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).
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Note
Key Takeaway: While the Normal distribution is the default for teaching, the Student's t-distribution is often the default for practitioners doing real risk modeling.
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