Normal distribution
Definition · Level 11 · Pricing toolkit
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The bell-shaped spread of outcomes around an average, described by that average and its standard deviation σ. About 68% of outcomes fall within 1σ, and 95% within 2σ (1.96σ exactly).
Example
Heights with average 170 cm and σ = 10 cm: 68% of adults are between 160 and 180 cm.
Where Tradecraft teaches it
Level 11 · Pricing toolkit, in the lesson “Maths primer: eˣ, ln and the normal curve”: Continuous compounding, log returns and the normal curve: the three pieces of maths the rest of this level uses.
Related terms
- Continuous compoundingInterest added at every instant instead of once a year: 1 grows to e^(rT) after T years, and 1 due in T years is worth e^(−rT) today (e ≈ 2.718).
- Log returnA return measured as the ln of the price ratio, ln(S₁ ÷ S₀).
- N(x)The chance that a normal variable (average 0, σ = 1) lands below x: N(0) = 0.5, N(1) ≈ 0.841, N(1.645) ≈ 0.95, N(1.96) ≈ 0.975, N(2.33) ≈ 0.99.
- Natural logarithm (ln)The undo button for eˣ: ln(y) is the power of e that gives y.
- 0.4 ruleAt-the-money-forward call or put ≈ 0.4 × S × σ × √T (small dividends, modest σ√T), because 1/√(2π) ≈ 0.4.
- Arbitrage boundsPrice limits any option must respect, or someone locks in a riskless profit: call ≤ S, European put ≤ K·e^(−rT), call ≥ max(0, S − K·e^(−rT))…