OPTIONS GREEKS ANALYSIS

Gamma • Vanna • Volga • Aggregate Dealer Vanna

Educational summary based on the analysis of Market Maker hedging mechanisms

1. Gamma Analysis

Analysis of the original document on delta-hedging mechanisms and the role of Gamma.

1.1 Definition and Role of Gamma

• Gamma = the speed at which delta changes as the underlying price moves. Correct definition.

Mathematically:

Γ = ∂Δ / ∂S

(second derivative of the option price with respect to S)

• If gamma is high → the dealer must continuously rebalance the delta-neutral hedge. Correct.
• High concentration of open interest at a given strike → the hedging flow becomes significant and can create “magnetic” or “explosive” levels. Standard concept (gamma walls / pin risk). No error.

1.2 Long Gamma vs Short Gamma

This is the most important part and it is correct.

MM Position	Gamma	Dynamic Hedging	Market Effect
Long options	Positive	Buy when price falls Sell when price rises	Compresses moves (low volatility)
Short options	Negative	Sell when price falls Buy when price rises	Amplifies moves (high volatility)

• Long Gamma: “they hedge against the move → tends to compress the market”. Correct: they buy on dips and sell on rallies → mean-reverting effect.
• Short Gamma: “they go with the move. Buying rallies and selling dips → make moves more explosive”. Correct: this is the classic behavior that amplifies trends (feedback loop).

The slide on page 4 of the original document perfectly summarizes this dynamic.

1.3 Delta Hedging Table

It is 100% correct:

Investor	Market Maker	Required Hedge
Long Call	Short Call	Buying Underlying
Short Call	Long Call	Sell Underlying
Long Put	Short Put	Sell Underlying
Short Put	Long Put	Buying Underlying

Mathematical rationale (delta signs):

• Long call → Δ > 0
• Long put → Δ < 0
• Opposite position of the MM → inverted delta sign → hedge with the underlying of the opposite sign.

1.4 Initial Flow

When an investor buys a call → the dealer is short the call (negative delta) → to remain neutral they buy the underlying. Correct. Then, as delta changes (due to gamma), they must continue to rebalance.

1.5 Conclusion on the Gamma Analysis

No mathematical or conceptual errors regarding gamma and delta-hedging. The document is educationally clear and faithful to the theory.

The only notes (not errors) are:

• It is purely qualitative (it does not show the Black-Scholes formulas for Δ and Γ).
• It assumes that market makers are predominantly short gamma (the typical situation when end clients buy options), which is realistic but not always true 100% of the time.

2. Vanna and Volga Analysis

Vanna and Volga are second-order Greeks related to volatility. They complete the gamma picture: while gamma measures how delta changes as the underlying price moves, Vanna and Volga measure how delta and vega change when implied volatility moves.

2.1 Mathematical Definitions

Let V be the option price, S the underlying and σ the implied volatility:

Vanna

Vanna = ∂²V / ∂S ∂σ = ∂Δ / ∂σ = ∂ν / ∂S

Measures how much delta changes when volatility changes (or how much vega changes when the price changes).

Volga (also called Vomma)

Volga = ∂²V / ∂σ² = ∂ν / ∂σ

Measures how much vega changes when volatility changes (it is the “convexity” of vega with respect to σ).

2.2 Black-Scholes Formulas (European Call)

With the usual notations d₁ and d₂:

d₁ = [ln(S/K) + (r − q + σ²/2)T] / (σ√T)

d₂ = d₁ − σ√T

Vanna:

Vanna = −e^(−qT) · n(d₁) · (d₂ / σ)

(where n(·) is the standard normal density function)

Volga:

Volga = S · e^(−qT) · n(d₁) · √T · (d₁ d₂ / σ)

(equivalent to Vega × (d₁ d₂ / σ)). The same formulas apply to puts (Vanna and Volga of a call and put with the same strike are identical).

2.3 Practical Interpretation

Greek	What it measures	Typical effect on the Market Maker
Gamma	Δ changes with S	Continuous rebalancing on price moves
Vanna	Δ changes with σ	Rebalancing when volatility moves (even if price is unchanged)
Volga	Vega changes with σ	Management of the vega position when vol “moves a lot”

• Positive Vanna: if volatility rises, delta increases (the position becomes longer).
• Negative Vanna: if volatility rises, delta decreases (the position becomes shorter).

In equities (especially indices) aggregate dealer Vanna is often relevant because of the skew (negative correlation between price and volatility).

2.4 Impact on Market Maker Hedging

With Vanna and Volga the gamma picture is completed:

• Gamma → hedging flow generated by price moves.
• Vanna → hedging flow generated by moves in implied volatility. Example: if dealers are net “short vanna” and vol rises, their deltas become more negative → they must buy the underlying even if the price has not moved.
• Volga → measures how “stressful” it is to manage a short-vega book when volatility explodes (analogy: short gamma on price vs short volga on volatility).

In practice, market moves are not only “gamma-driven”. When volatility re-prices rapidly (macro events, 0DTE, etc.), vanna-hedging flows can become dominant and generate sudden accelerations or reversals.

2.5 Relation to the Vanna-Volga Pricing Method

In FX (and sometimes in equities) the Vanna-Volga method exists: one takes the Black-Scholes price and adjusts it by adding the “cost” of a portfolio that replicates the option’s Vanna and Volga, using market prices of Risk Reversals and Butterflies. It is an empirical way to incorporate the volatility smile without using full stochastic-volatility models.

2.6 Summary

• Gamma → sensitivity of delta to the underlying price.
• Vanna → sensitivity of delta to volatility.
• Volga → sensitivity of vega to volatility.

Together they form the “triangle” of the most important second-order risks for anyone managing an options book. Gamma dominates price moves, Vanna dominates changing volatility regimes, and Volga measures the convexity (and therefore the stress) of the vega position.

3. Aggregate Dealer Vanna Analysis

Aggregate dealer Vanna (often called VEX – Vanna Exposure) measures how much the net delta of all market makers changes when implied volatility moves. It is the sum, across all strikes and all expiries, of the Vanna of the positions held by dealers.

In practice it answers the question: “If implied volatility rises (or falls) by 1 point, how many billions of dollars of futures/shares must dealers buy or sell to remain delta-neutral?”

3.1 Why It Matters

While gamma generates hedging flows when price moves, Vanna generates hedging flows when volatility moves, even if the price remains unchanged.

These flows are mechanical, not discretionary, and can become dominant especially:

• after macro events (FOMC, CPI, earnings);
• during “vol crush” episodes (collapse in volatility);
• when there is large open interest in OTM options.

3.2 Typical Behavior (Most Common Scenario)

In most equity market regimes (especially SPX/SPY), dealers end up net positive Vanna. This stems from the fact that clients buy protection (puts) and dealers are short those options, combined with the typical negative equity skew.

Volatility Move	Dealer Flow (Positive Vanna)	Market Effect
IV falls (vol crush)	Buy underlying/futures	Bullish tailwind → “Vanna Rally”
IV rises	Sell underlying/futures	Amplifies sell-offs

The classic “Vanna Rally”

1. A stress event arrives → clients buy puts → dealers become short puts and hedge by selling futures.
2. The event passes (or the worst expectations do not materialize) → implied volatility collapses.
3. The deltas of the OTM puts decrease (Vanna effect).
4. Dealers no longer need to be as short → they buy back the futures.
5. This mechanical buying pushes prices higher → volatility falls even further → positive feedback loop.

This explains many of the “rallies with no news” that are observed after FOMC, CPI or after a peak of fear.

3.3 When Vanna Becomes Dangerous

If the market falls enough to push many puts in-the-money, aggregate Vanna can change sign or behavior. In that case:

• the rise in volatility forces dealers to sell even more;
• a negative feedback loop is created (vol up → selling → lower prices → even higher vol).

This is one of the mechanisms that contribute to downside accelerations and liquidity dislocations.

3.4 Relation to Gamma and Charm

The three second-order Greeks work together:

• Gamma → sensitivity to price moves (dominant intraday and near strikes with large OI).
• Vanna → sensitivity to volatility moves (dominant in volatility-regime changes).
• Charm → sensitivity to the passage of time (dominant near expiries and in the afternoon).

When Gamma, Vanna and Charm all point in the same direction, dealer hedging flows become very powerful and can overwhelm fundamental flows.

3.5 Operational Summary

• Positive aggregate Vanna + falling IV → strong mechanical support for prices (most frequent and “bullish” scenario).
• Positive aggregate Vanna + rising IV → additional selling pressure (amplifies drawdowns).
• Negative aggregate Vanna → inverted behavior and potentially destabilizing (less common).

Aggregate Vanna is therefore one of the most useful tools to understand why sometimes the market “rises on its own” after an event or, conversely, why a sell-off accelerates disproportionately relative to the news.