Probability density fitted to real return data. Boltzmann (orange) has heavier tails than Normal (blue) — it catches crash risk that Normal misses.
Boltzmann / Laplace
Normal / Gaussian (dashed)
10,000 simulated 1-day outcomes for a $1M portfolio sampled from the Boltzmann distribution. Red = worst 1%, amber = worst 1–5%.
Worst 1% — 99% VaR
1–5% — 95% VaR
Normal range
BOLTZMANN VaR — conservative
NORMAL VaR — optimistic baseline