Which of the following is true for a call option on a non-dividend-paying stock?

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BlackTom AI · Accepted Answer

Question restatement: Which of the following is true for a call option on a non-dividend-paying stock?

Option 1: If the option has a delta of 0.5, then the current stock price is equal to the strike price of the option.
- Analysis: Delta for a vanilla call is N(d1), where d1 = [ln(S/K) + (r + 0.5*sigma^2)*T] / (sigma*sqrt(T)). For delta to be exactly 0.5, d1 must be 0, which implies ln(S/K) = - (r + 0.5*sigma^2)*T. Since (r + 0.5*sigma^2) and T are nonnegative, the right-hand side is nonpositive, meaning ln(S/K) < 0, or S < K. Therefore a delta of 0.5 does not occur at S = K in general; with positive interest rate and volatility, S is typically below K when delta is 0.5. This makes the statement false in typical Black-Scholes settings for a non-dividend-paying stock.

Option 2: If the option has a delta of 0.5, then the current stock price is less than the strike price of the option.
- Analysis: As just noted, delta = N(d1) = 0.5 implies d1 = 0. From the d1 expression, d1 = 0 leads to ln(S/K) = - (r + 0.5*sigma^2)*T, which is negative, so S/K < 1, i.e., S < K. In other words, when delta is exactly 0.5, the stock price is below the strike price. This aligns with the mathematical relationship in Black-Scholes for a non-dividend-paying stock and a positive time to expiration. Hence this statement is true.

Option 3: If the option has a delta of 0.5, then the current stock price is greater than the strike price of the option.
- Analysis: By the same reasoning as above, delta = 0.5 implies S < K (since d1 = 0 forces ln(S/K) to be negative). Therefore the proposition that S > K under delta 0.5 is incorrect. This option is false.

Summary of reasoning: In Black-Scholes terms, delta of a call is N(d1). Setting delta to 0.5 requires d1 = 0, which imposes ln(S/K) = - (r + 0.5*sigma^2)T < 0, given positive r, sigma, and T. Consequently, S < K when delta = 0.5 for a non-dividend-paying stock. The first and third statements contradict this relationship by asserting equality or a greater-than relationship, respectively, and are therefore incorrect.

2025FallDYN-T-FIN530-86763-86762 Quiz 6 - "Black-Scholes-Merton Model and Dynamic Hedging"

Which of the following is true for a call option on a non-dividend-paying stock?

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