Answer-first summary for fast verification
Answer: $7.00
A forward-start option on an ATM call can be valued by scaling the known ATM call price at the start date. - The 6-month ATM call price when \(S=K=100\) is **$7.51**. - Because the strike is set to the stock price at \(T_1\), the option value at \(T_1\) is proportional to \(S(T_1)\). - Discounting back one year using the dividend yield gives: \[ \text{Price at } T_0 \approx 7.51 \times e^{-0.07} \approx 7.00 \] So the nearest price is **A. $7.00**.
Author: Manit Arora
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Q-729.2 The exhibit below shows the call option prices for various times to maturity, three months, six months, nine months, 1.0 year, 1.25 years, and 1.5 years for an at-the-money European call option while the stock and strike price are both $100.00, the stock's volatility () is 30.0%, the risk-free rate is 4.0% and the stock pays a continuous dividend yield of 7.0%:
European call option prices with various times to maturity (T)
| Stock (S0) | $100.00 | $100.00 | $100.00 | $100.00 | $100.00 | $100.00 |
|---|---|---|---|---|---|---|
| Strike (K) | $100.00 | $100.00 | $100.00 | $100.00 | $100.00 | $100.00 |
| Volatility, σ | 30.0% | 30.0% | 30.0% | 30.0% | 30.0% | 30.0% |
| Riskfree rate, r | 4.00% | 4.00% | 4.00% | 4.00% | 4.00% | 4.00% |
| Time to maturity, T, years | 0.25 | 0.50 | 0.75 | 1.00 | 1.25 | 1.50 |
| Div Yield, q | 7.00% | 7.00% | 7.00% | 7.00% | 7.00% | 7.00% |
| Implied PV lump-sum, D | $1.75 | $3.51 | $5.27 | $7.04 | $8.81 | $10.58 |
| BSM call price, c = | $5.53 | $7.51 | $8.88 | $9.92 | $10.76 | $11.45 |
The stock price is $100.00 today. Consider a forward-start option that is a contract to buy, one year from today, a six-month to expiration at-the-money (ATM) call option; i.e., year, years. Which is nearest to the price of this forward-start option?
A
$7.00
B
$9.25
C
$10.68
D
$11.45
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