What is the lower bound of an American call option with 14 months to expiry and a strike price of 9. The underlying stock is currently trading at 15.1, and the risk-free rate is 6%. Do not enter the dollar sign "$". Your answer should have at least two decimal places.

Question

BlackTom AI · Accepted Answer

For an American call on a non-dividend-paying stock the relevant lower bound is the present-value-adjusted intrinsic value, i.e. C ≥ S − K e^(−rT). Early exercise is not optimal without dividends, so the American call value is at least the European lower bound S − K e^(−rT) (and also ≥ intrinsic value S − K).

Step-by-step reasoning and calculation:
1. Identify inputs:
   - Current stock price S = 15.10
   - Strike K = 9.00
   - Annual risk-free rate r = 0.06
   - Time to expiration T = 14 months = 14/12 = 7/6 ≈ 1.1666667 years
2. Compute rT: rT = 0.06 × (7/6) = 0.07.
3. Compute discount factor e^(−rT) = e^(−0.07). Numerically,
   e^(−0.07) ≈ 0.93239382.
4. Compute present value of strike: K e^(−rT) = 9 × 0.93239382 ≈ 8.39154438.
5. Compute the lower bound: S − K e^(−rT) = 15.10 − 8.39154438 = 6.70845562.
6. Round to two decimal places: 6.71.

Why this is correct:
- For calls on non-dividend-paying stocks, early exercise yields no benefit, so the American call must be worth at least the discounted strike-adjusted intrinsic value S − K e^(−rT). Because e^(−rT) < 1, S − K e^(−rT) is larger than the simple intrinsic value S − K (= 6.10), so 6.71 is the appropriate, tighter lower bound.

Why other simple values are incorrect:
- Using intrinsic value S − K = 6.10 understates the lower bound because it ignores the time value (discounting the strike).
- Any lower number would violate arbitrage bounds.

Final answer confirmed: 6.71

What is the lower bound of an American call option with 14 months to expiry and a strike price of 9. The underlying stock is currently trading at 15.1, and the risk-free rate is 6%. Do not enter the dollar sign "$". Your answer should have at least two decimal places.数值题

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