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119 lines
7.0 KiB
Markdown
119 lines
7.0 KiB
Markdown
# LOGNORMDIST: Log-Normal Distribution CDF
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> *The log-normal CDF models variables whose logarithm is normal — the natural distribution of prices and multiplicative processes.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Numeric |
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| **Inputs** | Source (close) |
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| **Parameters** | `mu` (default 0.0), `sigma` (default 1.0), `period` (default 14) |
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| **Outputs** | Single series (Lognormdist) |
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| **Output range** | Varies (see docs) |
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| **Warmup** | `period` bars |
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| **PineScript** | [lognormdist.pine](lognormdist.pine) |
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- The Log-Normal Distribution CDF transforms a min-max normalized price into the cumulative distribution function of the log-normal distribution, pro...
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- **Trading note:** Log-normal distribution; models multiplicative processes like returns. Foundation of Black-Scholes.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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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.
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## Historical Context
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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.
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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 $[0, 1]$.
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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 $1.5 \times 10^{-7}$.
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## Architecture and Physics
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The computation follows a three-phase pipeline:
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**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 $\ln(0)$.
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**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.
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**Phase 3: Normal CDF** evaluates $\Phi(z)$ using the Abramowitz and Stegun approximation with 5 polynomial coefficients:
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$$\Phi(z) = 1 - \phi(|z|) \cdot (b_1 t + b_2 t^2 + b_3 t^3 + b_4 t^4 + b_5 t^5)$$
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where $t = 1/(1 + 0.2316419|z|)$ and $\phi(z) = e^{-z^2/2}/\sqrt{2\pi}$.
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**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.
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## Mathematical Foundation
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If $X \sim \text{LogNormal}(\mu, \sigma^2)$, then $\ln(X) \sim N(\mu, \sigma^2)$, and the CDF is:
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$$F(x; \mu, \sigma) = \Phi\!\left(\frac{\ln x - \mu}{\sigma}\right), \quad x > 0$$
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where $\Phi$ is the standard normal CDF.
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**Moments of the log-normal distribution:**
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$$E[X] = e^{\mu + \sigma^2/2}$$
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$$\text{Var}(X) = (e^{\sigma^2} - 1) \cdot e^{2\mu + \sigma^2}$$
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$$\text{Skew} = (e^{\sigma^2} + 2)\sqrt{e^{\sigma^2} - 1}$$
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**Standard normal CDF** (Abramowitz and Stegun 7.1.26):
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$$\Phi(z) = 1 - \frac{e^{-z^2/2}}{\sqrt{2\pi}} \sum_{i=1}^{5} b_i t^i, \quad t = \frac{1}{1 + 0.2316419|z|}$$
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with $b_1 = 0.319381530$, $b_2 = -0.356563782$, $b_3 = 1.781477937$, $b_4 = -1.821255978$, $b_5 = 1.330274429$.
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**Parameter constraints**: `period` $> 0$, $\sigma > 0$, $\mu \in \mathbb{R}$. Output is bounded $[0, 1]$.
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```
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LOGNORMDIST(source, period, mu, sigma):
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// Phase 1: min-max normalization
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min_val = min(source[0..period-1])
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max_val = max(source[0..period-1])
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range = max_val - min_val
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x = range > 0 ? (source - min_val) / range : 0.5
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safe_x = max(1e-10, x)
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// Phase 2: log-standardization
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z = (ln(safe_x) - mu) / sigma
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// Phase 3: standard normal CDF
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return normalCdf(z)
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```
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## Performance Profile
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### Operation Count (Streaming Mode)
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Log-Normal CDF = Normal CDF of (ln(x) - mu) / sigma — one log() plus an erfc() evaluation.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Input validation (x > 0; sigma > 0) | 2 | 2 cy | ~4 cy |
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| log(x) | 1 | 8 cy | ~8 cy |
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| z = (log(x) - mu) / sigma | 1 | 4 cy | ~4 cy |
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| Normal CDF via erfc (rational approximation) | 1 | 15 cy | ~15 cy |
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| NaN guard + state update | 1 | 2 cy | ~2 cy |
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| **Total** | **O(1)** | — | **~33 cy** |
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O(1) — reduces to Normal CDF after log transform. erfc() rational approximation dominates; log() is secondary cost.
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### Batch Mode (SIMD Analysis)
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| Operation | Vectorizable? | Notes |
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| :--- | :---: | :--- |
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| log(x) | Partial | _mm256_log_pd with SVML |
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| z normalization | Yes | Vector FMA |
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| erfc() | No | Rational polynomial; scalar |
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Limited vectorization — erfc blocks full SIMD. With SVML log: partial vectorization for the transform step.
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## Resources
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- Galton, F. "The Geometric Mean, in Vital and Social Statistics." Proc. Royal Society, 1879.
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- Aitchison, J. & Brown, J.A.C. "The Lognormal Distribution." Cambridge University Press, 1957.
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- Black, F. & Scholes, M. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, 1973.
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- Abramowitz, M. & Stegun, I. "Handbook of Mathematical Functions." NBS Applied Mathematics Series 55, 1964. Formula 7.1.26.
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- Limpert, E., Stahel, W. & Abbt, M. "Log-normal Distributions across the Sciences: Keys and Clues." BioScience, 2001. |