Black-Scholes-Mertonbeginner

Z-table and $N(x)$

A z-table lists pre-computed values of the standard normal CDF $N(x) = P(Z \leq x)$ . In BSM, you compute $d_1$ and $d_2$ , then look up $N(d_1)$ and $N(d_2)$ to finish the call or put price. The CDF has no elementary antiderivative, so tabulated values (or numerical routines) are the practical way to evaluate it.

Why it matters

Think of the z-table as a pre-computed answer key for the bell-curve integral. A z value like 0.43 means 0.43 standard deviations above the mean. The row gives the first decimal (0.4) and the column gives the second (0.03), and the cell shows the area to the left, $N(0.43) = 0.6664$ . For negative $z$ , use the symmetry identity $N(-x) = 1 - N(x)$ rather than a second table.

Formulas

Look-up rule

N(z) \text{ from table where row} = z \text{ to first decimal, column} = z \text{ to second decimal}

Negative argument

N(-z) = 1 - N(z)

Lets a one-sided table cover negative values without a second sheet.

Worked examples

Scenario

BSM has produced $d_1 = 0.75$ . Find $N(d_1)$ .

Solution

Locate row 0.7 and column 0.05. The cell reads 0.7734, so $N(0.75) = 0.7734$ . In a call with $S_0 =$ K = 50$, that means a delta of roughly 0.77 shares per option.

Scenario

Find $N(-0.32)$ using a one-sided z-table.

Solution

Look up $N(0.32) = 0.6255$ , then apply symmetry, $N(-0.32) = 1 - 0.6255 = 0.3745$ . In a BSM put, this would correspond to $N(-d_1)$ multiplying $S_0$ .

Common mistakes

✗You must memorise the z-table. In exams, the table is provided. The skill to practise is reading the row and column correctly and applying $N(-x) = 1 - N(x)$ .
✗Interpolation is always required. Most undergraduate problems use $z$ values to two decimals, so a direct look-up works. Interpolation only matters when you need a value such as $N(0.435)$ to four decimals.

Revision bullets

•Pre-computed values of $N(x)$
•Row, first decimal of $z$ ; column, second decimal
•Symmetry, $N(-x) = 1 - N(x)$
•Provided in exams, no need to memorise
•Inputs $d_1$ and $d_2$ come from BSM

Quick check

Using a one-sided z-table that lists only positive arguments, $N(-0.5)$ is best found by

Connected topics

Normal Dist.d1 & d2

In learning paths

Pricing Models Path

Sources

NIST e-Handbook, Normal Distribution
National Institute of Standards and Technology. "Normal Distribution." NIST/SEMATECH e-Handbook of Statistical Methods.
Authoritative reference for the standard normal CDF and tabulated values.
Hull (2022), Appendix B
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Includes the standard normal table used throughout the book's BSM and Greek calculations.
Abramowitz & Stegun (1972), §26.2
Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, 1972.
Classic source for high-accuracy polynomial approximations to $N(x)$, the basis for most software implementations.

How to cite this page

Dr. Phil's Quant Lab. (2026). Z-table and $N(x)$. Derivatives Atlas. https://phucnguyenvan.com/concept/z-table

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