Second-order Greeks

Definition

Second-order Greeks measure how the FIRST-order Greeks themselves move — the risk of the risk. The platform tracks four: VANNA (delta’s sensitivity to volatility / vega’s to spot), CHARM (delta’s decay with time), VOMMA (vega’s sensitivity to volatility), and SPEED (gamma’s sensitivity to spot). They matter because hedges built on first-order Greeks drift as conditions move — and near expiry or in stress, they drift fast enough to dominate.

How it works / structure

• Vanna (∂Δ/∂σ = ∂vega/∂S): OTM options gain delta as IV rises; books hedged flat at one IV are directional at another. At index scale, dealer vanna flows (re-hedging as IV moves) are a documented mechanical coupling between vol and spot — part of why vol-up/spot-down moves feed themselves in equity indexes.
• Charm (∂Δ/∂t): OTM deltas bleed toward 0 and ITM toward ±1 as expiry approaches; overnight delta drift on short-dated books is charm — a position hedged at the close is not hedged at the open (opt-0dte-mechanics makes charm intraday- scale).
• Vomma (∂vega/∂σ): OTM options have vega that grows as vol rises — long-wing structures gain vega exactly when vol explodes (why tails “convex in vol”), short wings the reverse.
• Speed (∂Γ/∂S): gamma’s cliff steepness near the strike at expiry; the formal name for the expiry gamma zone’s treachery (greek-gamma).
• Engine usage: full-repricing scenarios (risk-scenario-analysis) capture all of these implicitly — the named Greeks are the DIAGNOSTIC decomposition explaining WHY the scenario P&L looks the way it does (port-portfolio-greeks aggregation).

When it applies

Books with short-dated options (charm/speed scale inversely with time), skewed wing positions (vanna/vomma live in the wings — opt-volatility-skew), delta-hedging programs whose rebalance triggers should anticipate drift (mgmt-delta-hedging), and index-level flow analysis where dealer second-order re-hedging is a market-mechanics input.

Risk profile & failure modes

• First-order illusion of safety: “delta-neutral, vega-neutral” books can carry large vanna/vomma — neutral at today’s surface, exposed to any surface move; the 2018-style vol events repriced exactly these residuals.
• Decomposition overconfidence: Taylor-expansion Greeks approximate small moves; large moves need full repricing — the Greeks explain, scenarios measure.
• Complexity for its own sake: retail-scale defined-risk books rarely need this layer; the platform surfaces it where position size and expiry proximity make it material, not as decoration.

Evidence & limits

Definitions and derivations are standard derivatives mathematics (Hull is the reference exposition). Dealer-flow effects attributed to vanna/charm at index level are supported by practitioner research and documented hedging mechanics; precise flow magnitudes are estimates. The Greeks themselves are model outputs — they inherit opt-pricing-models assumptions.

Falsifiable-thesis examples

Illustrations only, not signals:

• “This book’s overnight delta drift (charm estimate) stays under 0.5% of notional per session through expiry week” — falsified by the daily hedge-slippage record.
• “Adding vanna-aware rebalance triggers reduces the hedged book’s P&L variance vs delta-band-only triggers this quarter” — falsified by the paired replay.

Cross-references

• First-order parents: greek-delta, greek-gamma, greek-vega
• Where they concentrate: opt-volatility-skew (wings), opt-0dte-mechanics (time compression)
• Measurement vs diagnosis: risk-scenario-analysis (full repricing), port-portfolio-greeks
• Hedging application: mgmt-delta-hedging