Normal Distribution

Analytical option pricing (Black-Scholes-Merton, Bachelier) and Value-at-Risk (VaR) calculations require standard normal probability functions evaluated to double-precision machine epsilon.

Mathematical Functions

1. Probability Density Function (PDF)

$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}$

2. Cumulative Distribution Function (CDF)

$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{1}{2} u^2} \, du = \frac{1}{2} \left[ 1 + \text{erf}\left( \frac{x}{\sqrt{2}} \right) \right]$

SabiQuant uses Hart’s high-precision rational Chebyshev approximation with absolute error $|\epsilon| < 10^{-16}$ .

3. Inverse CDF / Quantile ( $\Phi^{-1}(p)$ )

Maps cumulative probability $p \in (0, 1)$ to standard score $z$ :

$z = \Phi^{-1}(p)$

Implemented via Peter John Acklam’s rational approximation algorithm with refinement, accurate to $1.15 \times 10^{-9}$ across central and extreme tail regions ( $10^{-30} \le p \le 1 - 10^{-30}$ ).

Rust Code Example

use sabiquant::math::distributions::NormalDistribution;

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let standard_normal = NormalDistribution::standard();

    // Evaluate PDF at x = 0
    let pdf_0 = standard_normal.pdf(0.0);
    println!("Standard Normal PDF(0): {:.6}", pdf_0); // 1 / sqrt(2*pi) ≈ 0.398942

    // Evaluate CDF at z = 1.96 (approx 97.5% two-tailed confidence level)
    let cdf_196 = standard_normal.cdf(1.959964);
    println!("CDF(1.96):              {:.4}%", cdf_196 * 100.0); // 97.50%

    // Inverse CDF: 99% 1-day Value at Risk multiplier
    let p_var99 = 0.99;
    let z_var99 = standard_normal.inverse_cdf(p_var99)?;
    println!("99% VaR Critical Value: {:.4}", z_var99); // 2.3263

    // Arbitrary Mean and Standard Deviation (e.g. Asset Returns)
    let asset_dist = NormalDistribution::new(0.08, 0.15)?; // 8% return, 15% vol
    println!("Prob(Return < 0%):      {:.2}%", asset_dist.cdf(0.0) * 100.0);

    Ok(())
}