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 › Market Risk Measurement and Management
Free FRM Market Risk Measurement and Management Practice Questions & Answers
400 exam-style Market Risk Measurement and Management questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 211
The 'volatility of volatility' (vol-of-vol) parameter in stochastic volatility models affects:
Select an option first.
Correct answer: B — The curvature (convexity) of the volatility smile — higher vol-of-vol produces more pronounced smile (U-shaped), while lower vol-of-vol produces flatter smile
Explanation: B is correct: in models like Heston, the vol-of-vol parameter ξ determines how much the variance fluctuates, which controls the shape of the implied volatility smile. Higher ξ → more curvature in the smile (larger smile). The correlation ρ between returns and volatility controls the skew/asymmetry. A is wrong — the level is controlled by the mean variance θ. C is wrong — skew is controlled by the correlation ρ. D is wrong — time decay is theta.
Question 212
The 'VIX futures' market allows traders to:
Select an option first.
Correct answer: B — Gain direct exposure to future implied volatility — allowing hedging of volatility risk without taking a view on market direction
Explanation: B is correct: VIX futures are contracts on the future value of the VIX. Buying VIX futures hedges against volatility spikes (VIX rises during market crashes). A is wrong — S&P 500 options determine VIX, but VIX futures are separate. C is wrong — selling VIX futures would harvest volatility risk premium. D is wrong — VIX is equity volatility, not rates.
Question 213
The 'volatility term structure' is inverted (downward-sloping) when:
Select an option first.
Correct answer: B — Short-term implied volatility exceeds long-term implied volatility — typically during market stress when near-term uncertainty is elevated relative to long-term expectations
Explanation: B is correct: inverted term structure occurs in high-volatility regimes (e.g. during a market crash). Short-dated IV spikes above long-dated IV, reflecting near-term uncertainty with expected mean reversion to lower volatility over longer horizons. A is wrong — inverted means short-term options are more expensive. C is wrong — expectation of rising vol would create normal (upward) structure. D is wrong — low VIX suggests normal (upward) term structure.
Question 214
The 'forward implied volatility' between two expiries T₁ and T₂ is derived from:
Select an option first.
Correct answer: B — The no-arbitrage relationship: σ²_fwd × (T₂ − T₁) = σ²_spot(T₂) × T₂ − σ²_spot(T₁) × T₁ — forward variance must equal the difference in total variance
Explanation: B is correct: forward implied variance is derived from the calendar arbitrage-free relationship between spot implied variances at different maturities. It represents the market-implied variance expected to prevail between dates T₁ and T₂. A is wrong — both spot IVs are needed together with the formula. C is wrong — GARCH is a statistical model. D is wrong — the forward price is separate from the forward vol.
Question 215
The 'correlation smile' in FX options is the plot of implied correlation between currency pairs against:
Select an option first.
Correct answer: B — The maturity of options — different maturities imply different expected correlations between currency pairs via consistency conditions
Explanation: B is correct: for a currency trio (USD/EUR/GBP), the implied correlation between, say, EUR/USD and GBP/USD is extracted from the triangle of implied volatilities. The correlation implied by options at different maturities forms a term structure ('correlation smile' or 'correlation surface'). A is wrong — FX smile is IV vs strike; FX correlation smile is vs maturity. C is wrong — spot rate is not the dimension. D is wrong — moneyness is for the smile not the correlation structure.
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 216
The 'GARCH option pricing' model extends the B-S framework by:
Select an option first.
Correct answer: B — Incorporating GARCH dynamics for the underlying's variance into option pricing — the conditional variance changes the option's 'effective' volatility and produces a volatility smile
Explanation: B is correct: Duan (1995) extended GARCH to option pricing by deriving the risk-neutral dynamics from the GARCH process. This produces state-dependent pricing where the option 'effective' volatility depends on current GARCH variance, generating implied volatility smiles consistent with observed market data. A is wrong — risk-neutral density is derived from GARCH dynamics. C is wrong — GARCH gives conditional (not historical average) volatility. D is wrong — GARCH has continuous-time limits.
Question 217
The 'regime-switching' volatility model allows:
Select an option first.
Correct answer: B — The volatility process to switch between distinct states (e.g. low-volatility and high-volatility regimes) with a Markov transition matrix governing state changes
Explanation: B is correct: regime-switching models (Hamilton 1989; applied to volatility by Hamilton and Susmel) assume the data-generating process switches between K states, each with different parameters (e.g. different σ). The switching is governed by a Markov chain with transition probabilities. A is wrong — regime-switching has time-varying volatility. C is wrong — the regime (latent state) also determines volatility. D is wrong — calendar effects are a different extension.
Question 218
The 'Bayes information criterion' (BIC) is used in GARCH model selection to:
Select an option first.
Correct answer: B — Penalise model complexity: BIC = −2ln(L) + k×ln(n), where k = number of parameters and n = sample size — selecting the model that balances fit and parsimony
Explanation: B is correct: BIC (and AIC) are information criteria for model selection. BIC penalises more heavily for extra parameters than AIC. The model with the lowest BIC is preferred. For GARCH family models, BIC typically selects low-order models like GARCH(1,1) over higher-order specifications. A is wrong — BIC penalises complexity. C is wrong — higher-order GARCH can have lower BIC. D is wrong — maximum likelihood without penalty is not BIC.
Question 219
In the 'variance gamma' model for option pricing:
Select an option first.
Correct answer: B — Stock returns are modelled as Brownian motion with a random time change driven by a gamma process — capturing both skewness and excess kurtosis while remaining analytically tractable
Explanation: B is correct: the Variance Gamma model (Madan et al. 1998) represents stock price movements as arithmetic Brownian motion (with drift θ and vol σ) time-changed by a gamma process with rate 1/ν. This gives a pure jump process (no diffusion limit) with three parameters controlling vol, skewness, and kurtosis. A is wrong — variance is stochastic. C is wrong — it has both continuous and jump components. D is wrong — VG is not a GARCH model.
Question 220
The 'conditional correlation' between two assets during a crisis period compared to normal periods is typically:
Select an option first.
Correct answer: B — Higher — crisis periods show correlation spikes as assets co-move more strongly when market-wide factors dominate idiosyncratic factors
Explanation: B is correct: empirically, correlations increase during market stress (e.g. 2008 GFC showed equity correlations approaching 1 across sectors). This 'correlation breakdown' means diversification fails precisely when needed. A is wrong — correlations change significantly. C is wrong — diversification decreases. D is wrong — correlations are high, not zero.
Question 221
The 'BEKK-GARCH' model is a multivariate GARCH that ensures:
Select an option first.
Correct answer: B — Positive definiteness of the conditional covariance matrix — the BEKK parameterisation H_t = C'C + A'ε_{t-1}ε'_{t-1}A + B'H_{t-1}B ensures H_t > 0 for all t
Explanation: B is correct: the BEKK (Baba-Engle-Kraft-Kroner) model uses matrix-valued ARCH and GARCH coefficients to model the full conditional covariance matrix, with the quadratic form structure guaranteeing positive definiteness. A is wrong — different assets have different variances. C is wrong — BEKK models correlations. D is wrong — correlations are time-varying.
Question 222
The 'Cholesky decomposition' is used in Monte Carlo simulation to:
Select an option first.
Correct answer: B — Generate correlated random variables — decomposing the covariance matrix Σ = LL' where L is lower triangular, then multiplying independent random variables by L to obtain correlated variables
Explanation: B is correct: to simulate correlated returns z = L × ε where ε ~ N(0,I), giving z ~ N(0,LL'=Σ). Cholesky decomposition is efficient and requires Σ to be positive definite. A is wrong — matrix inversion is a different operation. C is wrong — normality testing uses different methods. D is wrong — GARCH uses MLE.
Question 223
The 'spectral density' of a time series in the frequency domain for risk analysis:
Select an option first.
Correct answer: B — Decomposes the total variance into contributions from different frequencies — identifying cycles or periodic patterns in returns or volatility
Explanation: B is correct: the spectral density f(ω) satisfies γ(0) = ∫₋π^π f(ω)dω (total variance). Peaks at frequency ω indicate periodic patterns at period 2π/ω. A is wrong — autocorrelation is the time-domain equivalent. C is wrong — EVT is about tail probability. D is wrong — GARCH is a time-domain model.
Question 224
The 'HAR (Heterogeneous AutoRegressive) model' for realised volatility uses:
Select an option first.
Correct answer: B — Daily, weekly, and monthly realised variance components as predictors — capturing the heterogeneous beliefs of traders with different investment horizons
Explanation: B is correct: HAR-RV (Corsi 2009): RV_t = β₀ + β_D RV_{t-1} + β_W RV̄_{t-1,5} + β_M RV̄_{t-1,22} + ε_t. The three components capture short-term, medium-term, and long-term traders' contributions to volatility. A is wrong — only daily lags misses the multi-scale structure. C is wrong — HAR-RV is not a GARCH model. D is wrong — three lag components are used.
Question 225
The 'Jarque-Bera test' is used to test:
Select an option first.
Correct answer: B — Whether a distribution is normal — using the test statistic JB = n(S²/6 + (K−3)²/24) where S = skewness and K = kurtosis
Explanation: B is correct: JB = n × (S²/6 + (K−3)²/24) ~ χ²(2) under H₀ (normality). Financial returns typically fail this test due to excess skewness (S ≠ 0) and kurtosis (K ≠ 3). A is wrong — Levene/Brown-Forsythe tests variance equality. C is wrong — unit root tests test stationarity. D is wrong — t-tests or LR tests test parameter significance.
Question 226
The 'Engle-Granger test' is used in multivariate volatility modelling to:
Select an option first.
Correct answer: B — Test whether two non-stationary time series are cointegrated — sharing a long-run equilibrium relationship despite individual unit root behaviour
Explanation: B is correct: cointegration between financial time series (e.g. long-run relationship between stock prices of companies in the same sector) is tested using the Engle-Granger two-step procedure. If cointegrated, an error-correction model (ECM) should be used. A is wrong — Engle (1982) ARCH LM test is for GARCH effects. C is wrong — Ljung-Box tests autocorrelation. D is wrong — information criteria select GARCH order.
Question 227
The 'risk reversal' in FX options gives information about:
Select an option first.
Correct answer: B — The skew of the implied volatility smile — defined as IV(25-delta call) − IV(25-delta put), it measures the asymmetry in demand for upside vs downside protection
Explanation: B is correct: risk reversal (RR) = IV(25Δ call) − IV(25Δ put). For equity options, RR < 0 (put skew), meaning OTM puts have higher IV than OTM calls. For a given currency pair, positive RR means calls are more expensive (market fears currency appreciation). A is wrong — interest rate differential gives the forward premium/discount. C is wrong — the butterfly spread gives the total level of the smile. D is wrong — the ATM forward is separately quoted.
Question 228
The 'strangle' in FX options is used to measure:
Select an option first.
Correct answer: B — The curvature of the implied volatility smile — the strangle (average of OTM call and put IV minus ATM IV) measures the smile's butterfly spread, indicating how much OTM options cost relative to ATM
Explanation: B is correct: butterfly/strangle = (IV(25Δ call) + IV(25Δ put))/2 − IV(ATM). A positive strangle means OTM options are expensive relative to ATM — indicating a pronounced smile. A is wrong — currency risk is a broader concept. C is wrong — CIP determines forward rates. D is wrong — forward premium is from interest rates.
Question 229
The 'VVIX' index measures:
Select an option first.
Correct answer: B — The volatility of VIX — how much the VIX itself is expected to fluctuate over the next 30 days, derived from VIX options
Explanation: B is correct: VVIX (CBOE Volatility of Volatility Index) measures the 30-day implied volatility of VIX using VIX options. High VVIX means the market expects VIX itself to be volatile — 'vol of vol'. A is wrong — VIX² would be implied variance, not VVIX. C is wrong — historical VIX variance is backward-looking. D is wrong — VIX/S&P correlation is a different measure.
Question 230
The 'Parzen kernel' in ARCH LM testing is used to:
Select an option first.
Correct answer: B — Estimate the variance of a Ljung-Box type test statistic for heteroskedasticity — the Engle ARCH LM test regresses squared residuals on their lags
Explanation: B is correct: The Engle (1982) ARCH LM test regresses ê²_t on ê²_{t-1},...,ê²_{t-q} and tests whether the coefficients are jointly zero. The test statistic is n × R² ~ χ²(q) under H₀ (no ARCH effects). The kernel reference is more relevant to non-parametric variance estimation. B describes the test correctly.
Question 231
The 'bivariate GARCH' model with DCC for two assets estimates:
Select an option first.
Correct answer: B — Marginal variances for each asset (via GARCH) and their time-varying conditional correlation (via DCC) — providing a comprehensive picture of joint risk
Explanation: B is correct: DCC GARCH: (1) fit univariate GARCH to each series; (2) standardise residuals; (3) model the dynamic correlation of standardised residuals. This two-step approach is computationally efficient and captures both volatility clustering and correlation clustering. A is wrong — DCC also models correlations. C is wrong — marginal variances are also modelled. D is wrong — DCC provides time-varying correlations.
Question 232
The 'Gumbel copula' is particularly useful for modelling:
Select an option first.
Correct answer: B — Upper tail dependence (joint gains) — both variables tend to be simultaneously extreme in their upper tails, suitable for modelling joint risks in commodity markets where supply shocks drive co-movement
Explanation: B is correct: the Gumbel copula has upper tail dependence (λ_U > 0) but no lower tail dependence (λ_L = 0). This is the complement of the Clayton copula. It may be appropriate for modelling scenarios where both variables are simultaneously high (e.g. both commodity prices surge). A is wrong — lower tail dependence is Clayton. C is wrong — Gaussian copula has zero tail dependence. D is wrong — t copula has symmetric tail dependence.
Question 233
The 'maximum likelihood estimation' (MLE) for GARCH parameters finds parameters by:
Select an option first.
Correct answer: B — Maximising the log-likelihood function: L = Σ(−½log(2π) − ½log(σ²_t) − r²_t/(2σ²_t)) — finding parameters that make the observed data most probable
Explanation: B is correct: for GARCH with normal innovations, the log-likelihood is the sum of individual log-densities of the conditional normal distributions. Numerical optimisation (BFGS, etc.) maximises this function over (ω, α, β). A is wrong — OLS minimises squared residuals (for linear models). C is wrong — GMM uses moment conditions. D is wrong — BIC selects among fitted models.
Question 234
The 'Ljung-Box Q-test' applied to GARCH residuals tests:
Select an option first.
Correct answer: B — Whether there is remaining autocorrelation in the standardised residuals (returns/conditional standard deviation) — a well-specified GARCH model should have white-noise standardised residuals
Explanation: B is correct: after GARCH estimation, standardised residuals ẑ_t = r_t/σ_t should be i.i.d. if the model is correctly specified. The Ljung-Box test checks for autocorrelation in {ẑ_t} and {ẑ²_t}. Significant autocorrelation in ẑ²_t suggests the GARCH model does not fully capture volatility clustering. A is wrong — t-tests or LR tests check parameter significance. C is wrong — Jarque-Bera tests normality. D is wrong — stationarity testing uses unit root tests.
Question 235
Under extreme value theory, the tail index α (= 1/ξ) determines:
Select an option first.
Correct answer: B — The tail weight — lower α (higher ξ) indicates heavier tails and slower probability decay; for α < 2, the variance is infinite; for α < 1, the mean is infinite
Explanation: B is correct: the tail index α controls how fast the tail probability decays. For equity returns, α is typically 3-5 (finite mean and variance). For some markets, α < 2 is observed, implying infinite theoretical variance. A is wrong — threshold selection is separate. C is wrong — tail index is one parameter. D is wrong — the threshold is u, chosen independently.
Question 236
The 'bias-corrected' Hill estimator addresses:
Select an option first.
Correct answer: B — The downward bias of the Hill estimator for small samples — the bias-corrected version adds a second-order term to improve accuracy
Explanation: B is correct: the standard Hill estimator is consistent but biased in finite samples. Corrections (e.g. Peng 1998, Caers and Dykmans) add second-order terms based on the estimated decay rate of the tail probability. A is wrong — Hill estimates are positive by construction. C is wrong — GARCH bias is addressed by different methods. D is wrong — realised kernel addresses microstructure noise.
Question 237
The 'conditional skewness' of a distribution measures:
Select an option first.
Correct answer: B — The asymmetry of the conditional distribution at a given conditioning event — in GARCH models with skewed innovations, conditional skewness can be modelled explicitly
Explanation: B is correct: some GARCH models (e.g. GARCH with skewed-t innovations) model conditional skewness — the asymmetry of the return distribution given current volatility. This is useful when the distribution of shocks is itself asymmetric, not just the dynamics. A is wrong — mean-to-median ratio is a simple skewness indicator. C is wrong — VaR-ES gap is the ES gap. D is wrong — that is conditional kurtosis.
Question 238
The 'risk neutral' measure Q is derived from the physical measure P through:
Select an option first.
Correct answer: B — A change of measure using the Radon-Nikodym derivative dQ/dP = exp(−λ'dW − ½||λ||²dt) where λ is the market price of risk vector — discounting by the risk premium
Explanation: B is correct: the Girsanov theorem governs the change of measure between P and Q. Under Q, risky assets earn the risk-free rate (risk premiums are absorbed into the measure change). The market price of risk λ = (μ − r)/σ per unit of volatility determines the Radon-Nikodym derivative. A is wrong — risk premiums are captured in dQ/dP. C is wrong — under Q, drift becomes r, not P-drift plus r. D is wrong — under Q, drift becomes r, not zero.
Question 239
The 'multivariate t copula' with ν degrees of freedom approaches the Gaussian copula as:
Select an option first.
Correct answer: B — ν → ∞ — the t distribution approaches the normal as degrees of freedom increase, so the t copula approaches the Gaussian copula with the same correlation matrix
Explanation: B is correct: t(ν) → N(0,1) as ν → ∞ (central limit theorem). Similarly, the multivariate t copula converges to the multivariate normal copula as ν → ∞. This makes the Gaussian copula a special (limiting) case of the t copula. A is wrong — at ν=0 the t distribution is undefined. C is wrong — at ν=1 the Cauchy distribution has no finite moments. D is wrong — ν is a degrees-of-freedom parameter, not related to asset count.
Question 240
The 'ARCH effect' test (Engle 1982) detects whether:
Select an option first.
Correct answer: B — Squared returns exhibit autocorrelation — indicating time-varying conditional variance (volatility clustering) that warrants GARCH modelling
Explanation: B is correct: the ARCH LM test regresses ê²_t on lagged ê²_{t-1},...,ê²_{t-q}. If the R² is large (test statistic significant), squared returns are autocorrelated, indicating ARCH effects and the need for a GARCH-type model. A is wrong — autocorrelation in returns is tested by Ljung-Box on {r_t}. C is wrong — normality is tested by Jarque-Bera. D is wrong — constant variance is the null hypothesis.
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.