The Greeks are sensitivities
Each Greek is the sensitivity of an option’s value to one input: delta to the underlying price, gamma to delta itself, vega to volatility, theta to time, and rho to interest rates.
Geometrically, delta is the slope of the value curve and gamma is its curvature:
Delta is the slope of the value curve; gamma is its curvature. The Greeks are simply how value responds to each market factor.
Two ways to compute them
- Analytic (closed-form): for options priced with a formula like Black-76, the Greeks have closed-form expressions, fast and exact for that model.
- Finite difference (bump-and-revalue): nudge an input by a small amount, revalue, and measure the change. This works for any instrument, including those priced by Monte Carlo, at the cost of extra revaluations.
Cross-gamma in commodities
Commodity books are exposed to many related contracts, so a single option’s delta to one contract can change when a different contract moves. Cross-gamma captures that interaction, essential for spread and calendar positions, and surfaced in the Greeks mart.
From Greeks to risk
Aggregated across the book, the Greeks feed parametric VaR, hedging decisions, and P&L attribution, which is why they must be computed on the same live positions the desk trades.
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