Why Delta Is Not the Probability of an Option Expiring in the Money Simply Explained

The Black-Scholes model is a cornerstone of modern financial mathematics, providing a framework for pricing European call and put options. It assumes a risk-neutral environment, where all assets grow at the risk-free rate, $r$, eliminating arbitrage opportunities. This assumption enables the pricing of options based solely on their probabilistic payoffs.

The Black-Scholes Formula

The price of a European call option under the Black-Scholes model is given by:

$$C=S\cdot N(d_1)-X\cdot e^{-rT}\cdot N(d_2),$$

where:

• $C$: Price of the call option
• $S$: Current stock price
• $X$: Strike price of the option
• $r$: Risk-free interest rate
• $T$: Time to maturity
• $N(d_1)$ and $N(d_2)$: Cumulative distribution functions of the standard normal distribution evaluated at $d_1$ and $d_2$

The parameters $d_1$ and $d_2$ are defined as:

$$d_1=\frac{\ln (S/X)+(r+{\sigma }^2/2)\cdot T}{\sigma \cdot \sqrt{T}},$$

$$d_2=d_1-\sigma \cdot \sqrt{T},$$

where:

• $\sigma$: Volatility of the underlying stock
• $\ln$: Natural logarithm
• $\sqrt{T}$: Square root of the time to maturity

Understanding $N(d_1)$ and $N(d_2)$

A common source of confusion in the Black-Scholes model arises from the distinct roles of $N(d_1)$ and $N(d_2)$, both of which are cumulative distribution functions of the standard normal distribution.

$N(d_1)$: Delta and Hedge Ratio

In the Black-Scholes framework, $N(d_1)$ represents the Delta of the call option, which measures the sensitivity of the option price to changes in the stock price. Mathematically, Delta is defined as:

$$\mathrm{\Delta }=\frac{\partial C}{\partial S},$$

where $\mathrm{\Delta }$ quantifies how much the option price changes for a small change in the stock price. $N(d_1)$ serves as the hedge ratio, indicating the number of shares of the underlying stock required to hedge the option position effectively in a risk-neutral world.

It is important to note that $N(d_1)$ is not the probability of the option expiring in the money (ITM). Instead, it is a risk-adjusted measure reflecting the exposure of the option to the underlying stock.

$N(d_2)$: Risk-Neutral Probability of Expiring ITM

In contrast to $N(d_1)$, $N(d_2)$ represents the risk-neutral probability that the option will expire ITM. It estimates the likelihood that the stock price will exceed the strike price at expiration, taking into account the risk-free rate, volatility, and time to maturity. This probability is adjusted under the risk-neutral measure to ensure the absence of arbitrage.

$$N(d_2)=P(S_T>X\mid \text{Risk-neutral world}),$$

where $S_T$ denotes the stock price at expiration.

Practical Interpretation

While $N(d_1)$ and $N(d_2)$ both stem from the cumulative normal distribution, their interpretations are distinct:

Understanding the distinction between $N(d_1)$ and $N(d_2)$ is crucial for accurate application of the Black-Scholes model. Misinterpreting these terms can lead to errors in hedging and pricing.

Limitations of the Black-Scholes Model

Despite its widespread use, the Black-Scholes model has several limitations:

• Constant Volatility: The model assumes constant volatility, which may not hold in real markets where volatility can be stochastic or vary with time.
• No Dividends: The model does not account for dividends unless explicitly adjusted.
• European Options Only: The model applies strictly to European options, which can only be exercised at expiration.

Extensions to the Black-Scholes framework, such as stochastic volatility models or jump-diffusion processes, address these limitations by incorporating more realistic market dynamics.