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Free Questions › FRM › Quantitative Analysis
Free FRM Quantitative Analysis Practice Questions & Answers
400 exam-style Quantitative Analysis questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 361
The fundamental difference between historical simulation and Monte Carlo simulation for VaR is:
Select an option first.
Correct answer: B — Historical simulation uses actual observed past returns (non-parametric, limited by history), while Monte Carlo generates unlimited synthetic scenarios from a specified parametric distribution, allowing exploration of scenarios not in historical data, but requires distributional assumptions
Explanation: B
Question 362
Liquidity-adjusted VaR (LVaR) extends standard VaR by:
Select an option first.
Correct answer: B — Adding the expected cost of liquidating positions: LVaR = VaR + ½ × (Ask - Bid) × Position size × (1 + κ/√n) where the spread and its uncertainty are incorporated into the risk measure
Explanation: B
Question 363
The gamma-vega interaction in an option portfolio creates:
Select an option first.
Correct answer: B — A second-order sensitivity called 'vomma' (dVega/dσ) and 'vanna' (dDelta/dσ or dVega/dS) — when both the underlying price AND volatility change simultaneously, these cross-Greeks determine the total P&L beyond the first-order delta-vega approximation
Explanation: B
Question 364
The joint distribution of correlated normal returns can be modelled using:
Select an option first.
Correct answer: B — A multivariate normal distribution characterised by a mean vector μ and covariance matrix Σ, where linear combinations are also normally distributed — allowing portfolio risk to be computed as σ²_p = w'Σw
Explanation: B
Question 365
Filtered Historical Simulation (FHS) for VaR combines:
Select an option first.
Correct answer: B — Historical simulation with a GARCH-type volatility model — standardising historical returns by current volatility estimates to produce volatility-adjusted scenarios that better reflect current market conditions
Explanation: B
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Question 366
The variance-covariance (parametric) approach to portfolio VaR assumes:
Select an option first.
Correct answer: B — Portfolio returns are normally distributed (given normally distributed individual returns with linear correlations), so portfolio VaR = z_α × √(w'Σw) where w is the weight vector and Σ is the covariance matrix
Explanation: B
Question 367
The Black-Scholes-Merton model assumes:
Select an option first.
Correct answer: B — Log-normally distributed stock prices, constant volatility, continuous trading, no dividends, and a constant risk-free rate
Explanation: B
Question 368
Implied volatility is:
Select an option first.
Correct answer: B — The volatility that, when input to an option pricing model, produces the model's theoretical price equal to the observed market price
Explanation: B
Question 369
The volatility smile in equity options refers to:
Select an option first.
Correct answer: B — The pattern where implied volatility tends to be higher for out-of-the-money puts and calls than for at-the-money options
Explanation: B
Question 370
Gamma of an option measures:
Select an option first.
Correct answer: B — The rate of change of delta per unit change in the underlying price
Explanation: B
Question 371
An in-the-money call option has:
Select an option first.
Correct answer: B — Intrinsic value greater than zero — the stock price exceeds the strike price
Explanation: B
Question 372
Time value of a call option is:
Select an option first.
Correct answer: B — Total option price minus intrinsic value (max(S-K,0)) — it represents the additional value from the possibility of further price appreciation before expiry
Explanation: B
Question 373
The Black-Scholes formula requires the following inputs:
Select an option first.
Correct answer: B — Current stock price (S), strike price (K), risk-free rate (r), time to expiration (T), and implied volatility (σ)
Explanation: B
Question 374
The intrinsic value of an ITM put option with stock price $40 and strike $50 is:
Select an option first.
Correct answer: B — $10
Explanation: B
Question 375
The binomial option pricing model creates a risk-free portfolio by:
Select an option first.
Correct answer: B — Combining a specific number of shares (the delta) with a long or short option position to create a portfolio whose value is identical in all states of the world, allowing risk-free discounting
Explanation: B
Question 376
In the binomial option pricing model, the risk-neutral probability is:
Select an option first.
Correct answer: B — A probability derived from no-arbitrage conditions that allows discounting at the risk-free rate, where p = (e^{rΔt} - d) / (u - d)
Explanation: B
Question 377
The delta-gamma approximation for a portfolio's P&L is:
Select an option first.
Correct answer: B — P&L ≈ Δ×ΔS + (1/2)×Γ×(ΔS)²
Explanation: B
Question 378
The Black-Scholes formula for a European call includes N(d1) and N(d2). N(d2) represents:
Select an option first.
Correct answer: B — The risk-neutral probability that the call will expire in the money
Explanation: B
Question 379
For a European put option using Black-Scholes, as the underlying stock price increases:
Select an option first.
Correct answer: D — Both B and C are correct
Explanation: D
Question 380
The vega of a short option position is:
Select an option first.
Correct answer: B — Negative — the position loses value as volatility increases
Explanation: B
Question 381
An Asian option on the average price is cheaper than an equivalent European option because:
Select an option first.
Correct answer: B — The average of a series of prices is less volatile than any individual price, reducing the expected option payoff variance
Explanation: B
Question 382
The delta of a forward contract on a non-dividend paying stock is:
Select an option first.
Correct answer: C — 1.0 — the forward perfectly tracks the stock
Explanation: C
Question 383
Early exercise of an American put option becomes more likely when:
Select an option first.
Correct answer: B — The option is deep in the money, interest rates are high, and there is little time remaining — the time value from waiting is less than the interest earned on the intrinsic value received immediately
Explanation: B
Question 384
The risk-neutral valuation principle states that:
Select an option first.
Correct answer: B — Under no-arbitrage conditions, derivatives can be priced by taking the expected payoff under the risk-neutral measure and discounting at the risk-free rate, regardless of investor risk preferences
Explanation: B
Question 385
The relationship between the price of an option and its delta, gamma, theta, and rho is captured by:
Select an option first.
Correct answer: B — The Black-Scholes PDE: rC = rS×Δ + (1/2)σ²S²×Γ + Θ, which must be satisfied by any derivative security in the BSM framework
Explanation: B
Question 386
The no-arbitrage principle ensures that:
Select an option first.
Correct answer: B — Two portfolios with identical future payoffs must have the same present value — any deviation creates a risk-free profit opportunity that arbitrageurs will exploit until prices correct
Explanation: B
Question 387
Implied volatility is said to be 'the market's forecast of future volatility' because:
Select an option first.
Correct answer: B — The implied volatility backed out from option prices represents the market's risk-neutral expectation of volatility over the option's life — incorporating all available information about future uncertainty
Explanation: B
Question 388
The Black-Scholes model's assumption of constant volatility implies:
Select an option first.
Correct answer: A — The volatility smile should be flat (same implied vol at all strikes)
Explanation: A
Question 389
In the presence of a volatility skew, using BSM with a single implied volatility to price a portfolio of options will:
Select an option first.
Correct answer: B — Misprice options at different strikes — typically underpricing OTM puts relative to the skew-adjusted price
Explanation: B
Question 390
A quanto option is:
Select an option first.
Correct answer: B — An option where the payoff is in a fixed currency (e.g., USD) but based on an underlying denominated in a foreign currency (e.g., a Japanese index), eliminating currency risk
Explanation: B
More free FRM topics
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Liquidity and Treasury Risk Management400
Risk Management & Investment Management398
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