Gaussian vs Power Law

Distribution Theory

Gaussian Distribution: The bell curve, characterized by mean (μ) and standard deviation (σ). The probability density function is: f(x) = (1/σ√2π) × e^(-(x-μ)²/2σ²)

Power Law Distribution: Heavy-tailed distribution where P(X > x) ∝ x^(-α). Common in financial markets, natural disasters, and wealth distribution.

Student's t Distribution

Student's t bridges the gap between Gaussian and heavy-tailed distributions. It has one key parameter: degrees of freedom (ν or df).

• df → ∞: Converges to Gaussian (thin tails)
• df = 30: Nearly Gaussian
• df = 4-6: Moderately fat tails (typical for stocks)
• df = 3: Very fat tails (no finite variance for df ≤ 2)
• df = 1: Cauchy distribution (extremely fat tails)

Estimating df from Kurtosis

The kurtosis measures how heavy the tails are. For Student's t with df > 4:

This formula lets us estimate df from observed data:

• Kurtosis = 3 → df = ∞ (Gaussian)
• Kurtosis = 4 → df = 10
• Kurtosis = 5 → df = 7
• Kurtosis = 6 → df = 6
• Kurtosis = 9 → df = 5

Why This Matters

Key Insight: Under Gaussian assumptions, a 6σ event should happen once every 1.5 million years. With Student's t (df=4), the same 6σ event happens roughly every 6 years!

Most financial return data has kurtosis between 5-15, implying df of 3-7. This means traditional Gaussian risk models (VaR, Black-Scholes) systematically underestimate tail risk.

Geometric Brownian Motion

GBM models stock prices as: dS = μSdt + σSdW, where W is a Wiener process.

The solution is: S(t) = S(0) × exp((μ - σ²/2)t + σW(t))

Key Properties:

• Log returns are normally distributed
• Prices are always positive
• Volatility is constant (unrealistic in practice)

Fat-Tailed Shocks: Using Student's t distribution instead of normal produces more realistic extreme movements.

Random Walk & Efficient Markets

A random walk is the mathematical formalization of taking successive random steps.

Key Properties:

• Each step is independent
• Expected position after n steps: E[X_n] = n(2p-1) where p is up probability
• Variance grows linearly with steps: Var(X_n) = 4np(1-p)

Efficient Market Hypothesis: If markets are efficient, price changes should be unpredictable - like a random walk. Past patterns provide no edge.

Leverage and Fat Tails

Research by Thurner, Farmer, and Geanakoplos shows how leverage amplifies market movements:

• Value Investors: Buy when prices fall, sell when they rise (stabilizing)
• Trend Followers: Buy rising prices, sell falling prices (destabilizing)
• Leverage Effect: When leveraged positions face margin calls, forced selling creates a cascade that amplifies the original shock

Key Insight: Even with Gaussian fundamental shocks, leverage creates power-law distributed returns through the feedback mechanism of margin calls.

GARCH and Volatility Clustering

GARCH (Generalized Autoregressive Conditional Heteroskedasticity) captures the empirical fact that volatility clusters:

σ²(t) = ω + α×ε²(t-1) + β×σ²(t-1)

• ω: Long-run variance
• α: How much yesterday's shock affects today's variance
• β: How persistent volatility is

Stylized Fact: α + β close to 1 means volatility is highly persistent - calm periods and stormy periods can last for extended times.

Black-Scholes and the Volatility Smile

The Black-Scholes model assumes log-returns are normally distributed. Under this assumption:

• Implied volatility should be constant across strikes
• Extreme moves are very rare
• Options are relatively cheap for out-of-the-money strikes

Reality (Volatility Smile):

• OTM puts trade at higher implied volatility (crash protection premium)
• The "smile" or "smirk" pattern exists because markets price in fat tails
• After 1987 crash, the smile became permanently embedded in options markets

Wide Moats and Put Selling

A "wide moat" refers to a durable competitive advantage that protects a company's profits:

• Network Effects: Visa, Mastercard, Meta
• Switching Costs: Adobe, Microsoft, Salesforce
• Cost Advantages: Costco, GEICO
• Intangible Assets: Coca-Cola, Disney

Why This Matters for Put Sellers:

• Wide moat stocks have lower permanent loss risk
• They recover faster after crashes
• Being "assigned at the bottom" is a buying opportunity, not a disaster
• No moat stocks may never recover to assignment price