Second-Order Greeks
Vanna, volga, charm, color — when they show up in real positions
A market maker has done a perfect Delta hedge on a portfolio of OTM puts. Their Delta exposure is mathematically zero. Overnight, the underlying drops 2% and implied volatility expands by 3 vol points — both moves the BSM model says should be hedged out.
The desk opens to find their P&L is sharply negative — not from Delta (zero by construction), not directly from vega in the way they expected, but from a third effect that emerges when the underlying and implied volatility move in the same direction. The trader's Delta neutrality was correct under the BSM assumption that vol is constant. In a world where vol moves with spot — which is exactly the world equity index options live in — the 'true' Delta is BSM Delta plus a correction term that depends on how vega itself shifts with spot.
That is the opening. Finishing a lesson is where it stops being interesting and starts being useful: the full lesson runs to 9 sections and ends with 6 practice questions. A free account is what opens the rest, and the other 255 lessons in the Academy with it. No card.
What this lesson covers
- 1Vanna — how Delta changes when implied vol changes
- 2Charm and color — the time-decay second-order Greeks
- 3Second-Order Greek Surfaces
- 4Second-order Greeks in closed form
- 5Vanna across moneyness — 90-day call ($200 strike, $200 stock, 30% IV)
- 6Volga (vomma) across moneyness — same 90-day call
- 7FX options markets and the structural importance of vanna and volga
- 8Where to see this on the platform
- 9Summary