In the Playground's Time Machine, you'll see a scenario broken into four dollar contributions:
- Δ contribution — dollars from the underlying moving
- Γ contribution — dollars from your delta shifting during the move
- Θ contribution — dollars from time passing
- Vega contribution — dollars from implied volatility changing
Plus a fifth number — the residual— which we'll come back to.
The formula behind each contribution
Each contribution comes from a first-order Taylor approximation. In plain terms:
- Δ contribution = initial Delta × dS
- Γ contribution = ½ × initial Gamma × dS²
- Θ contribution = initial Theta × days forward
- Vega contribution = initial Vega × Δ IV (in vol-points)
The Playground shows the formula next to each row, so you can watch the math happen with your actual numbers.
Why the residual exists
Greeks measure sensitivity at a specific point. As soon as the underlying moves, those sensitivities change too — gamma itself depends on price. Higher-order effects (like “the change in gamma as spot moves”) aren't captured by the first-order contributions. The residual is what's left over.
A small residual (a few dollars) means the first-order Greeks tell the whole story. A large residual (more than ~10% of actual P&L) means the shock was too big for the linear approximation — and the Playground shows an amber warning to tell you so.
Why this beats a payoff diagram
A payoff diagram shows you the destination. Decomposition shows you the journey. If a trade lost money, decomposition answers questions like: “Did it lose because the stock moved against me, or because IV crushed after I put the trade on?” Those are very different lessons.