LOGNORMDIST: Log-Normal Distribution CDF

Property	Value
Category	Numeric
Inputs	Source (close)
Parameters	`mu` (default 0.0), `sigma` (default 1.0), `period` (default 14)
Outputs	Single series (Lognormdist)
Output range	Varies (see docs)
Warmup	`period` bars

TL;DR

The Log-Normal Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the log-normal distribution, pro...
Parameterized by mu (default 0.0), sigma (default 1.0), period (default 14).
Output range: Varies (see docs).
Requires period bars of warmup before first valid output (IsHot = true).
Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Log-Normal Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the log-normal distribution, producing an output in [0, 1]. A random variable X is log-normally distributed when \ln(X) follows a normal distribution. This makes the log-normal CDF natural for financial data, where multiplicative returns (log-returns) are approximately normally distributed. The indicator min-max normalizes the source to (0, 1], takes the natural logarithm, standardizes by parameters \mu and \sigma, then evaluates the standard normal CDF. The result emphasizes values near the bottom of the recent range (where the logarithm diverges) and compresses values near the top.

Historical Context

The log-normal distribution was first described by Francis Galton (1879) and formalized by Donald McAlister (1879) in a paper read to the Royal Society. It gained prominence in finance through Louis Bachelier's thesis (1900) on price speculation and was later adopted as the foundation of the Black-Scholes option pricing model (1973), where stock prices are assumed to follow geometric Brownian motion, making the price at any future time log-normally distributed.

The log-normal assumption remains the default model in quantitative finance despite well-documented violations (fat tails, volatility clustering). Its mathematical tractability and the economic argument that prices cannot go negative (the log-normal support is (0, \infty)) make it a reasonable first approximation. The CDF form used here provides a probability integral transform: if the normalized price truly followed a log-normal distribution with parameters \mu and \sigma, the output would be uniformly distributed on [0, 1].

The implementation reduces the log-normal CDF to the standard normal CDF through the substitution z = (\ln x - \mu)/\sigma, then uses the Abramowitz and Stegun rational approximation (formula 7.1.26) for \Phi(z), achieving accuracy of approximately 1.5 \times 10^{-7}.

Architecture and Physics

The computation follows a three-phase pipeline:

Phase 1: Min-max normalization scans period bars for extrema, maps the current source to x \in [0, 1]. A floor of 10^{-10} is applied to prevent \ln(0).

Phase 2: Log-standardization computes z = (\ln x - \mu) / \sigma. With default \mu = 0, \sigma = 1, this simplifies to z = \ln(x). Since x \in (0, 1], z \in (-\infty, 0], so default parameters place most output in [0, 0.5]. Shifting \mu negative or increasing \sigma spreads the output across the full [0, 1] range.

Phase 3: Normal CDF evaluates \Phi(z) using the Abramowitz and Stegun approximation with 5 polynomial coefficients:

\Phi(z) = 1 - \phi(|z|) \cdot (b_1 t + b_2 t^2 + b_3 t^3 + b_4 t^4 + b_5 t^5)

where t = 1/(1 + 0.2316419|z|) and \phi(z) = e^{-z^2/2}/\sqrt{2\pi}.

Parameter effects: \mu shifts the inflection point of the S-curve along the logarithmic axis. \sigma controls the steepness: small \sigma produces a sharp transition, large \sigma produces a gradual one. For financial applications, \mu = -1, \sigma = 0.5 centers the CDF near the geometric midpoint of the [0, 1] range.

Mathematical Foundation

If X \sim \text{LogNormal}(\mu, \sigma^2), then \ln(X) \sim N(\mu, \sigma^2), and the CDF is:

F(x; \mu, \sigma) = \Phi\!\left(\frac{\ln x - \mu}{\sigma}\right), \quad x > 0

where \Phi is the standard normal CDF.

Moments of the log-normal distribution:

E[X] = e^{\mu + \sigma^2/2} \text{Var}(X) = (e^{\sigma^2} - 1) \cdot e^{2\mu + \sigma^2} \text{Skew} = (e^{\sigma^2} + 2)\sqrt{e^{\sigma^2} - 1}

Standard normal CDF (Abramowitz and Stegun 7.1.26):

\Phi(z) = 1 - \frac{e^{-z^2/2}}{\sqrt{2\pi}} \sum_{i=1}^{5} b_i t^i, \quad t = \frac{1}{1 + 0.2316419|z|}

with b_1 = 0.319381530, b_2 = -0.356563782, b_3 = 1.781477937, b_4 = -1.821255978, b_5 = 1.330274429.

Parameter constraints: period > 0, \sigma > 0, \mu \in \mathbb{R}. Output is bounded [0, 1].

LOGNORMDIST(source, period, mu, sigma):
    // Phase 1: min-max normalization
    min_val = min(source[0..period-1])
    max_val = max(source[0..period-1])
    range = max_val - min_val
    x = range > 0 ? (source - min_val) / range : 0.5
    safe_x = max(1e-10, x)

    // Phase 2: log-standardization
    z = (ln(safe_x) - mu) / sigma

    // Phase 3: standard normal CDF
    return normalCdf(z)

Performance Profile

Operation Count (Streaming Mode)

Log-Normal CDF = Normal CDF of (ln(x) - mu) / sigma — one log() plus an erfc() evaluation.

Operation	Count	Cost (cycles)	Subtotal
Input validation (x > 0; sigma > 0)	2	2 cy	~4 cy
log(x)	1	8 cy	~8 cy
z = (log(x) - mu) / sigma	1	4 cy	~4 cy
Normal CDF via erfc (rational approximation)	1	15 cy	~15 cy
NaN guard + state update	1	2 cy	~2 cy
Total	O(1)	—	~33 cy

O(1) — reduces to Normal CDF after log transform. erfc() rational approximation dominates; log() is secondary cost.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
log(x)	Partial	_mm256_log_pd with SVML
z normalization	Yes	Vector FMA
erfc()	No	Rational polynomial; scalar

Limited vectorization — erfc blocks full SIMD. With SVML log: partial vectorization for the transform step.

Resources

Galton, F. "The Geometric Mean, in Vital and Social Statistics." Proc. Royal Society, 1879.
Aitchison, J. & Brown, J.A.C. "The Lognormal Distribution." Cambridge University Press, 1957.
Black, F. & Scholes, M. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, 1973.
Abramowitz, M. & Stegun, I. "Handbook of Mathematical Functions." NBS Applied Mathematics Series 55, 1964. Formula 7.1.26.
Limpert, E., Stahel, W. & Abbt, M. "Log-normal Distributions across the Sciences: Keys and Clues." BioScience, 2001.

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