Black-Scholes-Mertonbeginner

Normal Distribution Intuition

The standard normal distribution is the bell curve with mean zero and variance one. Its cumulative distribution function (CDF) $N(x) = P(Z \leq x)$ gives the probability that a standard normal variable $Z$ falls at or below $x$ . In BSM, $N(d_1)$ and $N(d_2)$ are the two pieces of the call formula. $N(d_2)$ is the risk-neutral probability that the call finishes in the money, and $N(d_1)$ is the option's delta, the hedge ratio in shares of stock per option (Hull, 2022, §15.6).

Why it matters

BSM does not assume that stock prices are normal. It assumes that log returns $\ln(S_T / S_0)$ are normal, which makes $S_T$ lognormal. That distinction matters because lognormal prices cannot go negative, but they can grow unboundedly on the upside. The bell curve says that returns far from the mean are rare. A 2-sigma daily move on the S&P 500 has roughly a 2.3% chance per day under a normal assumption, and the empirical frequency is higher, which is why traders watch the fat tails.

Formulas

Standard normal CDF

N(x) = P(Z \leq x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-z^2/2}\, dz

No elementary antiderivative, so values are tabulated or computed numerically.

Symmetry

N(-x) = 1 - N(x)

The bell curve is symmetric about zero, which lets a one-sided z-table cover negative arguments.

Worked examples

Scenario

Read the following values from a standard normal table, $N(0)$ , $N(1)$ , $N(-1)$ , $N(1.96)$ .

Solution

$N(0) = 0.5000$ . $N(1) = 0.8413$ , so 84.13% of the mass lies at or below one standard deviation above the mean. $N(-1) = 1 - 0.8413 = 0.1587$ . $N(1.96) = 0.9750$ , the textbook value behind the 95% two-sided confidence interval.

Scenario

Inside BSM, what do $N(d_1)$ and $N(d_2)$ mean numerically?

Solution

Suppose $d_1 = 0.62$ and $d_2 = 0.32$ . Then $N(d_1) = 0.7324$ , so a call writer hedges with 0.7324 shares per option. $N(d_2) = 0.6255$ , the risk-neutral probability the call finishes in the money. The two are different because $N(d_1)$ also reflects the conditional expected payoff of the stock given exercise.

Common mistakes

✗BSM assumes normal stock prices. It assumes lognormal stock prices, equivalently normal log returns. A normal price model would allow negative prices, which is nonsense for limited-liability equity.
✗ $N(d_2)$ is the real-world probability of exercise. It is the risk-neutral probability. The real-world probability uses the actual drift $\mu$ rather than the risk-free rate $r$ and is typically higher for calls on stocks with positive risk premia.

Revision bullets

•Standard normal has mean 0 and variance 1
• $N(x)$ is the CDF, area to the left of $x$
•BSM assumes lognormal prices, normal log returns
•Symmetry, $N(-x) = 1 - N(x)$
• $N(d_2)$ is risk-neutral probability of exercise

Quick check

$N(-1.5)$ equals

BSM assumes that

Connected topics

Z-table d1 & d2

In learning paths

Pricing Models Path

Sources

Hull (2022), §15.6
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Explains how $N(d_1)$ and $N(d_2)$ enter the BSM call formula and the meaning of each as a hedge ratio and risk-neutral probability.
Black & Scholes (1973), JPE
Black, Fischer, and Myron Scholes. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81, no. 3 (1973): 637-654.
Original BSM paper, where the lognormal price assumption and the normal CDF first appear as the building blocks of an option price.
NIST Engineering Statistics Handbook
National Institute of Standards and Technology. "Normal Distribution." NIST/SEMATECH e-Handbook of Statistical Methods.
Authoritative reference for the standard normal CDF, including tabulated values and the symmetry identity used in z-tables.

How to cite this page

Dr. Phil's Quant Lab. (2026). Normal Distribution Intuition. Derivatives Atlas. https://phucnguyenvan.com/concept/normal-distribution

← Back to the atlas See in the network →