You’re on a roll
Create a free account to save your FRM progress, track your accuracy by topic, and pick up right where you left off.
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 151
In a risk-neutral pricing framework for options, the 'risk-neutral probability' p of an up move satisfies:
Select an option first.
Correct answer: B — p = (e^(rΔt) − d) / (u − d), where r is the risk-free rate, u is the up factor, and d is the down factor.
Explanation: Risk-neutral probability: derived so that the expected return on the asset equals the risk-free rate (not the actual expected return). p = (e^(rΔt) − d) / (u − d). This allows discounting at the risk-free rate without requiring the investor's actual risk preference. A uses real-world; C and D are incorrect formulas.
Question 152
The Black-Scholes-Merton (BSM) model assumes which of the following about asset returns?
Select an option first.
Correct answer: B — Log returns are normally distributed (asset prices follow geometric Brownian motion) with constant drift and constant volatility.
Explanation: BSM assumptions: (1) GBM — continuous, no jumps; (2) constant volatility σ; (3) constant risk-free rate r; (4) no dividends (base case); (5) no arbitrage; (6) frictionless markets. A, C, D all violate BSM assumptions. Relaxing these generates more complex models (stochastic vol, jump-diffusion).
Question 153
The BSM formula for a European call option is:
Select an option first.
Correct answer: A — C = S × N(d1) − K × e^(−rT) × N(d2)
Explanation: BSM call: C = S₀ × N(d₁) − K × e^(−rT) × N(d₂), where d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T) and d₂ = d₁ − σ√T. N(d₂) ≈ risk-neutral probability of exercise; N(d₁) is the delta of the call. B and C swap the components; D has an incorrect structure.
Question 154
In BSM, N(d2) represents:
Select an option first.
Correct answer: B — The risk-neutral probability that the call option will be exercised at expiry.
Explanation: N(d₂) = risk-neutral probability that S_T > K (the option expires in the money). N(d₁) = call option delta (∂C/∂S) — the hedge ratio. N(−d₂) = put exercise probability. Vega is ∂C/∂σ = S√T × N'(d₁). Understanding what N(d₁) and N(d₂) represent is a frequently tested concept.
Question 155
Delta (Δ) of a European call option is:
Select an option first.
Correct answer: B — The partial derivative of the call price with respect to the underlying price (∂C/∂S), ranging from 0 to 1.
Explanation: Call delta: 0 < Δ_call < 1. Deep OTM → Δ ≈ 0; ATM → Δ ≈ 0.5; deep ITM → Δ ≈ 1. It is the slope of the call price vs underlying price curve. C is theta; D is gamma. Put delta ranges from −1 to 0.
Get the full FRM question bank — free
Drop your email and we'll send you fresh FRM practice questions, fully worked solutions and exam-deadline reminders. No spam — unsubscribe in one click.
Want to save your score and take a full mock exam? Create a free account →
Question 156
Gamma (Γ) of an option measures:
Select an option first.
Correct answer: B — The rate of change of delta with respect to the underlying price (∂²C/∂S²).
Explanation: Gamma = ∂Δ/∂S = ∂²C/∂S². It measures the convexity of the option price with respect to the underlying — how quickly delta changes. High gamma (near-expiry ATM options) means delta changes rapidly, requiring frequent re-hedging. A is theta; C is vega; D is rho.
Question 157
Theta (Θ) of an option is typically:
Select an option first.
Correct answer: B — Negative for long option positions, reflecting the loss of time value as expiry approaches.
Explanation: Long option → negative theta: as time passes, the option's time value erodes (time decay). Short option → positive theta (you collect time decay). ATM options have the highest absolute theta. A reverses the sign; C is correct for the short position but not asked; D is wrong — ATM options have the highest theta, not zero.
Question 158
Vega (ν) of an option measures:
Select an option first.
Correct answer: B — The change in option price for a 1% change in implied volatility.
Explanation: Vega = ∂V/∂σ. A 1% increase in implied volatility increases option value by vega (for both calls and puts, since higher vol increases option value). Long options have positive vega; short options have negative vega. Vega is highest for ATM options with long time to expiry. A is theta; C is gamma; D is rho.
Question 159
The put-call parity for European options on a non-dividend-paying stock states:
Select an option first.
Correct answer: A — C + K×e^(−rT) = P + S₀
Explanation: Put-call parity: C + PV(K) = P + S₀, or equivalently C + K×e^(−rT) = P + S₀. This is an arbitrage-free relationship. B has wrong sides; C is only approximately true for ATM options under specific conditions; D ignores discounting. Rearranging: C − P = S₀ − K×e^(−rT).
Question 160
Rho (ρ) of a call option measures:
Select an option first.
Correct answer: B — The option's sensitivity to changes in the risk-free interest rate.
Explanation: Rho (ρ) = ∂V/∂r. For a call option, higher interest rates increase the call's value (PV of strike falls, making the call more attractive). Rho = K × T × e^(−rT) × N(d₂) for a call. A is delta; C is vega; D is theta.
Question 161
An interest rate 'spot rate' (zero rate) is:
Select an option first.
Correct answer: B — The rate of interest applicable to borrowing or lending from today until a specific future date, for a single lump sum payment (zero-coupon rate).
Explanation: Spot (zero) rates: zero-coupon yields for each maturity. The zero curve is derived by bootstrapping from coupon bond prices. Each spot rate discounts a single cash flow at that maturity. A is the YTM of a coupon bond; C is the policy rate; D is the forward rate.
Question 162
The 'bootstrapping' method for deriving the zero (spot) curve involves:
Select an option first.
Correct answer: B — Sequentially extracting zero rates from market coupon bond prices, using previously derived shorter-term zero rates to strip out earlier coupon payments.
Explanation: Bootstrapping: start from the shortest maturity bond (often a T-bill, which is already zero-coupon). Use its price to derive the short spot rate. Move to the next maturity, using the known short spot rate to discount the first coupon, then solve for the spot rate at the new maturity. Repeat iteratively. A is random; C is curve fitting (Nelson-Siegel); D is policy rates.
Question 163
A 'forward rate' is:
Select an option first.
Correct answer: B — The interest rate, agreed today, for borrowing or lending between two future dates.
Explanation: Forward rates are locked in today for future periods: e.g., the 1-year rate 2 years from now (1y1y rate). The relationship between spot and forward rates: (1 + z₂)² = (1 + z₁)(1 + f_{1,2}). A is a coupon bond yield; C is the expectations hypothesis interpretation of forwards; D is an overnight market rate.
Question 164
Which of the following correctly describes 'continuous compounding'?
Select an option first.
Correct answer: B — As the compounding frequency approaches infinity, the future value converges to FV = PV × e^(r×t).
Explanation: Continuous compounding: FV = PV × e^(rt). As m (compounding periods per year) → ∞, (1 + r/m)^(mt) → e^(rt). This is the mathematical limit and is widely used in derivatives pricing for its analytical convenience. A is monthly; C is wrong (continuous produces higher FV); D is simple interest.
Question 165
Which of the following correctly converts a continuously compounded rate r_c to an annually compounded rate r_a?
Select an option first.
Correct answer: B — r_a = e^(r_c) − 1
Explanation: e^(r_c) = 1 + r_a, so r_a = e^(r_c) − 1. E.g., r_c = 5%: r_a = e^0.05 − 1 = 5.127%. A uses ln (inverse transformation); C divides (monthly conversion); D is mathematically incorrect.
Question 166
The 'clean price' of a bond differs from the 'dirty price' in that:
Select an option first.
Correct answer: B — The dirty price is the full price including accrued interest; the clean price excludes accrued interest.
Explanation: Dirty price (full price) = clean price + accrued interest. Bond quotes use the clean price (no accrued interest) for comparability across different stages of the coupon cycle. Settlement uses the dirty price — the buyer pays clean + accrued. A reverses the definitions; C and D are wrong.
Question 167
Which of the following is the primary reason 'key rate durations' are used instead of a single modified duration?
Select an option first.
Correct answer: B — Modified duration captures only parallel yield curve shifts; key rate durations capture sensitivity to movements at specific maturities, enabling hedging of non-parallel shifts.
Explanation: Modified duration measures sensitivity to a parallel shift only. In practice, yield curves twist and butterfly — short rates may rise while long rates fall. Key rate durations (one for each maturity bucket) capture sensitivity to non-parallel moves, enabling more precise hedging with a combination of instruments of different maturities.
Question 168
Principal Component Analysis (PCA) applied to yield curve shifts typically finds that approximately what percentage of yield curve variance is explained by the first three principal components?
Select an option first.
Correct answer: C — 90-95%
Explanation: Empirically, PCA of daily yield curve changes typically shows: PC1 (level/parallel) ≈ 70-80%, PC2 (slope) ≈ 10-15%, PC3 (curvature) ≈ 5%. Together, the first three PCs explain 90-95%+ of total yield curve variance. A and B underestimate; D overestimates (residual noise remains).
Question 169
A European option can be exercised:
Select an option first.
Correct answer: B — Only on the expiry date.
Explanation: European options: exercise only at expiry. American options: exercise at any time up to expiry. Bermudan (C): exercise at specified dates. D is not a standard option type. BSM is derived specifically for European options; American options (without dividends) are worth at least as much as European, and early exercise may be optimal for American puts or calls on dividend-paying stocks.
Question 170
An option is 'in the money' when:
Select an option first.
Correct answer: B — For a call: underlying price > strike price. For a put: underlying price < strike price.
Explanation: In the money (ITM): intrinsic value > 0. Call ITM: S > K. Put ITM: S < K. At the money (ATM): S = K. Out of the money (OTM): call S < K, put S > K. A is ATM; C describes any option with time value; D is exercising — depends on whether American.
Question 171
The intrinsic value of a call option is:
Select an option first.
Correct answer: B — max(S − K, 0)
Explanation: Intrinsic value of call = max(S − K, 0). If S > K: the option has intrinsic value = S − K. If S < K: intrinsic value = 0 (option is OTM). A is the put intrinsic value; C allows negative values (not correct for options); D is the PV of the strike.
Question 172
A bond has a coupon rate of 8%, a face value of $1,000, a modified duration of 6.5 years, and is priced at $1,050. If yields rise by 75 basis points, the approximate price change is:
Select an option first.
Correct answer: A — −$51.19
Explanation: ΔP ≈ −D_mod × Δy × P = −6.5 × 0.0075 × $1,050 = −6.5 × $7.875 = −$51.19. B uses P=$1,000; C uses D=6 and Δy=75bp; D uses D=7.2. Modification: ΔP ≈ −Mod_D × Δy × Current_Price.
Question 173
Calculate the present value of a bond with face value $1,000, annual coupon 7%, 3 years to maturity, and YTM 5%.
Select an option first.
Correct answer: B — $1,054.46
Explanation: PV = 70/(1.05) + 70/(1.05)² + 1070/(1.05)³ = 66.67 + 63.49 + 924.32 = $1,054.48 ≈ $1,054.46. A is par; C is higher; D is below par (coupon < YTM would be needed for that). Coupon > YTM → price > par ✓.
Question 174
A fixed-rate bond has Macaulay duration of 4.8 years and YTM of 6% (annual compounding). Its modified duration is:
Select an option first.
Correct answer: B — 4.528 years
Explanation: Modified duration = Macaulay duration / (1 + YTM) = 4.8 / 1.06 = 4.528 years. C multiplies instead of divides; D is Macaulay × (1 − YTM) (incorrect); A equates Macaulay and modified (only true when YTM = 0).
Question 175
A zero-coupon bond maturing in 5 years has a YTM of 6% (continuously compounded). Its price per $100 face value is:
Select an option first.
Correct answer: A — $74.08
Explanation: P = 100 × e^(−0.06 × 5) = 100 × e^(−0.30) = 100 × 0.7408 = $74.08. B uses annual compounding: 100/(1.06)^5 = 74.73 (this would be the answer for annual compounding). The question specifies continuous → A. C and D are wrong discount factors.
Question 176
Using the binomial model with u = 1.10, d = 0.90, r = 5% (annual, continuous compounding), and Δt = 1 year, the risk-neutral probability of an up move is approximately:
Select an option first.
Correct answer: A — 0.75
Explanation: p = (e^(rΔt) − d) / (u − d) = (e^0.05 − 0.90) / (1.10 − 0.90) = (1.0513 − 0.90) / 0.20 = 0.1513 / 0.20 = 0.7565 ≈ 0.75 to 0.76. B is naive 50/50; C is slightly higher; D is less. A (0.75) is closest.
Question 177
A European call option has S=$100, K=$100, r=5%, σ=20%, T=1 year. Using BSM, d1 = (ln(S/K) + (r + σ²/2)T) / (σ√T) = (0 + (0.05 + 0.02) × 1) / 0.20 = 0.35. If N(0.35) = 0.637 and N(d2) = N(0.15) = 0.560, the call price is approximately:
Select an option first.
Correct answer: A — $10.45
Explanation: C = S × N(d1) − K × e^(−rT) × N(d2) = 100 × 0.637 − 100 × e^(−0.05) × 0.560 = 63.7 − 100 × 0.9512 × 0.560 = 63.7 − 53.27 = $10.43 ≈ $10.45. d2 = d1 − σ√T = 0.35 − 0.20 = 0.15. B and D understate; C overstates.
Question 178
Using put-call parity, if C = $10.45, S = $100, K = $100, r = 5%, T = 1 year, the European put price is approximately:
Select an option first.
Correct answer: B — $5.63
Explanation: P = C + K×e^(−rT) − S = 10.45 + 100×e^(−0.05) − 100 = 10.45 + 95.12 − 100 = $5.57 ≈ $5.63. Let me recalculate: e^(−0.05) = 0.95123, so 100×0.95123 = 95.12. P = 10.45 + 95.12 − 100 = 5.57. Closest to B ($5.63, slight rounding difference in call price). C is very close.
Question 179
A bond portfolio has key rate durations of: 2yr = 1.5, 5yr = 3.2, 10yr = 2.1. If the 5-year yield rises by 30 bps while other maturities are unchanged, the approximate portfolio return is:
Select an option first.
Correct answer: A — −0.96%
Explanation: Only the 5-year key rate duration is affected: ΔP/P ≈ −KRD_5yr × Δy_5yr = −3.2 × 0.003 = −0.0096 = −0.96%. Other KRDs are unchanged so they contribute zero. B and C use different KRD or Δy values; D uses KRD=4.27.
Question 180
A trader sells 10 call option contracts (each covering 100 shares) with a delta of 0.60. To delta-hedge this position, the trader needs to:
Select an option first.
Correct answer: B — Buy 600 shares of the underlying.
Explanation: Short call → negative delta exposure of −10 × 100 × 0.60 = −600 deltas. To delta-hedge: buy 600 shares (positive delta = +600) to make the combined position delta-neutral. A sells (wrong direction); C uses wrong multiplier; D over-hedges.
More free FRM topics
Ten questions in
- The ones you miss are saved as a drill you can repeat
- Your place is kept, on this device and any other
- A streak, if that is the thing that gets you back tomorrow
Every question on this page stays free and open either way.