Why Delta Is Not the Probability of an Option Expiring in the Money in Simple Terms

The Black-Scholes model is a widely used framework for pricing European call and put options. It operates under the assumption of a risk-neutral world, where all assets grow at the risk-free rate, r. This allows the pricing of options using the formula for a European call option:

C = S * N(d1) - X * e^(-r * T) * N(d2)

where C is the price of the call option, S is the current stock price, X is the strike price of the option, r is the risk-free rate, T is the time to maturity, N(d1) and N(d2) are the cumulative distribution functions of the standard normal distribution evaluated at d1 and d2, respectively, and e is the base of the natural logarithm.

The parameters d1 and d2 are given by:

d1 = [ln(S / X) + (r + (σ^2) / 2) * T] / (σ * √T)

d2 = d1 - σ * √T

where σ is the volatility of the underlying stock, ln is the natural logarithm, and √T is the square root of the time to maturity.

Understanding N(d1) and N(d2)

In the Black-Scholes framework, N(d1) and N(d2) serve distinct purposes and should not be confused. Although both are cumulative distribution functions of the standard normal distribution, their roles in option pricing differ.

N(d1) is closely related to Delta, which measures the sensitivity of the option price to changes in the stock price. Delta acts as a hedge ratio, indicating the amount of the underlying asset needed to hedge the option position. Specifically, Delta quantifies how much the option price changes for a small change in the stock price (∂C/∂S). In the Black-Scholes model, Delta is represented as N(d1). 

However, N(d1) is not the probability of the option ending in the money (ITM). Instead, it serves as a risk-adjusted measure that helps determine how many shares of the underlying stock should be held to hedge the option effectively under the risk-neutral assumption. Think of N(d1) more as a factor that adjusts the hedge ratio, rather than a straightforward probability.

To clarify, N(d1) represents how the option price is expected to change as the stock price moves, given the exposure to the underlying stock in a risk-neutral world. While some might loosely interpret N(d1) as a “probability,” its main function is to guide the hedging strategy by providing the Delta.

N(d2), in contrast, is associated with the risk-neutral probability of the option expiring ITM. It represents the adjusted probability that the stock price will be greater than the strike price at expiration, under the assumptions of the Black-Scholes model. This interpretation of N(d2) as a probability is distinct from that of N(d1). While N(d1) relates to the sensitivity of the option price (Delta), N(d2) estimates the likelihood of an ITM payoff at maturity.

Distinguishing N(d1) from N(d2)

Understanding the difference between N(d1) and N(d2) is crucial for correctly applying the Black-Scholes model. Although both terms play a role in option pricing, they are fundamentally different. N(d1) is the Delta of the call option, representing the rate of change of the option price with respect to the stock price. 

It functions as a hedge ratio, indicating how much of the underlying stock you need to hold to neutralize the option’s sensitivity to price changes. N(d2), on the other hand, is the risk-neutral probability that the option will expire ITM. It takes into account the option’s strike price, current stock price, time to maturity, volatility, and the risk-free rate to estimate the chance of a positive payoff under the risk-neutral framework.

Key Takeaways:

1. Black-Scholes Model: Prices European options in a risk-neutral world where assets grow at the risk-free rate, using N(d1) and N(d2) as key parameters.
2. N(d1) and Delta: N(d1) is linked to Delta, indicating how sensitive the option price is to stock price changes. Delta also acts as a hedge ratio, not the probability of ending ITM.
   
3. N(d2) as Risk-Neutral Probability: N(d2) represents the risk-neutral probability of the option expiring ITM, reflecting the likelihood of the stock price ending above the strike price.
   
4. Distinct Roles: N(d1) helps with hedging and sensitivity, while N(d2) estimates the probability of expiring ITM. Their distinct roles are crucial for accurate option pricing.