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Free FRM Financial Markets and Products Practice Questions & Answers
400 exam-style Financial Markets and Products questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 361
Under the Black-Scholes model, the price of a European put option can be derived from the call price using:
Select an option first.
Correct answer: B — Put-call parity: P = C + K × e^{−rT} − S₀
Explanation: B is correct: put-call parity: C + K × e^{−rT} = P + S₀, rearranging to P = C + K × e^{−rT} − S₀. This is the standard form. A presents the same equation rearranged as P = C − S₀ + K × e^{−rT} which is identical to B. Both A and B are mathematically equivalent. B is the conventional textbook form. C is wrong — call and put values don't sum to 1. D is wrong — not the correct formula.
Question 362
The 'credit triangle' relationship in credit markets states that:
Select an option first.
Correct answer: B — Credit spread ≈ PD × LGD — the yield spread approximately equals the probability of default multiplied by the loss given default, providing a simple link between credit spreads and default probability
Explanation: B is correct: the credit triangle approximation: credit spread ≈ PD × LGD (assuming small time intervals and continuous time). For example, a credit spread of 200 bps with LGD of 40% implies PD ≈ 200 bps / 40% = 500 bps = 5% per annum. B is correct. A is wrong — partial equation. C is wrong — risk-free rate is not in the triangle. D is wrong — that is a different relationship.
Question 363
The 'immunisation' strategy using bonds fails when:
Select an option first.
Correct answer: B — There are non-parallel yield curve shifts (twists, humps) — duration matching only protects against parallel shifts; convexity matching helps but non-parallel shifts can still create mismatches
Explanation: B is correct: classic duration immunisation only protects against parallel shifts in the yield curve (all maturities change by the same amount). In reality, the yield curve can twist (short rates rise while long rates fall), steepen, flatten, or change shape in complex ways. These non-parallel movements cause mismatches even in a duration-matched portfolio. B is correct. A is wrong — small parallel shifts are exactly what immunisation handles. C is correct by definition of immunisation. D is wrong — coupon rates affect how to achieve the target duration, not whether immunisation works.
Question 364
A 'stripped' bond (Treasury STRIP) is created by:
Select an option first.
Correct answer: B — Separating a coupon-bearing Treasury bond into individual zero-coupon components — each coupon payment and the principal become separate tradeable zero-coupon bonds
Explanation: B is correct: Treasury STRIPS (Separate Trading of Registered Interest and Principal of Securities) are created by stripping a Treasury bond into its component cash flows — each individual coupon payment and the final principal payment become separate zero-coupon bonds traded independently. STRIPS are used to construct the Treasury spot (zero) curve. A is wrong — both coupons and principal are stripped. C is wrong — STRIPs are a Treasury program. D is wrong — indentures are for corporate bonds.
Question 365
The price of a bond with multiple coupons is calculated as the:
Select an option first.
Correct answer: B — Present value of all future cash flows (coupons + principal), discounted at the yield to maturity
Explanation: B is correct: bond price = Σ[C/(1+y)^t] + F/(1+y)^n, where C is the coupon, y is the yield, F is the face value, and n is the number of periods. This is the fundamental bond pricing formula — the present value of a stream of fixed cash flows. A is wrong — undiscounted sum ignores time value. C is wrong — product is not used. D is wrong — not the correct formula.
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Question 366
The 'cheapest to deliver' (CTD) option in Treasury futures creates a delivery option embedded in the short position. This option's value:
Select an option first.
Correct answer: A — Reduces the futures price relative to the theoretical no-CTD price
Explanation: A is correct: since the short has the option to deliver the cheapest bond, this option has positive value for the short. Because the short benefits from this option, they are willing to accept a lower futures price. The futures price is therefore below the theoretical forward price of the cheapest bond — the delivery option reduces the futures price. A is correct. B is wrong — the lower futures price is the mechanism, not a higher price. C is wrong — it clearly affects pricing. D is wrong — conversion factors standardise delivery but are not equal to option value.
Question 367
A 'spread option' pays based on:
Select an option first.
Correct answer: B — The difference between two rates or prices (e.g. the spread between 10-year and 2-year Treasury yields), providing exposure to the relationship between the two underlyings
Explanation: B is correct: a spread option's payoff depends on the spread between two asset prices or rates: max(0, S₁ − S₂ − K). Common examples include: yield spread options (10Y-2Y), commodity spread options (crack spread — refinery margin), and credit spread options. B is correct. A is wrong — that is a vanilla option. C is wrong — that would be a product option. D is wrong — that is a basket or average option.
Question 368
Under the SABR (Stochastic Alpha Beta Rho) model for interest rate volatility:
Select an option first.
Correct answer: B — Both the forward rate and its volatility are stochastic — the model captures the volatility smile and is widely used for calibrating cap/floor and swaption volatilities
Explanation: B is correct: the SABR model (Hagan et al.) assumes both the forward rate F and its instantaneous volatility α are stochastic processes, with a correlation parameter ρ between the two. The β parameter controls the backbone shape (β=0 is normal SABR; β=1 is lognormal SABR). SABR is standard for calibrating the implied volatility surface for interest rate options. A is wrong — constant vol is BSM. C is wrong — SABR has stochastic volatility. D is wrong — ρ is the key correlation parameter.
Question 369
An 'excess spread' in a securitisation structure represents:
Select an option first.
Correct answer: B — The difference between the yield on the collateral pool and the total coupon paid to all ABS tranches — a first line of defence against losses
Explanation: B is correct: excess spread = gross yield on collateral − (senior and mezzanine coupon payments + servicing fees). It is the recurring income buffer that can absorb small losses before principal is impaired. Excess spread is typically captured in a reserve account to cover future losses. A is wrong — subordination is structural credit enhancement, separate from excess spread. C is wrong — that is the credit spread of the senior tranche. D is wrong — over-collateralisation is a separate structural feature.
Question 370
The 'effective annual rate' (EAR) and 'annual percentage rate' (APR) differ in that:
Select an option first.
Correct answer: B — EAR compounds interest to show the true annual cost/return; APR is a simple (non-compounding) rate that may understate the true cost when compounding occurs more than once per year
Explanation: B is correct: APR is a simple annualised rate (= periodic rate × number of periods), which understates the true cost when compounding occurs. EAR (or effective annual rate) accounts for intra-year compounding: EAR = (1 + APR/m)^m − 1 where m is the number of compounding periods. For continuous compounding: EAR = e^r − 1. A is wrong — they differ when compounding occurs. C is wrong — EAR ≥ APR with equality at annual compounding. D is wrong — both apply to savings and loans.
Question 371
In the binomial pricing model for American options, a 'risk-neutral' world assumption means:
Select an option first.
Correct answer: B — Assets earn the risk-free rate on average in the binomial tree — this allows the no-arbitrage option price to be derived without knowing real-world probabilities or investor risk preferences
Explanation: B is correct: in a risk-neutral world, all assets earn the risk-free rate. This is a mathematical convenience — the real world may have risk-averse investors, but by the no-arbitrage argument, options must be priced the same in the risk-neutral world as in the real world. Risk-neutral probabilities are chosen to make the expected return equal to the risk-free rate, allowing discounting at r. A is wrong — risk-neutral does not mean investors don't require compensation; it is a pricing technique. C is wrong — risk-neutral probabilities are not real-world probabilities. D is wrong — options have non-zero premiums.
Question 372
The term 'prepayment speed' in MBS markets is measured by:
Select an option first.
Correct answer: B — CPR (Conditional Prepayment Rate) — the annualised percentage of the outstanding mortgage balance prepaid per year, or equivalently by the monthly SMM (Single Monthly Mortality)
Explanation: B is correct: CPR is the standard measure of prepayment speed in MBS analysis. CPR = 1 − (1 − SMM)^12, where SMM is the single-month mortality. A CPR of 10% means 10% of the outstanding pool balance is prepaid over one year. The PSA model expresses prepayment as a percentage of the benchmark CPR schedule. A is wrong — CPR is a rate, not dollar amount. C is wrong — CPR measures prepayment (full payoff), not default. D is wrong — WALA measures loan seasoning, not prepayment speed.
Question 373
An inverse floater bond pays a coupon of:
Select an option first.
Correct answer: B — A fixed rate minus a floating rate (e.g. 15% − SOFR) — the coupon increases when rates fall and decreases when rates rise, creating a bond with high effective duration
Explanation: B is correct: an inverse floater's coupon = cap rate − floating rate. For example, 15% − SOFR. When rates fall, the coupon rises; when rates rise, the coupon falls or hits a floor of zero. Because the coupon is negatively correlated with rates, inverse floaters have very high effective duration and are very sensitive to interest rate changes. They are often created by splitting a fixed-rate bond into a standard floater and an inverse floater. A is wrong — that is a standard floater. C is wrong — that is a fixed-rate bond. D is wrong — IO strips are different.
Question 374
The 'no-arbitrage' principle in derivatives pricing states that:
Select an option first.
Correct answer: B — Two portfolios with identical future cash flows in all states of the world must have the same price today — any price discrepancy would allow riskless profit
Explanation: B is correct: the law of one price (no-arbitrage) states that two portfolios with identical payoffs must have identical current prices. If they differed, an investor could buy the cheaper portfolio and sell the more expensive one, earning a riskless profit. This principle underlies all derivatives pricing — call option prices, forward prices, and swap rates are derived using replicating portfolios. A is wrong — prices can be zero or negative for some derivatives. C is wrong — no-arbitrage is about relative pricing, not absolute fundamental value. D is wrong — empirically, short-term deviations occur but are quickly arbitraged away.
Question 375
When calibrating an interest rate model to market data, which instruments are typically used?
Select an option first.
Correct answer: B — Market prices of liquid instruments — government bonds (to calibrate the yield curve), caps/floors (to calibrate at-the-money vol), and swaptions (to calibrate swaption vol surface) — ensuring the model prices these instruments correctly
Explanation: B is correct: a well-calibrated interest rate model must simultaneously: (1) fit the current yield curve (using zero-coupon bond prices/yields); (2) fit at-the-money cap/floor volatilities; and (3) fit swaption volatility surfaces. Different models excel at different aspects — HJM fits the entire yield curve; LIBOR Market Model (BGM) fits cap/floor volatilities; SABR adds smile. B captures the comprehensive calibration approach. A is wrong — historical data is for risk management, not pricing calibration. C and D are partial.
Question 376
The 'credit linkage' in a first-to-default basket CDS means:
Select an option first.
Correct answer: B — The basket CDS pays out upon the first credit event among all reference entities — the payout terminates the contract, leaving subsequent defaults unprotected
Explanation: B is correct: in a first-to-default (FTD) basket, the protection buyer pays a premium and receives protection against the first credit event among a basket of (say) 5-10 names. Once the first name defaults and the contract pays out, the CDS terminates — no further protection. This makes FTD baskets heavily exposed to correlation: high correlation reduces the FTD value (all names likely to default together); low correlation increases value (many independent chances for one to default). B is correct. A is wrong — not all names are simultaneously protected. C is wrong — correlation is a key input. D is wrong — that would be an nth-to-default basket.
Question 377
The 'term structure of volatility' (volatility surface) for options shows:
Select an option first.
Correct answer: B — How implied volatility varies across both strike prices (volatility smile/skew) and time to expiry (volatility term structure) — forming a 2-dimensional surface
Explanation: B is correct: the volatility surface plots implied volatility (derived from market option prices using BSM) as a function of two dimensions: (1) strike price or moneyness (horizontal, showing the smile/skew); and (2) time to expiry (depth, showing the term structure). The surface is used for interpolation and calibration of more sophisticated models. A is wrong — BSM assumes constant vol; the surface reveals this is violated. C is wrong — term structure of vol can be upward or downward sloping. D is wrong — both strike and maturity dimensions are needed.
Question 378
A 'condor' options strategy involves:
Select an option first.
Correct answer: B — Buying a call at a very low strike, selling a call at a moderately low strike, selling a call at a moderately high strike, and buying a call at a very high strike — capping both maximum gain and loss
Explanation: B is correct: a long condor uses four options at four different strikes: buy K1, sell K2, sell K3, buy K4 (K1 < K2 < K3 < K4). Maximum profit occurs if the underlying stays between K2 and K3 at expiry. Maximum loss is the net premium paid. It is a lower-cost alternative to a butterfly spread, with a wider profit zone. A is wrong — that is a bull call spread. C is wrong — that is a short straddle combined with a long strangle. D is wrong — two calls is a spread or straddle.
Question 379
The 'lognormal distribution' assumption for stock prices in BSM implies that:
Select an option first.
Correct answer: B — Returns (log price changes) are normally distributed — this means stock prices can never go negative (lognormal prices) but allows for large positive moves and heavy negative tail
Explanation: B is correct: geometric Brownian motion (GBM) assumes continuously compounded returns (log returns) are normally distributed. Since stock prices = exp(log returns), prices follow a lognormal distribution — they are always positive and right-skewed (potential for large gains). The downside tail is limited by zero. A is wrong — prices are lognormal, not normal. C is wrong — in the real world, stocks earn more than the risk-free rate; in risk-neutral pricing, they earn the risk-free rate on average. D is wrong — BSM assumes constant volatility.
Question 380
The 'duration' of a floating-rate bond resets to approximately zero at each coupon reset date because:
Select an option first.
Correct answer: B — At each reset, the coupon is aligned with the current market rate — the bond's next coupon exactly equals the market rate, so the present value of remaining cash flows always equals par at reset dates, giving minimal rate sensitivity
Explanation: B is correct: immediately after each reset date, the coupon rate equals the current market rate, so the bond prices at par. The effective duration is approximately the time to the next reset (e.g. 0.5 years for semi-annual resets) rather than the bond's full maturity. Between reset dates, the duration is the fraction of time to the next reset. A is wrong — FRNs do have coupon payments. C is wrong — notional stays fixed. D is wrong — price moves between resets as rates change.
Question 381
The 'credit spread duration' of a bond portfolio measures:
Select an option first.
Correct answer: B — The sensitivity of the portfolio's value to a parallel shift in credit spreads — allowing portfolio managers to manage credit spread risk
Explanation: B is correct: credit spread duration (or spread duration) measures how much the portfolio's value changes for a 1 bps change in credit spreads across all bonds. It is distinct from interest rate duration (sensitivity to risk-free rates). Portfolio managers use credit spread duration to manage their exposure to the credit cycle independently of interest rate risk. A is wrong — that is interest rate duration/DV01. C is wrong — average maturity is different from duration. D is wrong — that is equity beta.
Question 382
The 'LIBOR Market Model' (LMM or BGM model) is widely used for pricing interest rate derivatives because:
Select an option first.
Correct answer: B — It models forward LIBOR (or SOFR) rates directly — making it consistent with market conventions for caps and swaptions and allowing calibration to observed volatility surfaces
Explanation: B is correct: the Brace-Gatarek-Musiela (BGM) or LIBOR Market Model models forward rates (not the short rate) that correspond directly to market instruments like LIBOR-based FRAs and caps. This makes calibration to market volatilities natural and intuitive. Under LMM, the volatility structure can be specified to match cap and swaption market prices simultaneously. A is wrong — LMM is a multi-factor model. C is wrong — LMM is a full term structure model. D is wrong — multiple models price bond options.
Question 383
The payoff of a 'butterfly spread' using calls (long K1, short 2×K2, long K3 with equal spacing) is:
Select an option first.
Correct answer: B — A tent-shaped payoff — maximum profit when the underlying equals K2 at expiry, losses limited to net premium paid when underlying is below K1 or above K3
Explanation: B is correct: the butterfly spread has a 'tent' or inverted V-shaped payoff. Maximum profit at K2 (middle strike): (K2 − K1) − net premium. As the underlying moves away from K2 (in either direction), profit decreases, eventually reaching the maximum loss (net premium) when below K1 or above K3. B is correct. A is wrong — a straddle has a V-shaped payoff. C is wrong — above K3, the position is flat at maximum loss. D is wrong — payoff depends on underlying price.
Question 384
Prepayment risk for IO (interest-only) strips and PO (principal-only) strips behave symmetrically in that:
Select an option first.
Correct answer: B — They have opposite sensitivities — IO strips benefit from slow prepayment (more interest payments); PO strips benefit from fast prepayment (principal returned sooner)
Explanation: B is correct: IO and PO strips are created by separating the interest and principal cash flows of an MBS. IO value depends on how long the principal remains outstanding (more principal = more interest), so it benefits from slow prepayment. PO value depends on the timing of principal receipt (purchased at a discount), so it benefits from fast prepayment (face value received sooner). They are natural complements. A is wrong — IO is hurt by fast prepayment. C is wrong — rising rates reduce prepayment (good for IO, bad for PO). D is wrong — they are highly sensitive to prepayment.
Question 385
A 'strangle' option strategy involves:
Select an option first.
Correct answer: B — Buying a call with a higher strike and a put with a lower strike — both OTM at inception. Profits from large moves in either direction but requires a bigger move than a straddle to profit due to lower premium cost
Explanation: B is correct: a long strangle = buy OTM call (strike K_C > S) + buy OTM put (strike K_P < S). The premium is lower than a straddle (both options are OTM), but the underlying needs to move further to generate a profit. The strategy profits from high volatility. A is wrong — that is a straddle. C is wrong — selling both is a short strangle. D is wrong — that is not a standard strangle.
Question 386
The 'Merton model' for corporate debt and equity uses the insight that:
Select an option first.
Correct answer: B — Equity is a call option on the firm's assets with a strike equal to the face value of debt — equity holders have limited liability and benefit from asset appreciation above the debt level
Explanation: B is correct: Merton (1974) modelled a firm's equity as a European call option on the firm's total assets: equity holders have the right to 'buy back' the firm from debt holders by paying face value of debt at maturity. If assets < debt, equity is worthless (option expires OTM). This framework allows corporate bond pricing using option pricing theory. A is wrong — equity and debt are linked through the firm's total asset value. C is wrong — default risk affects debt value. D is wrong — Merton's original model assumed no bankruptcy costs.
Question 387
The 'credit delta' (CS01) measures:
Select an option first.
Correct answer: B — The change in the value of a credit instrument (bond or CDS) for a 1 basis point change in credit spread
Explanation: B is correct: CS01 (credit spread 01) or credit delta = change in value for 1 bps move in credit spreads. For a long bond position, a 1 bps widening of credit spread reduces value by approximately duration × bond price × 0.0001. CS01 is used for managing credit spread risk in bond and CDS portfolios. A is wrong — that is equity options delta. C is wrong — default probability is PD. D is wrong — that is interest rate DV01.
Question 388
Under the 'real-world' (P-measure) versus 'risk-neutral' (Q-measure) probability framework:
Select an option first.
Correct answer: B — P-measure uses actual (historical/physical) probabilities and investors' required risk premiums; Q-measure (risk-neutral) is the probability measure under which all assets earn the risk-free rate, used for pricing derivatives without knowing risk preferences
Explanation: B is correct: the real-world measure (P) reflects actual probabilities of market outcomes and investors' risk premiums. The risk-neutral measure (Q) is a mathematical construct where all assets are assumed to earn the risk-free rate — it is used for derivative pricing because it makes discounting at the risk-free rate consistent with no-arbitrage. Under Q, the expected stock return is r (not the actual return μ). B is correct. A is wrong — they give different probabilities but the same no-arbitrage prices. C and D are wrong.
Question 389
The 'duration' of an amortising bond is:
Select an option first.
Correct answer: B — Shorter than a bullet bond of the same stated maturity — because principal is returned throughout the bond's life (not just at maturity), reducing the average time of cash flows
Explanation: B is correct: an amortising bond returns principal gradually over its life. Since principal payments occur before final maturity, the weighted average time of all cash flows (duration) is shorter than for a bullet bond of the same stated maturity. This is why mortgages and ABS with amortising structures have shorter durations than their stated maturities suggest. A is wrong — amortisation shortens duration. C is wrong — shorter duration due to earlier principal receipts. D is wrong — duration is well-defined.
Question 390
A 'convertible bond' gives the holder the right to:
Select an option first.
Correct answer: B — Convert the bond into a specified number of the issuer's common shares — combining features of a straight bond and a call option on the underlying equity
Explanation: B is correct: a convertible bond contains an embedded call option on the issuer's equity. The bondholder receives fixed coupon payments (like a straight bond) but can convert to shares at the conversion ratio if the stock price rises sufficiently. Convertibles have a floor value (straight bond value) and equity participation. The issuer benefits from lower coupon costs. A is wrong — currency choice is a separate feature. C is wrong — that is a puttable feature. D is wrong — that is a callable feature (issuer's option).
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