Probability has a long list of named distributions, but in practical quantitative finance, eight of them carry the load. Knowing the shape, the moments, and the use cases — and where each fails — separates a working analyst from one who blindly defaults to 'assume normal'.
1. Normal (Gaussian)
X ~ N(μ, σ²)f(x) = (2πσ²)^(-1/2) exp(-(x - μ)² / (2σ²))
The default for daily log returns over short windows. CLT-justified. Closed-form integrals, closed-form quantiles via Φ⁻¹. Symmetric, mesokurtic (kurtosis = 3). Black-Scholes assumes lognormal terminal prices, equivalently normal log returns.
2. Lognormal
X is lognormal if log X is normal. Used for stock prices, asset values, loss-given-default. Strictly positive (impossible to go below zero). Right-skewed; mean exceeds median. Geometric Brownian motion implies lognormal terminal prices.
log X ~ N(μ, σ²)E[X] = exp(μ + σ²/2)Var(X) = (exp(σ²) - 1) exp(2μ + σ²)
3. Student-t
Heavy-tailed alternative to normal. Parameter ν (degrees of freedom) controls tail weight: small ν → fat tails; ν → ∞ → normal. Used for daily returns when normality is too thin. Variance only exists for ν > 2; kurtosis only for ν > 4.
Empirical t-fit for equity returns
Daily SPX and NSE-20 returns are typically well-fit by Student-t with 4-6 degrees of freedom. This implies a kurtosis around 9-12 — three to four times the normal — and tail probabilities at ±4σ that are roughly 100× higher than the normal would predict. This is why VaR models that assume normality systematically under-estimate tail risk.
4. Bernoulli and binomial
Bernoulli(p): single success/failure (default/no-default). Binomial(n, p): sum of n independent Bernoullis. E[X] = np, Var(X) = np(1-p). Used for default counts in homogeneous credit portfolios; binomial trees in option pricing.
5. Poisson
Number of independent events in a fixed interval, with rate λ. PMF: P(X = k) = exp(-λ) λ^k / k!. E[X] = Var(X) = λ. Used to model jump arrivals (Merton jump-diffusion), default events in low-frequency portfolios, large-loss tail events.
6. Exponential
Waiting time between Poisson events. PDF f(x) = λ exp(-λx). Memoryless: P(X > s + t | X > s) = P(X > t). Used for inter-default times, time-to-trigger in barrier options.
7. Beta
On [0, 1] with parameters α, β. Flexible shape — uniform, U-shaped, peaked. Used for recovery rates (LGD), default probabilities under Bayesian priors, anything bounded and continuous.
8. Generalised Pareto
The limiting distribution of excesses over a high threshold (Pickands-Balkema-de Haan theorem). Foundation of extreme-value theory (Module 12). Used for VaR/CVaR estimation in the tail.
Comparison table
- Support [0, ∞): lognormal, exponential, Pareto, gamma.
- Support R: normal, Student-t.
- Support [0, 1]: beta.
- Discrete on {0, 1, ..., n}: binomial. Discrete on {0, 1, ...}: Poisson, negative binomial.
Exercise
You model daily portfolio returns as Student-t with ν = 5, σ = 1.5% (annualised vol about 24%). Compute approximately the 1-day 99% VaR and compare to the normal-based 99% VaR with the same σ.