Black-Scholes Formula and European Option Pricing › Anatomy of the Black-Scholes Formula

The five inputs to the Black-Scholes formula each affect option prices in intuitive ways. Understanding these relationships builds intuition for how options behave and why traders care about each parameter.

• Stock price ($S$): Higher stock prices increase call values and decrease put values. This is the most direct relationship: a call gives you the right to buy at $K$, so the higher $S$ is relative to $K$, the more valuable that right. If the stock is already trading at $120 and your strike is $100, the option has $20 of intrinsic value plus whatever time value remains. Mathematically, the stock price enters the formula directly in the first term and indirectly through $d_1$ and $d_2$.

• Strike price ($K$): Higher strikes decrease call values and increase put values when the other inputs are fixed. A call has a positive expiry payoff only when the stock finishes above its strike. For example, calls struck at \100$and$$110$have different exercise thresholds. A purchased call's profit also depends on its premium and financing; ignoring financing, expiry break-even is the strike plus the premium. The strike enters both the discounted payment and the log-moneyness in$d_1$and$d_2$.

• Time to expiration ($\tau = T-t$): More remaining time often raises an option's value, but the effect depends on the option type and inputs. A European put can fall in value as remaining maturity increases, even when interest rates are positive. Near expiration, at-the-money options can experience especially rapid time decay. Theta captures this local calendar-time effect with other inputs fixed.

• Risk-free rate ($r$): Higher rates increase call values and decrease put values. The intuition is that buying a call is like a leveraged position in the stock. You get stock exposure while keeping your capital earning interest elsewhere. Higher rates make this implicit financing benefit more useful. Additionally, higher rates reduce the present value of the strike payment, making it cheaper in today's dollars to acquire the stock at expiration.

• Volatility ($\sigma$): Higher volatility increases both call and put values with the other inputs fixed. Under the model's pricing measure, changing volatility leaves the terminal stock-price mean $Se^{r\tau}$ unchanged and changes the dispersion of outcomes. The convex vanilla payoffs benefit from that extra dispersion: a call struck at \100$pays$$20$if the stock finishes at$$120$and zero if it finishes at$$80$. These are payoff examples, not probabilities assigned to exact endpoints. Greater volatility does not make every upper-tail event more likely. At very low volatility the call approaches$\max(S-Ke^{-r\tau},0)$, so its value still depends on carry and remaining maturity.

The quantities $d_1$ and $d_2$ are standardized measures of "moneyness," expressing how far in- or out-of-the-money the option is, adjusted for time and volatility. These quantities serve as the arguments to the cumulative normal distribution function, converting the lognormal distribution of stock prices into a tractable form.

Consider $d_2$ first:

$$d_2 = \frac{\ln(S/K) + (r - \frac{\sigma^2}{2})\tau}{\sigma\sqrt{\tau}}$$

where:

• $S$: stock price
• $K$: strike price
• $r$: risk-free rate
• $\sigma$: volatility
• $\tau$: time to expiration

To understand $d_2$, we need to recall how the stock price evolves under geometric Brownian motion. The numerator $\ln(S/K) + (r - \frac{\sigma^2}{2})\tau$ is the expected log-return of the stock under the risk-neutral measure, relative to the strike. The term $r - \sigma^2/2$ appears rather than just $r$ because of the difference between arithmetic and geometric means. When a variable follows geometric Brownian motion with drift $\mu$, its expected logarithm grows at rate $\mu - \sigma^2/2$, not $\mu$. This correction factor, sometimes called the Itô correction or convexity adjustment, arises from the nonlinear relationship between a variable and its logarithm.

Recall that under geometric Brownian motion with drift $r$, the expected log-price is:

$$\mathbb{E}[\ln S_T] = \ln S + (r - \frac{\sigma^2}{2})\tau$$

where:

• $\mathbb{E}[\ln S_T]$: expected log stock price at time $T$
• $\ln S$: log of current stock price
• $r$: risk-free rate
• $\sigma$: volatility
• $\tau$: time to expiration

So the numerator equals $\mathbb{E}[\ln S_T] - \ln K = \mathbb{E}[\ln(S_T/K)]$. This is the expected value of the log of the ratio of the terminal stock price to the strike. When this is positive, the stock is expected to end up above the strike on average. The denominator $\sigma\sqrt{\tau}$ is the standard deviation of this log-return. The volatility of the log-price after time $\tau$ is $\sigma\sqrt{\tau}$, which reflects the diffusion of prices over time. Thus $d_2$ measures how many standard deviations the expected log-price exceeds the log-strike, which is exactly the input you'd need to find $\mathbb{P}(S_T > K)$ using the normal distribution.

The quantity $d_1$ exceeds $d_2$ by $\sigma\sqrt{\tau}$. Completing the square in the lognormal density shows that $\mathbb{E}^{\mathbb Q}[S_T\mathbf1_{\{S_T>K\}}]=Se^{r\tau}N(d_1)$. This is the stock receipt weighted across all outcomes, with zero contribution when exercise does not occur. To obtain the expectation conditional on exercise, divide by $N(d_2)$ when that probability is positive.

The technical explanation involves a change of measure: for this non-dividend stock, $N(d_1)$ is the probability of the call event $S_T>K$ under the stock-numeraire or share measure. The density change weights outcomes by the terminal stock price, increasing the probability of this upper-tail event. It decreases the probability of the complementary lower-tail event $S_T<K$; the comparison is not the same for put exercise.