An investor holds a portfolio of European call options on a non-dividend-paying stock. The current stock price is $50, the strike price is $55, the risk-free rate is 3% (continuously compounded), and the time to expiration is 6 months. The volatility of the stock is 25%. Which of the following statements about the value of the call option is most accurate?
- AThe call option must be worth at least $5.
- BThe call option must be worth at least $0.
- CThe call option must be worth at least $50 - $55e^(-0.03*0.5).
- DThe call option must be worth at least $50 - $55.
Show answer & explanationAnswer & explanation
Correct answer: C. The call option must be worth at least $50 - $55e^(-0.03*0.5).
The lower bound for the price of a European call option on a non-dividend-paying stock is given by max(0, S0 - Xe^(-rT)). Here, S0 = $50, X = $55, r = 0.03, T = 0.5. So, the lower bound is max(0, 50 - 55e^(-0.03*0.5)) = max(0, 50 - 55 * 0.98507) = max(0, 50 - 54.17885) = max(0, -4.17885) = 0. However, the question asks for 'at least' and option C provides the exact formula before taking the maximum with zero.
Why the other options are wrong
- A. Incorrect. This would be the intrinsic value if the stock price were $60, not $50.
- B. While true, this is a trivial lower bound and not the most accurate or informative one.
- D. Incorrect. This ignores the time value of money (discounting the strike price) and is just the intrinsic value for an in-the-money option, which this is not.
European Call Option Lower Bound (Non-Dividend)
The theoretical minimum value (lower bound) for a European call option on a non-dividend-paying stock is the greater of zero or the current stock price minus the present value of the strike price.
- Formula: c >= max(0, S0 - Xe^(-rT)).
- S0 = Current stock price, X = Strike price, r = Risk-free rate (continuously compounded), T = Time to expiration.
- This bound ensures no arbitrage opportunities exist.
Memory trick: Bounds: No Option is Cheaper Than Its 'Bare Minimum' Value!