Gamma, vanna & volga
Explain curvature, cross-risk and smile exposure.
01Compute gamma, vanna and volga
02Interpret second-order P&L
03Diagnose unstable finite differences
Define the exposure before compressing it into a metric.
Second-order Greeks explain why a locally hedged book still gains or loses when state variables move together.
Gamma prices spot curvature.
Vanna couples spot and volatility.
Volga prices volatility convexity.
Fix portfolio, scenarios, horizon, and legal terms.
Large moves, sticky-delta smile dynamics and vol shocks make cross terms material to P&L.
risk reversals
straddles
barriers
Mixed derivatives state which variables and surface dynamics are held fixed.
Aggregate with an explicit measure and convention.
Second-order P&L
Curvature and cross terms complete the local quadratic explain.
Open in AnalyticsShort derivation
From information set to computable quantity
Each line states the information, measure and unit before manipulating the expression.
- 01
Write multivariate Taylor
Expand V(S+ΔS,σ+Δσ) through quadratic order.
- 02
Identify diagonal curvature
Gamma and volga multiply squared shocks with one-half factors.
- 03
Identify cross curvature
Vanna multiplies the joint spot-volatility move.
- 04
Reconcile P&L
Compare first-order and quadratic explains against full repricing.
The result is valid only under the filtration, measure and discretization just made explicit.
Inputs
Γ=∂²V/∂S²Vanna=∂²V/∂S∂σ
Assumptions and limits
- Quadratic expansion still fails for jumps and barriers.
- Higher derivatives amplify calibration noise.
Volga
Volga measures how vega itself changes with volatility.
Open in AnalyticsReconcile valuation, risk, and model limitations.
Use analytic formulas where available and symmetric two-dimensional bumps for mixed derivatives.
Declare sticky-strike or sticky-delta before bumping spot; the surface rule changes vanna materially.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Typed domain validation
- Deterministic seeded computation
- Readout plus invariant
Gamma, vanna & volga
Reproduce the governing quantity, then challenge it with an invariant.
import numpy as np def deltavapproxdeltad(x: np.ndarray) -> float: x = np.asarray(x, dtype=float) assert np.isfinite(x).all() return float(np.mean(x)) sample = np.array([0.8, 1.0, 1.2])value = deltavapproxdeltad(sample)assert sample.min() <= value <= sample.max()print(f"value={value:.6f}")Move the state. Challenge the equation.
Gamma, vanna & volga
Move spot, volatility and horizon. Prices, desk-unit Greeks and hedge residuals share one pricing state.
Local option sensitivity across spot, in declared desk units.
- Gamma
Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.
Turn exposure into a controlled decision.
“Delta hedged is not curvature hedged.”
surface dynamics
joint shocks
Declare sticky-strike or sticky-delta before bumping spot; the surface rule changes vanna materially.
RISKgamma
vanna
volga
- Validate market state and timestamp
- Recompute the baseline
- Run a controlled perturbation
- Explain P&L and residuals
Production failure modes
- Silent convention or measure changes
- Unstable numerics hidden by plausible prices
09MACRO CONNECTIONOpen the transmission channel.
Transmission from state to valuation
The causal chain separates the economic shock from the modelling response.
transmitsmoves two state axes
transmitscreates nonlinear P&L
outputreveals model/surface dynamics
10COMMON PITFALLSOpen the failure checklist.
Omitting one-half on diagonal Taylor terms
Computing vanna under an unstated smile rule
11SOURCES / FURTHER READINGOpen sources and continue the track.
Measure theory, simulation and computational-finance lectures
The lesson uses original prose and a fresh typed implementation; the linked material is a research map, not copied product code.
- Source
- Computational Finance Course
- Author
- L. A. Grzelak
- Ref
- main