QFC Quantitative Finance Modeling & Valuation 2 — Questions and Answers
Question 1: In a binomial option pricing model, what happens to the option price as the number of time steps approaches infinity?
- It approaches the Black-Scholes price (Correct answer)
- It converges to zero
- It diverges to infinity
- It equals the intrinsic value only
Correct answer: It approaches the Black-Scholes price
As the number of binomial steps increases, the discrete binomial model converges to the continuous Black-Scholes model.
Question 2: The Gordon Growth Model (GGM) values a stock as D1 / (r - g). Which condition must hold for this formula to be valid?
- The growth rate g must be less than the discount rate r (Correct answer)
- The dividend D1 must be positive and g must equal r
- The firm must have no debt
- The payout ratio must be 100%
Correct answer: The growth rate g must be less than the discount rate r
GGM requires r > g; if g ≥ r, the denominator is non-positive and the model produces a nonsensical or infinite value.
Question 3: Which sensitivity measure captures the rate of change of an option's delta with respect to the underlying asset price?
- Gamma (Correct answer)
- Vega
- Theta
- Rho
Correct answer: Gamma
Gamma (Γ) is the second derivative of the option price with respect to the underlying price, measuring how fast delta changes.
Question 4: In discounted cash flow (DCF) valuation, the terminal value using the perpetuity growth method is most sensitive to which input?
- The assumed long-run growth rate (Correct answer)
- The tax rate applied in year 1
- The capital expenditure in year 5
- The depreciation method chosen
Correct answer: The assumed long-run growth rate
Small changes in the perpetuity growth rate dramatically alter terminal value because it appears in the denominator (WACC – g).
Question 5: A zero-coupon bond with a face value of $1,000 matures in 5 years. If the continuously compounded rate is 4%, what is its present value?
- $818.73 (Correct answer)
- $821.93
- $800.00
- $854.60
Correct answer: $818.73
PV = 1000 × e^(−0.04×5) = 1000 × e^(−0.20) ≈ $818.73 under continuous compounding.
Question 6: Which of the following best describes the concept of 'risk-neutral valuation' in derivative pricing?
- Pricing derivatives as if all investors are indifferent to risk, using risk-free rates for discounting (Correct answer)
- Assuming the market is perfectly efficient with no arbitrage
- Pricing based on the actual probability distribution of the underlying
- Using the CAPM beta to adjust expected returns
Correct answer: Pricing derivatives as if all investors are indifferent to risk, using risk-free rates for discounting
Risk-neutral valuation replaces real-world probabilities with risk-neutral probabilities and discounts at the risk-free rate to preclude arbitrage.
Question 7: A callable bond trades at a lower price than an otherwise identical non-callable bond. What explains this difference?
- The issuer's call option has value, which is subtracted from the straight bond price (Correct answer)
- Callable bonds always have lower credit ratings
- Callable bonds pay higher coupons to compensate investors
- The call feature reduces duration, raising price
Correct answer: The issuer's call option has value, which is subtracted from the straight bond price
Callable bond price = straight bond price − value of the embedded call option the issuer holds, so it trades at a discount.
QFC Quantitative Finance Modeling & Valuation 2
In a binomial option pricing model, what happens to the option price as the number of time steps approaches infinity?