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Question 1. Which of the following best defines historical volatility? A) The market‑implied volatility from option prices B) The standard deviation of past asset returns C) The expected future variance under the risk‑neutral measure D) The volatility of a volatility index Answer: B Explanation: Historical volatility is computed as the statistical standard deviation of observed past returns. Question 2. Implied volatility differs from historical volatility because it is: A) Measured from past price data B) Extracted from current option prices C) Always larger than realized volatility D) Independent of the underlying asset price Answer: B Explanation: Implied volatility is the volatility level that, when input into an option pricing model, reproduces the market price of the option. Question 3. The relationship between variance (σ²) and volatility (σ) is: A) σ = √σ² B) σ² = σ⁴ C) σ = σ² D) σ² = σ / 2 Answer: A Explanation: Variance is the square of volatility; therefore volatility is the square‑root of variance.
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Question 4. When scaling volatility from daily to annual terms, which rule is applied? A) Multiply by √ B) Divide by √ C) Multiply by 252 D) Divide by 252 Answer: A Explanation: Annualized volatility = daily volatility × √(number of trading days per year), typically 252. Question 5. Realized variance over a period is calculated as: A) The sum of squared daily log returns B) The average of daily price changes C) The square of the realized volatility D) The integral of implied volatility over time Answer: A Explanation: Realized variance is the sum (or average) of squared log returns observed over the observation window. Question 6. In a variance swap, the floating leg pays:** A) The difference between realized variance and the variance strike B) The realized volatility multiplied by a notional C) The fixed variance strike only D) The underlying asset price at maturity Answer: A
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D) max(S_T – K, 0) Answer: C Explanation: A log contract pays the natural logarithm of the ratio of terminal to initial price. Question 10. In the static replication of a variance swap, the weight of an option with strike K is proportional to:** A) The first derivative of the option price with respect to K B) The second derivative of the option price with respect to K C) The option’s delta D) The option’s vega Answer: B Explanation: The replication uses a continuum of options weighted by the second derivative of option prices (i.e., the risk‑neutral density). Question 11. Which statement correctly describes a volatility swap? A) It exchanges realized variance for a fixed variance strike. B) It exchanges realized volatility for a fixed volatility strike. C) It can be perfectly replicated by a static options portfolio. D) Its payoff is linear in the underlying price. Answer: B Explanation: A volatility swap pays the difference between realized volatility and a predetermined volatility strike. Question 12. The main reason a volatility swap cannot be perfectly replicated by static options is:**
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A) Volatility is the square root of variance, a nonlinear transformation. B) Options are only available at discrete strikes. C) The underlying asset pays dividends. D) Market participants are risk‑averse. Answer: A Explanation: Because volatility is a concave function of variance, Jensen’s inequality prevents exact static replication. Question 13. The convexity adjustment needed to price a volatility swap relative to a variance swap accounts for:** A) The difference between E[σ] and √E[σ²] B) The impact of interest rates on option prices C) The bid‑ask spread of the underlying asset D) The correlation between underlying and interest rates Answer: A Explanation: The convexity adjustment corrects for the fact that the expected volatility is not the square root of expected variance. Question 14. Which Taylor‑series based approximation is most frequently used to convert a fair variance strike (K_{\text{var}}) into a fair volatility strike (K_{\text{vol}})? A) (K_{\text{vol}}\approx K_{\text{var}}^{2}) B) (K_{\text{vol}}\approx \sqrt{K_{\text{var}}}) C) (K_{\text{vol}}\approx \frac{K_{\text{var}}}{2}) D) (K_{\text{vol}}\approx \ln(K_{\text{var}})) Answer: B Explanation: Because volatility is the square‑root of variance, a first‑order approximation uses (K_{\text{vol}}\approx\sqrt{K_{\text{var}}}).
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D) The underlying asset’s forward price. Answer: C Explanation: The model‑independent replication shows that expected variance equals a weighted integral of option prices across strikes. Question 18. In the context of variance swaps, what does the term “static replication” refer to? A) A dynamic delta‑hedge updated daily. B) Holding a fixed portfolio of European options until maturity. C) Using only the underlying asset to replicate the payoff. D) Replicating the swap with a single forward contract. Answer: B Explanation: Static replication uses a time‑invariant set of options whose payoff matches the variance swap’s payoff at expiry. Question 19. Which of the following is NOT a typical input when calibrating a Heston stochastic volatility model for pricing variance swaps? A) Long‑run variance level (\theta). B) Mean‑reversion speed (\kappa). C) Correlation (\rho) between asset and variance Brownian motions. D) The underlying asset’s dividend yield volatility. Answer: D Explanation: Dividend yield volatility is not a standard Heston parameter; the model focuses on spot variance dynamics.
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Question 20. A corridor variance swap only accrues variance when the underlying price lies within a predefined range ([L,U]). Which effect does this feature have on the swap’s sensitivity to extreme price moves? A) Increases sensitivity because variance is always counted. B) Decreases sensitivity because out‑of‑range moves are ignored. C) Has no effect; the payoff is identical to a standard variance swap. D) Turns the swap into a volatility swap. Answer: B Explanation: By excluding variance outside ([L,U]), the contract reduces exposure to large jumps that move the price outside the corridor. Question 21. Which Greek measures the sensitivity of a variance swap’s price to a change in the underlying asset’s forward price? A) Delta B) Vega C) Gamma D) Theta Answer: A Explanation: Even though the payoff depends on variance, the static replication involves a strip of options, giving the swap a non‑zero delta. Question 22. The VIX index is best described as:** A) The price of a VIX futures contract. B) The square root of the forward‑starting variance swap rate on the S&P 500. C) The implied volatility of a single at‑the‑money S&P 500 option. D) The historical volatility of the S&P 500 over the past 30 days.
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B) 365 (calendar days) C) √ D) √ Answer: A Explanation: Daily squared returns are summed and then multiplied by 252 to express variance on an annual basis. Question 26. Which market risk is most directly mitigated by entering a variance swap? A) Directional price risk of the underlying asset. B) Credit risk of the counter‑party. C) Uncertainty about future volatility levels. D) Liquidity risk of the underlying market. Answer: C Explanation: Variance swaps provide pure exposure to realized variance, isolating volatility risk from directional price moves. Question 27. The “discreteness risk” in variance swap pricing arises because:** A) The underlying price follows a continuous diffusion. B) Observations are made at discrete intervals (e.g., daily) rather than continuously. C) The swap notional is not adjusted for time. D) The variance strike is reset daily. Answer: B Explanation: The theoretical replication assumes continuous quadratic variation, but actual contracts observe variance at discrete times, introducing a small bias.
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Question 28. Which of the following best describes the impact of a sudden price jump on a variance swap’s payoff? A) Jumps have no effect because variance is measured from log returns. B) Jumps increase realized variance dramatically, benefiting the holder of the variance receiver. C) Jumps reduce realized variance because they truncate the observation window. D) Jumps only affect volatility swaps, not variance swaps. Answer: B Explanation: A large jump creates a large squared return, raising realized variance and thus the payoff to the party receiving variance. Question 29. In the context of a variance swap, the “variance notional” (N_{\text{var}}) is expressed in units of:** A) Currency per variance point (e.g., USD per 1%²). B) Currency per volatility point (e.g., USD per 1%). C) Underlying shares per variance point. D) Basis points of the underlying price. Answer: A Explanation: The notional multiplies the variance difference, so it is quoted per variance point (often per 1%²). Question 30. Which of the following adjustments is necessary when pricing a variance swap on an asset that pays a known dividend yield (q)? A) Subtract (q) from the variance strike. B) Use forward prices that incorporate the dividend yield. C) Increase the notional by a factor of ((1+q)). D) No adjustment is needed because dividends do not affect variance.
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A) (K_{\text{vol}}\approx \sqrt{K_{\text{var}}} - \frac{1}{8} \frac{\text{Var}(\sigma^{2})}{K_{\text{var}}^{3/2}}) B) (K_{\text{vol}}\approx K_{\text{var}}^{2} + \frac{1}{2}) C) (K_{\text{vol}}\approx \frac{1}{\sqrt{K_{\text{var}}}}) D) (K_{\text{vol}}\approx \ln(K_{\text{var}})) Answer: A Explanation: The second‑order expansion adds a convexity correction term involving the variance of variance. Question 34. Which of the following best captures the “vanna” exposure of a volatility swap? A) Sensitivity of the swap’s price to changes in the underlying price. B) Sensitivity to simultaneous changes in the underlying price and implied volatility. C) Sensitivity to the passage of time. D) Sensitivity to the risk‑free rate. Answer: B Explanation: Vanna measures cross‑sensitivity of delta to volatility (or vice‑versa), relevant for volatility‑linked products. Question 35. In practice, why might a dealer charge a “convexity premium” when quoting a volatility swap price? A) To compensate for the cost of funding the notional. B) Because the fair volatility strike is higher than the square‑root of the fair variance strike due to Jensen’s inequality. C) To reflect the dividend yield of the underlying. D) To offset the bid‑ask spread of the underlying options. Answer: B
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Explanation: The convexity premium adjusts for the fact that (E[\sigma] > \sqrt{E[\sigma^{2}]}) when the variance distribution is skewed. Question 36. Which of the following is true about the relationship between VIX futures and variance swaps on the S&P 500? A) VIX futures are priced exactly the same as variance swaps. B) VIX futures provide a forward contract on the square root of variance, while variance swaps provide the variance itself. C) VIX futures are unrelated to variance swaps. D) VIX futures replicate a corridor variance swap. Answer: B Explanation: VIX futures trade the expected future volatility (square root of variance), whereas variance swaps settle on variance. Question 37. A “forward variance” contract is essentially:** A) A variance swap with its start date set in the future. B) A volatility swap with a forward start. C) An option on variance. D) A variance swap with a floating strike. Answer: A Explanation: Forward variance contracts lock in the variance that will be realized over a future period, similar to a forward-start variance swap. Question 38. In a non‑equity market (e.g., FX), which additional factor must be considered when pricing a variance swap? A) The foreign‑currency risk‑free rate. B) The dividend yield of the underlying.
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Question 41. Which risk is most heightened when the underlying market experiences low liquidity in out‑of‑the‑money options used for variance‑swap replication? A) Counter‑party risk. B) Model risk. C) Hedging error due to discrete rebalancing. D) Execution risk and slippage on the static options portfolio. Answer: D Explanation: Poor liquidity leads to larger bid‑ask spreads and execution costs for the required OTM options. Question 42. In a “gamma swap,” the payoff is proportional to:** A) The realized variance of the underlying. B) The cumulative gamma exposure of a dynamically hedged portfolio. C) The square root of realized variance. D) The realized skewness of the underlying. Answer: B Explanation: Gamma swaps capture the integrated gamma (second‑order sensitivity) of a portfolio over time. Question 43. Which of the following best explains why variance swaps are considered “model‑independent” in valuation? A) Their price can be derived solely from a continuous strip of European options, without assuming any dynamics for volatility. B) They require a specific stochastic volatility model for pricing. C) Their payoff depends only on the terminal price. D) They are priced using historical variance only.
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Answer: A Explanation: The replication argument uses only observable market option prices, making the valuation independent of any particular stochastic model. Question 44. Which of the following statements about “forward‑starting variance swaps” is correct? A) They have a fixed start date equal to today. B) Their payoff depends on variance realized after the forward start date, not from today. C) They cannot be hedged with static options. D) They are identical to standard variance swaps. Answer: B Explanation: Forward‑starting swaps lock in a variance contract that begins at a future date, allowing participants to hedge future variance exposure. Question 45. The “volatility term structure” typically displays which of the following shapes for equity indices? A) Flat across all maturities. B) Upward sloping (long‑dated vol > short‑dated). C) Downward sloping (short‑dated vol > long‑dated). D) Random with no discernible pattern. Answer: C Explanation: Equity indices often exhibit a downward‑sloping term structure, reflecting higher near‑term implied vol due to uncertainty. Question 46. Which of the following is a primary driver of the “volatility smile” observed in option markets? A) Constant interest rates.
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Question 49. When a variance swap is settled in cash, the cash amount paid is:** A) The realized variance multiplied by the underlying price. B) The notional times the difference between realized variance and variance strike. C) The variance strike multiplied by the notional, regardless of realized variance. D) The underlying price at maturity. Answer: B Explanation: Cash settlement equals (N_{\text{var}} \times (\text{Realized Variance} - K_{\text{var}})). Question 50. Which of the following is true about the “vega” of a variance swap? A) It is zero because the swap payoff does not depend on implied volatility. B) It is positive for the variance receiver and negative for the payer. C) It equals the delta of the underlying asset. D) It is infinite due to the static replication. Answer: B Explanation: Since the fair variance strike depends on the implied volatility surface, changes in implied vol affect the swap’s value, giving it non‑zero vega. Question 51. In a “double‑no‑touch” variance swap, variance is accrued only when:** A) The underlying price stays within a pre‑specified band for the entire life of the swap. B) The price touches either barrier at least once. C) The price never touches either barrier during the observation period. D) The price crosses a single barrier exactly once. Answer: C
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Explanation: “Double‑no‑touch” requires the underlying to avoid both barriers; variance is only counted when the condition holds. Question 52. Which of the following best describes “model risk” for volatility derivatives? A) The risk that the underlying asset defaults. B) The risk that the chosen pricing model mis‑estimates the fair strike due to incorrect dynamics. C) The risk of counter‑party default. D) The risk that interest rates change. Answer: B Explanation: Model risk refers to errors arising from using an inappropriate or mis‑calibrated model for pricing and hedging. Question 53. A “variance forward” differs from a variance swap in that:** A) It has a fixed payoff at inception. B) It is settled at a future date with a pre‑agreed variance level, rather than exchanging realized variance. C) It pays the square root of variance. D) It is only traded on exchanges. Answer: B Explanation: A variance forward locks in a forward variance level to be exchanged at a later date, similar to a forward contract. Question 54. Which of the following is a typical use of variance swaps by a portfolio manager? A) To obtain directional exposure to the underlying asset’s price. B) To hedge the vega risk of an options‑heavy portfolio.
