What is the upper bound of an American put option with 7 months to expiry and a strike price of $ 6. The underlying stock is currently trading at $ 1.5, and the risk-free rate is 2.6%. Do not enter the dollar sign "$".

Question

BlackTom AI · Accepted Answer

An American put option can never exceed the strike price K in value, because immediate exercise yields at most K (when the underlying stock price goes to zero).

1) Relevant knowledge points and concepts
- Payoff from exercising a put immediately = max(K − S, 0). The maximum possible immediate payoff is K (when S = 0).
- An American option can be exercised early, so its price cannot exceed the largest possible immediate payoff, i.e., K.
- For comparison: the European put upper bound is K·e^{−rT} (discounted strike), which is ≤ K. The intrinsic value is K − S (a lower bound).

2) Complete reasoning step by step
- The put price P_American ≤ maximum immediate exercise payoff.
- Maximum immediate exercise payoff occurs when S = 0, giving payoff = K − 0 = K.
- Therefore P_American ≤ K.

3) Calculations (intermediate steps shown)
- Given K = 6, S = 1.5, r = 2.6% and T = 7/12 years.
- Intrinsic value = K − S = 6 − 1.5 = 4.5 (this is a lower bound, not the upper bound).
- European put discounted-strike upper bound = K·e^{−rT} = 6·e^{−0.026·(7/12)}
  = 6·e^{−0.0151667} ≈ 6·0.98495 ≈ 5.9097 (≈5.91), which is less than K.

4) Why the correct answer is correct
- Because early exercise is allowed, the American put can reach the full strike amount in the extreme case S → 0, so its theoretical maximum value is K = 6. No arbitrage argument allows the price to exceed K.

5) Why other plausible quantities are incorrect
- K·e^{−rT} (~5.91) is the European put’s discounted-strike upper bound, not the American put’s. Intrinsic value (4.5) is a lower bound. The stock price S (1.5) is unrelated as an upper bound.

6) Conclusion
- The upper bound of the American put is 6, which matches the provided answer.

What is the upper bound of an American put option with 7 months to expiry and a strike price of $ 6. The underlying stock is currently trading at $ 1.5, and the risk-free rate is 2.6%. Do not enter the dollar sign "$".数值题

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