The d1 Term in Black-Scholes Formula in Simple Terms

The Black-Scholes model is widely known for calculating the theoretical value of European-style options, assuming that stock prices follow a lognormal distribution. Within this model, a key component is $d₁$, a term that captures several important aspects of option pricing. Understanding $d₁$ is crucial for gaining insight into how options behave and are valued.

The Formula for d₁

The $d₁$ term is defined as:

$d₁=\frac{\ln (S₀/X)+(r+\frac{\sigma ²}{2})T}{\sigma \sqrt{T}}$

Where:

• $S₀$: Current stock price
• $X$: Option strike price
• $T$: Time to expiration (in years)
• $r$: Risk-free interest rate (annualized)
• $\sigma$: Volatility of the stock (annualized)

Key Components of d₁

• Natural Logarithm of the Stock to Strike Ratio: The term $\ln (S₀/X)$ measures the relative position of the stock price to the strike price. If $S₀>X$, the logarithm is positive, indicating the option is "in the money." If $S₀<X$, it is negative, meaning the option is "out of the money." The natural logarithm converts the ratio into a scale that reflects compounded returns over time.
• Risk-Free Rate Compensation: The term $rT$ accounts for the time value of money. It reflects the opportunity cost of holding cash instead of investing it in a risk-free asset over the option's lifespan.
• Volatility Adjustment: The term $(\sigma ²/2)T$ adjusts for the expected variance in stock price movements due to volatility $\sigma$. This factor arises from geometric Brownian motion, which governs stock price behavior in the Black-Scholes model.
• Denominator (Volatility Adjusted for Time): The denominator $\sigma \sqrt{T}$ standardizes $d₁$ by accounting for the stock's volatility and the time to expiration. It ensures that $d₁$ remains a dimensionless measure.

N(d₁): The Option’s Delta

The cumulative normal distribution function $N(d₁)$ gives the probability that a standard normal variable is less than $d₁$. In the Black-Scholes model, $N(d₁)$ represents the delta of a European call option, which measures the sensitivity of the option's price to changes in the underlying stock price.

Delta indicates how much the option's price is expected to change for a $1 change in the stock price. For example, if $N(d₁)=0.4$, the option's price would increase by $0.40 for every $1 increase in the stock price.

Common Misconception: N(d₁) vs. N(d₂)

While $N(d₁)$ represents the delta, $N(d₂)$ is the risk-neutral probability of the option expiring in the money. The distinction is critical:

• $N(d₁)$: Measures the sensitivity of the option price to changes in the stock price.
• $N(d₂)$: Represents the probability of the option being exercised profitably at expiration under a risk-neutral framework.

Key Insights

The $d₁$ term in the Black-Scholes model encapsulates the key factors affecting option pricing, including stock price, strike price, volatility, time, and interest rates. $N(d₁)$, as the delta, provides insights into the option’s sensitivity and serves as a cornerstone for hedging strategies. Distinguishing $N(d₁)$ from $N(d₂)$ helps clarify the distinct roles these terms play in understanding and managing options.