Delta
Delta = partial V / partial SDelta is the local first-order price response to the underlying.
Risk
Greeks and Risk Factors
Greeks measure sensitivity of prices to well-defined risk factors like spot, rates, and volatility.
Mental model
A Greek is a local map from a market-risk move to the corresponding price change.
Greeks such as delta, gamma, vega, and rho describe how derivative prices respond to small changes in underlying risk factors.
Risk factors are chosen coordinates (spot, curve pillars, vol points) that risk limits and aggregation are built on.
Explore the mechanism
Which market move is each sensitivity measuring, and over what local range is that approximation useful?
This curve shows option PV as a function of a single risk factor. At the current point you see the delta tangent and a faint delta+gamma approximation band. Use the controls to switch factor type and see how a small bump propagates into price.
Spot S = 100.0 · Δ ≈ 0.586 · Γ ≈ 9.81e-1
Numbers
Current Spot S = 100.0
PV ≈ 4.8151 · Δ ≈ 0.5864 · Γ ≈ 9.81e-1
Exact ΔPV (up bump) ≈ 1.0770
Δ-only ≈ 0.5864 · Δ+Γ ≈ 1.0770
Interpretation
Delta is the local slope of PV with respect to a chosen risk factor; gamma is the local curvature. For a tiny bump, delta alone explains the move; as the bump grows, curvature starts to matter and the delta+gamma approximation tracks the true PV better.
The same geometry applies whether your factor is spot, a rate pillar, or a vol point. The practitioner’s view: Greeks are not mystical, they are just derivatives along a chosen factor coordinate system. Good risk reports start by choosing sensible coordinates and then using these local slopes and curvatures consistently across the book.
UNDERSTAND
What to notice
In a library, the risk engine wraps pricers with bump logic or AAD to compute Greeks consistently across products.
A coherent factor set makes risk reports and limits comparable across desks and asset classes.
The quantitative structure
Notation
V- instrument value
S- spot or underlying level
sigma- volatility
Gamma
Gamma = partial^2 V / partial S^2Gamma measures how delta changes as spot moves.
Vega
Vega = partial V / partial sigmaVega measures local sensitivity to the model's volatility input.
A small example
Delta-gamma approximation
A position has delta 0.50 and gamma 0.02 per price unit. Spot rises by 2.
- Delta contribution: 0.50 * 2 = 1.00.
- Gamma contribution: 0.5 * 0.02 * 2^2 = 0.04.
- Approximate value change: 1.00 + 0.04 = 1.04.
ResultCurvature adds 0.04 beyond the first-order delta estimate.
Where it appears
- hedging option portfolios
- setting risk limits
- explaining daily valuation changes
Where the intuition breaks
What this view leaves out
- Greeks are local derivatives evaluated at a model state; they can change rapidly and omit jumps, cross-effects, liquidity, and model error.
Common mistakes
“A delta hedge removes all option risk.”
Not quite.Delta neutralizes a local first-order spot move; gamma, volatility, time, jumps, and rebalancing costs remain.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Greeks and Risk Factors
Greeks and Risk FactorsYOU ARE HERE
Built onBlack–Scholes Option PricingBlack–Scholes provides the canonical analytical setting for understanding option Greeks.
Supported byBlack-76 Pricing for Forwards and FuturesBlack-76 produces sensitivities of forward-option value to forward price, volatility, time and discounting.
SupportsP&L Explain and AttributionRisk-factor sensitivities provide the first-order building blocks for explaining daily valuation changes.
SupportsScenario and Stress EngineGreeks give local risk approximations, while scenario engines revalue under larger joint moves.
Validated byFinite-Difference Checks for GreeksBumping inputs and repricing provides an independent numerical check of reported sensitivities.
Related toDuration and Convexity (Intuition)Duration and convexity are fixed-income sensitivity measures analogous in spirit to first- and second-order Greeks.
Built on · Black–Scholes Option Pricing
Black–Scholes provides the canonical analytical setting for understanding option Greeks.
Explore concept →