Second-order Greeks: gamma, vanna, volga and the shapes they take

GRK · Chapter 213 min readAsked at Optiver, SIG, IMC, Akuna

Assumes Delta, gamma, vega and theta.

After this lesson you should be able to

  • Say what each second-order Greek measures and when it matters.
  • Describe how gamma and vega vary with moneyness and time.
  • Explain why short-dated gamma and long-dated vega are different businesses.

First-order Greeks tell you what happens for a small move. Second-order ones tell you how the first-order ones change — which is what actually determines whether a hedge holds up. Gamma is the famous one; vanna and volga are what make a volatility book behave unlike a delta book.

GreekMeasuresBites when
Gamma∂Δ/∂S\partial\Delta/\partial SThe underlying moves and your hedge goes stale
Vanna∂Δ/∂σ\partial\Delta/\partial\sigma, equally ∂ν/∂S\partial\nu/\partial SSpot and volatility move together — a selloff
Volga (vomma)∂ν/∂σ\partial\nu/\partial\sigmaVolatility itself moves a lot; it is long convexity in vol
Charm∂Δ/∂t\partial\Delta/\partial tDelta drifts overnight with no move at all
Speed∂Γ/∂S\partial\Gamma/\partial SLarge moves, where gamma itself is changing fast
Table 2.1 · The second-order Greeks. Vanna is the one that turns a "hedged" equity book into a directional position: when the index falls, implied volatility rises, and a book with vanna exposure finds its delta has changed without anyone trading.

Proposition 2.2

How gamma and vega vary

Both peak at the money and fall away in the wings, but they behave oppositely in time. Gamma is concentrated near expiry — it explodes as the option decides whether it is in or out — while vega grows with the square root of time, because a longer option has more variance to be uncertain about.

Holds when

  • Gamma at the money scales roughly as 1/T1/\sqrt{T}; vega scales as T\sqrt{T}.
  • A one-month and a one-year at-the-money option: the month has far more gamma, the year far more vega.
  • Deep in or out of the money, both collapse toward zero.

Short-dated gamma and long-dated vega are different jobs. Because the two exposures live at opposite ends of the curve, they are traded by different people for different reasons. A short-dated book is about realised volatility: you are hedging frequently, earning the moves, and paying theta, and your P&L is made in the hedging. A long-dated book is about implied volatility: you barely rehedge, and your P&L comes from the level of the surface repricing. Saying which one you mean when you claim to be "long volatility" is the distinction an options interviewer listens for.

Equation 2.3

Gamma and vega at the money

Both from φ(0)≈0.4\varphi(0) \approx 0.4, the standard normal density at zero — the same constant as the at-the-money price approximation.

Γ≈0.4SσT,ν≈0.4 ST\Gamma \approx \frac{0.4}{S\sigma\sqrt{T}}, \qquad \nu \approx 0.4\,S\sqrt{T}
ν\nu
Vega, per unit change in σ\sigma. Divide by 100 for "per volatility point".
Γ ν\Gamma\,\nu
Note their product is independent of TT — the two trade off exactly.

Example 2.4

A stock is at $100\$100 with 20%20\% volatility. Compare the at-the-money gamma and vega of a one-month option against a one-year one.

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Worked solution

  1. Formula
    Γ∝1T,ν∝T\Gamma \propto \frac{1}{\sqrt{T}}, \qquad \nu \propto \sqrt{T}
  2. Substitute
    T1=112,T2=1⇒T2/T1=12≈3.46T_1 = \tfrac{1}{12}, \quad T_2 = 1 \Rightarrow \sqrt{T_2/T_1} = \sqrt{12} \approx 3.46
  3. Solve
    Γ1m=3.46×Γ1y\Gamma_{1\text{m}} = 3.46 \times \Gamma_{1\text{y}}
  4. ν1y=3.46×ν1m\nu_{1\text{y}} = 3.46 \times \nu_{1\text{m}}
  5. Answer
    the month has 3.5× the gamma; the year has 3.5× the vega\text{the month has } 3.5\times \text{ the gamma; the year has } 3.5\times \text{ the vega}

Sanity check. Exactly reciprocal, which is the Γν\Gamma\nu relation above. If you want gamma you buy the front; if you want vega you buy the back; and you cannot have a lot of both in one option.

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Second-order Greeks: gamma, vanna, volga and the shapes they take · QuantMax