the floating interest rate — pays when rate > strike.
Explanation: Caplet = call on floating rate: payoff = max(SOFR−K,0)×N×α. From a bond perspective: caplet = put on a zero-coupon bond (rising rates → falling bond price → put on bond is ITM). B correctly describes the rate-option view. D describes the bond-option equivalent (also correct but less intuitive). A has wrong direction; C is American and applies to bonds.
Question 247
No-arbitrage forward on a dividend-paying stock (discrete PV of dividends = PV(D)):
Select an option first.
Correct answer: B — F = (S − PV(D)) × e^(rT)
Explanation: Discrete dividends: F = (S − PV(D)) × e^(rT). Subtract PV of dividends from spot (futures holders miss dividend income) then compound at r. A adds PV(D) incorrectly; C uses continuous dividend yield q (different approach); D divides (mathematically wrong).
Question 248
PCA on yield curve data: PC1 explains ~75% of variance. Efficient hedge for PC1 risk of a 10-year bond portfolio:
Select an option first.
Correct answer: B — A DV01-matched short position in Treasury futures — parallel shift hedge eliminates PC1 risk.
Explanation: PC1 is a parallel shift. To hedge PC1: match total portfolio DV01 with a short futures position (any maturity, as long as DV01 is matched). Since PC1 moves all yields together, a DV01-neutral position eliminates first-order exposure to this dominant risk factor. B is the most efficient description. A also works (same-maturity futures) but B is more general.
Question 249
Long bullet (5yr), short barbell (2yr+10yr, DV01-neutral). This trade profits when:
Select an option first.
Correct answer: B — A negative butterfly — 5-year yields fall relative to 2-year and 10-year wings.
Explanation: Long bullet / short barbell: profits when 5-year outperforms (yield falls more than wings) — a 'negative butterfly' (curvature decreases). A and D are parallel shifts: DV01-neutral → ≈zero P&L. C is a credit trade. A positive butterfly (5yr yield rises relative to wings) hurts the long bullet / short barbell position.
Question 250
Vega risk management requires:
Select an option first.
Correct answer: B — Trading options or volatility products — the underlying asset has zero vega.
Explanation: Vega can only be adjusted using options or volatility derivatives (variance swaps, VIX futures). The underlying asset has zero vega — delta hedging leaves vega exposure completely unchanged. Short options have negative vega (benefit from vol decline); long options have positive vega. A, C, D cannot change a portfolio's vega.
Question 251
Inverted volatility term structure (short-dated implied vol > long-dated) typically indicates:
Select an option first.
Correct answer: B — Near-term uncertainty is elevated — specific event risk or acute market stress expected to normalise over time.
Explanation: Inverted vol term structure: the market expects current high uncertainty to decay — near-term vol is elevated (specific event, acute stress) but long-run vol is lower (reversion to calm expected). Normal curve slopes upward (long-dated > short-dated) reflecting the uncertainty premium for longer-horizon uncertainty. A, C, D are wrong.
Question 252
Factors that most increase an interest rate cap's value (all else equal):
Select an option first.
Correct answer: C — Increasing implied volatility of the underlying rate.
Explanation: Cap (caplet) value drivers: (+) higher forward rate relative to strike (deeper ITM), (+) higher implied vol (wider distribution), (+) more time to expiry, (+) lower discount rate. Higher implied vol increases the caplet's value by increasing the probability of the rate exceeding the strike. A reduces value; B makes it further OTM; D makes it OTM.
Question 253
A bull flattener trade (long long-duration bonds, short short-duration bonds, DV01-neutral) profits when:
Select an option first.
Correct answer: B — Both long and short rates fall, but long rates fall more — the long-duration position gains more than the short-duration position loses.
Explanation: Bull flattener: long 10yr + short 2yr (DV01-neutral). Bull = rates fall; flattener = long rates fall more. The 10yr (higher duration) gains more than the 2yr (lower duration) loses. Net P&L positive. A is a steepener trade; C is a parallel shift (DV01-neutral → ≈zero P&L); D is a credit trade.
Question 254
The Black-76 model is used for pricing:
Select an option first.
Correct answer: B — Interest rate caps, floors, and swaptions — treating the forward rate or swap rate as the underlying.
Explanation: Black-76 (Black 1976): a variant of BSM where the forward price (not spot) is the underlying, allowing pricing of: (1) options on futures, (2) caplets/floorlets (forward LIBOR/SOFR as underlying), (3) swaptions (forward swap rate as underlying). A uses standard BSM; C uses Garman-Kohlhagen model; D requires American option methodology (binomial, Longstaff-Schwartz).
Question 255
Hedging vega of an options portfolio without changing delta or gamma:
Select an option first.
Correct answer: B — Buy/sell options for vega target, then delta-adjust with underlying, then gamma-adjust with a second option type, then re-adjust delta.
Explanation: Sequential Greek hedging: (1) trade options to neutralise vega → this changes delta and gamma; (2) restore delta neutrality with underlying (zero gamma/vega); (3) restore gamma neutrality with another option → this changes delta again; (4) re-adjust delta with underlying. Each Greek requires its own dedicated instrument. A, C, D cannot adjust vega.
Question 256
FRTB P&L attribution test compares:
Select an option first.
Correct answer: B — Risk-theoretical P&L (from model risk factors) versus actual P&L (full revaluation) — large differences indicate missing risk factors.
Explanation: P&L attribution test: if the risk model captures all relevant factors, the risk-theoretical P&L (from sensitivity-based repricing) should closely match the hypothetical P&L (full model revaluation using all factors, excluding fees). Persistent large differences → model missing key risk factors → desk loses IMA approval → must use standardised approach. This complements back-testing in FRTB model validation.
Question 257
A covered call position (long bond + short call on same bond) affects duration and convexity by:
Select an option first.
Correct answer: B — Reducing duration (negative delta of short call) and reducing convexity (negative gamma of short call).
Explanation: Short call: (1) negative delta → reduces portfolio effective duration; (2) negative gamma/convexity → portfolio convexity is reduced vs bond alone. The capped upside (when rates fall, bond price is called at par) is the economic manifestation of the negative convexity contributed by the short call. The covered call behaves like a shorter-duration, lower-convexity bond.
Question 258
MBS negative convexity increases as rates fall because:
Select an option first.
Correct answer: B — Prepayment incentive increases as rates fall below mortgage note rates — more borrowers refinance, shortening duration and capping price appreciation.
Explanation: MBS prepayment S-curve: as rates fall below the mortgage note rate, refinancing incentive grows nonlinearly. At very low rates, maximum prepayment speed is reached — MBS duration collapses to near zero, and price barely rises as yields fall (maximum negative convexity). The price-yield curve becomes concave. A, C, D are all wrong.
Question 259
After hedging PC1 (level) and PC2 (slope), remaining yield curve risk is primarily:
Select an option first.
Correct answer: B — PC3 (curvature/butterfly) risk — hedges for level and slope do not eliminate curvature.
Explanation: PCA hierarchy: PC1 (level, ~75%) + PC2 (slope, ~15%) + PC3 (curvature, ~5%) ≈ 95% of yield curve variance. After hedging PC1 (parallel shift via futures) and PC2 (slope via 2yr/10yr spread), the remaining ~5% curvature risk requires a butterfly hedge (long belly, short wings or vice versa). A is too optimistic; C and D are different risk types.
Question 260
As underlying price rises far above the strike for a call option, delta approaches:
Select an option first.
Correct answer: B — 1.0 — the call behaves like owning the underlying.
Explanation: Deep ITM call: d₁ → ∞, N(d₁) → 1, Δ → 1. The option is almost certain to be exercised at expiry — it moves dollar-for-dollar with the underlying. Delta = 1 means the call is a perfect synthetic long position in the stock. A applies to deep OTM; C applies only ATM; D has no basis in options theory.
Question 261
A receiver swaption gives the holder the right to:
Select an option first.
Correct answer: B — Receive fixed and pay floating — valuable when rates fall (locked-in fixed exceeds market rate).
Explanation: Receiver swaption: right to receive fixed and pay floating. Profitable when prevailing swap rates fall below the swaption strike (you receive an above-market fixed rate). Used by fixed-income investors or liability managers expecting rates to fall. Payer swaption (pay fixed, receive floating) profits when rates rise. A describes a pay-fixed swap; C and D are different instruments.
Question 262
The equity volatility skew (higher implied vol for OTM puts than OTM calls) is primarily caused by:
Select an option first.
Correct answer: B — Negatively skewed equity return distributions with fat left tails — crashes are more likely than log-normal models predict.
Explanation: Equity skew: return distributions are negatively skewed (crash risk) with fat left tails. OTM puts are in high structural demand (portfolio insurance). To match observed option prices, OTM puts require higher implied vol than BSM's log-normal assumption predicts. The risk-neutral distribution implied by options has heavier left tails than log-normal. Demand (A, D) is secondary to the distributional explanation. C is wrong.
Question 263
A long straddle's maximum loss equals:
Select an option first.
Correct answer: B — The total premium paid for both the call and put.
Explanation: Long straddle = long ATM call + long ATM put. Maximum loss occurs when underlying expires exactly at the strike — both options expire worthless, losing the entire combined premium. Profit when: S_T > K + Premium (call ITM) or S_T < K − Premium (put ITM). A, C, D are wrong. The maximum loss is bounded by the upfront premium paid.
Question 264
The BSM PDE satisfied by all derivative prices under BSM assumptions is:
Select an option first.
Correct answer: A — Theta + rS×Delta + ½σ²S²×Gamma = rV
Explanation: BSM PDE: ∂V/∂t + rS(∂V/∂S) + ½σ²S²(∂²V/∂S²) = rV. In Greek notation: Theta + rS×Delta + ½σ²S²×Gamma = rV. This must be satisfied by all derivatives priced under BSM assumptions. For a delta-hedged portfolio (Δ=0): Theta + ½σ²S²×Gamma = 0, explaining why long gamma positions offset theta cost. B, C, D have no mathematical basis.
Question 265
Realised vol exceeding implied vol benefits a long gamma position because:
Select an option first.
Correct answer: B — Gamma P&L (½Γ(ΔS)²) from daily moves exceeds theta decay (priced at implied vol) when realised vol > implied vol.
Explanation: Breakeven condition: ½Γ(ΔS)² = |Θ|×Δt. ΔS/S ≈ σ_realised per unit time; |Θ| is priced at σ_implied. If σ_realised > σ_implied: gamma P&L > theta cost → net positive. If σ_realised < σ_implied: net negative (theta bleeds). Long gamma = long realised vol, short implied vol. This is the fundamental economics of delta-hedged options books.
Question 266
A fixed income carry strategy involves:
Select an option first.
Correct answer: B — Borrowing at short-term rates and investing in higher-yielding long-dated bonds — profiting from the yield differential as long as the curve doesn't flatten enough to offset carry income.
Explanation: Carry trade: finance long-duration bonds in the repo market at short-term rates, earning the yield spread (carry). Risks: (1) curve flattening reduces the yield differential; (2) rising short rates increase financing costs; (3) MTM losses if yields rise. The carry is the day-one income; roll-down return is the additional P&L from the bond moving along the curve.
Question 267
Rolldown return in fixed income is distinct from yield income because:
Select an option first.
Correct answer: B — Rolldown captures price appreciation from a bond moving to shorter maturity along a positively sloped curve (lower yield → higher price) — independent of any overall yield change.
Explanation: Rolldown: on an upward-sloping curve, as time passes, a 5-year bond becomes a 4-year bond. If the curve is unchanged, the 4-year yield is lower than the 5-year → bond's yield falls → price rises (rolldown return). This is separate from coupon income (yield income). Rolldown is always positive on an upward-sloping curve. A, C, D are wrong.
Question 268
Gap risk in exotic options arises because:
Select an option first.
Correct answer: B — Sudden discontinuous price jumps (overnight gaps) cannot be hedged in advance — the hedge cannot be adjusted during the gap.
Explanation: Gap risk: delta-gamma hedging assumes continuous price movement. An overnight earnings surprise or geopolitical shock creates a price gap — no opportunity to rebalance between pre-gap and post-gap prices. Higher-order Greeks (speed, colour) create additional P&L from the gap that is not managed by standard frameworks. Barrier options near their barrier level are particularly exposed to gap risk. A, C, D are wrong or incomplete.
Question 269
Key rate duration hedging is superior to modified duration hedging for:
Select an option first.
Correct answer: B — Non-parallel shifts (slope changes, butterfly movements) where different maturities move by different amounts.
Explanation: Modified duration handles only parallel shifts (PC1). Key rate durations, one per maturity bucket, capture sensitivity to non-parallel moves (PC2: slope; PC3: curvature). Non-parallel shifts collectively explain ~20-30% of yield curve variance — significant for portfolios with concentrated maturity exposures. A is where modified duration works; C requires convexity; D needs credit DV01.
Question 270
A collar strategy (long stock + long OTM put + short OTM call) creates:
Select an option first.
Correct answer: B — Capped upside (call strike limits gains) and floored downside (put strike limits losses) at near-zero net premium.
Explanation: Collar: (1) long stock provides equity exposure; (2) long OTM put floors the downside (can sell at put strike if stock falls); (3) short OTM call caps the upside (if stock rises above call strike, gains are surrendered). The collar typically costs near zero net premium (put cost offset by call premium received). Classic use: an executive hedging concentrated stock position. A has wrong downside; C has wrong upside/downside; D is the unhedged stock.
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