What is the upper bound of a European call option with 8 months to expiry and a strike price of $ 9.5. The underlying stock is currently trading at $ 11.5, and the risk-free rate is 2.8%.

Question

BlackTom AI · Accepted Answer

For a non-dividend-paying underlying, no-arbitrage bounds give the maximum possible price of a European call as the current stock price S. Intuitively a call cannot be worth more than the stock itself because owning the stock dominates owning the call (you already have the underlying asset rather than just the right to buy it).

Relevant concepts:
- Upper bound for a European call on a stock with no dividends: C ≤ S.
- Lower (no-arbitrage) bound for a European call: C ≥ max(0, S − PV(K)), where PV(K) is the present value of the strike discounted at the risk-free rate.
- Present value with continuous discounting: PV(K) = K e^{-rT} (used below to illustrate the lower bound).

Step-by-step reasoning and calculations:
1. Identify inputs: S = $11.50, K = $9.50, r = 2.8% = 0.028, time T = 8 months = 8/12 = 2/3 ≈ 0.6666667 years.
2. Compute PV(K) (to show the lower bound, not the upper): rT = 0.028 × 0.6666667 ≈ 0.0186667. e^{-rT} = e^{-0.0186667} ≈ 0.9815. PV(K) = 9.50 × 0.9815 ≈ 9.324.
3. Lower bound = S − PV(K) = 11.50 − 9.324 ≈ 2.176 (≈ $2.18). This confirms a minimum plausible call price, but not the maximum.
4. Upper bound: by no-arbitrage C ≤ S. If a European call were priced above S, one could short the call and buy the stock for S, generating an immediate risk-free profit and no future loss — arbitrage would eliminate such pricing. Therefore the upper bound is S = $11.50.

Why this answer is correct:
- The call’s maximum possible value is the entire stock price because the call can never deliver more than owning the stock itself. Hence the upper bound equals S.

Why other plausible numbers are incorrect:
- S − PV(K) ≈ $2.18 is the lower bound, not the upper bound.
- PV(K) ≈ $9.32 is the present value of the strike (used in the lower bound), not an upper bound.
- Any number greater than S would permit arbitrage and thus cannot be an upper bound.

Conclusion: The upper bound of the European call is $11.50, which matches the provided answer.

What is the upper bound of a European call option with 8 months to expiry and a strike price of $ 9.5. The underlying stock is currently trading at $ 11.5, and the risk-free rate is 2.8%.数值题

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