N(d1) in the Black-Scholes-Merton model represents:

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Question restatement: N(d1) in the Black-Scholes-Merton model represents what?

Option A: Call option delta.
Option B: Hedge ratio.
Option C: Call option delta, hedge ratio, and probability.
Option D: Probability.

Option A analysis: In Black-Scholes, the call option delta is equal to N(d1). This quantity measures the sensitivity of the option price to small changes in the underlying asset price, and it is precisely the delta used in hedging practical positions. Therefore, N(d1) indeed corresponds to the call delta.

Option B analysis: The hedge ratio in a dynamic delta-hedging framework is commonly the option delta, which for a European call is N(d1). So this one is partially correct because the hedge ratio equals the delta for a plain-vanilla call, but the term ‘hedge ratio’ by itself is a broader concept. Since N(d1) is specifically the delta, calling it simply the hedge ratio without the delta context can be seen as incomplete though related.

Option C analysis: This option asserts that N(d1) equals the delta, the hedge ratio, and the probability. While N(d1) is the delta and thus serves as the hedge ratio in delta hedging, it is not the probability of finishing in the money. The probability of finishing in the money under the risk-neutral measure is N(d2), not N(d1). Therefore this option overstates what N(d1) represents.

Option D analysis: This option claims N(d1) is a probability. Although N(·) is a cumulative distribution function of the standard normal, N(d1) is not the probability of a specific event like finishing in the money; that interpretation is reserved for N(d2). Consequently this option mislabels the meaning of N(d1).

Putting the pieces together, the most accurate understanding is that N(d1) corresponds to the call option delta (and thus the hedge ratio in a delta-hedging framework), but it is not the probability of finishing in the money (which is N(d2)). The incorrect choices either misidentify N(d1) as a standalone probability or as an umbrella term for both delta and probability, which is not correct.

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

N(d1) in the Black-Scholes-Merton model represents:

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