Commmon Statistical Distributions in Finance

Since asset returns are random variables, we need probability distributions to model them.

1. Normal Distribution (Gaussian)

The "Bell Curve".

• Defined by Mean (Mu) and Variance (Sigma^2).
• Pros: Easy to use, mathematically simple (CLT - Central Limit Theorem often applies).
• Cons: Underestimates extreme risks (Thin tails). Does not capture the "crash risk" well.
• Usage: Used in basic VaR models and Portfolio Theory.

2. Log-Normal Distribution

If returns are Normal, then Prices are Log-Normal.

• Property: A variable is Log-Normal if its natural log is Normal.
• Why use it for Prices? Prices cannot be negative (Wait, oil did go negative once, but generally stocks don't). Normal distribution allows negative values (-Infinity to +Infinity), but Log-Normal is bounded at 0 (0 to +Infinity).
• Usage: Used for modeling Stock Prices (not returns).

3. Student's t-Distribution

Similar to Normal but with Fatter Tails.

• It has an extra parameter: Degrees of Freedom (v). Lower v = Fatter tails.
• As v approaches Infinity, it becomes a Normal distribution.
• Usage: Much better for modeling stock returns because it accounts for the higher probability of extreme events (Black Swans).

4. Poisson Distribution

Used for modeling "count" events.

• Usage: Modeling the number of defaults in a bond portfolio or the arrival of buy orders in HFT (High Frequency Trading).