pricing
Normal Distribution PDF
The standard normal probability density function — the famous bell curve. Used heavily in calculating Greeks like Gamma and Vega.
Formula
φ(x) = (1 / √(2π)) · e⁻ˣ²ᐟ²
Variables
- φ(x)
- Probability density at x
- x
- Number of standard deviations from mean
- π
- ≈ 3.14159
- e
- ≈ 2.71828 (Euler's number)
Worked Example
Inputs
- x
- 0 (at the mean)
Calculation Steps
- 1
φ(0) = (1 / √(2π)) · e⁰ - 2
φ(0) = 1 / 2.5066 = 0.3989
Result: φ(0) ≈ 0.3989 — the peak of the bell curve
Intuition
The peak at x=0 means outcomes near the mean are most probable. The density drops rapidly beyond ±2σ, which is why 95% of outcomes fall within 2 standard deviations.