Second-Order Greeks
Second-order Greeks are option sensitivity measures that capture how the first-order Greeks (delta, vega, theta) themselves change in response to changes in market variables, providing a more complete picture of an option position's risk dynamics—particularly under large or rapid market moves. The primary second-order Greeks are gamma (rate of change of delta with respect to the underlying price), vanna (sensitivity of delta to volatility, or vega to price), and volga (sensitivity of vega to volatility).
Key takeaways
- Gamma measures how delta changes per unit move in the underlying; positive gamma means the delta grows with rising prices (accelerating profits in a rally), while negative gamma means delta shrinks (accelerating losses).
- Vanna is the cross-Greek between delta and volatility—crucial for managing options books where both price and implied volatility move simultaneously, as in crises.
- Volga (also called 'vomma') measures the sensitivity of vega to changes in implied volatility; it is the primary driver of the volatility smile's curvature.
- Second-order Greeks are essential for dynamic hedging—delta hedging alone is insufficient for large moves; gamma hedging is required for precision.
- Market makers in options price second-order Greek risks into their bid-ask spreads, compensating for the cost of dynamically maintaining delta-neutral portfolios.
Explanation
First-order Greeks (delta, vega, theta, rho) provide a local linearization of an option's price sensitivity to market variables. But options pricing is non-linear: the relationship between an option's value and the underlying price is curved (due to the convexity of the payoff), not linear. This curvature means that first-order Greek estimates become inaccurate for large moves, making second-order Greeks essential for accurate P&L attribution and precise hedging.
Gamma is the most widely monitored second-order Greek. Mathematically, it is the second partial derivative of the option price with respect to the underlying price: Γ = ∂²C/∂S². Practically, it measures how much the option's delta changes per one-point move in the underlying. Long options (calls and puts) have positive gamma: as the stock rises, a long call's delta increases (accelerating the position's appreciation), while as the stock falls, the position's delta decreases (decelerating the loss). Short options positions have negative gamma: losses accelerate in volatile markets and gains decelerate in calm markets—the fundamental reason why option sellers are compensated by time decay (theta) for bearing negative gamma risk.
The relationship between gamma and theta is central to options market making. Delta-neutral option positions (where delta = 0) still have gamma and theta exposure. Long gamma / short theta positions profit from large moves (any direction) but lose time value daily—they are bets on realized volatility exceeding implied volatility. Short gamma / long theta positions profit from quiet, range-bound markets but suffer large losses during volatility spikes. The ratio of gamma P&L to theta cost is essentially the ratio of realized variance to implied variance, making options market making a continuous bet on the relationship between realized and implied volatility.
Vanna, the cross-Greek ∂Delta/∂σ = ∂Vega/∂S, becomes critical during stress events when both price and volatility move together (which they typically do—equity markets fall and volatility spikes simultaneously). A delta-hedged options book with significant vanna exposure will develop a net delta as volatility rises, requiring additional hedging transactions. Options desks at major banks explicitly monitor and manage vanna risk as part of their daily Greek reports, particularly in equity derivatives where the negative correlation between price and implied volatility is strongly persistent.
Volga (∂Vega/∂σ) measures the curvature of the option price with respect to implied volatility—it is to vega what gamma is to delta. Options with high volga benefit from volatility-of-volatility: a position with positive volga gains when implied volatility moves significantly in either direction, not just when it rises or falls. Volga is particularly important for exotic options (variance swaps, vol-of-vol options) and for structurally curvature-rich positions like strangles and butterflies. The volatility smile (the observation that out-of-the-money options trade at higher implied vols than ATM options) is partly explained by the market's demand for volga exposure—OTM options have more convexity with respect to volatility, commanding a premium.
Formula
Gamma = ∂²C/∂S² = N'(d1) / (S × σ × √T); Vanna = ∂Delta/∂σ = -N'(d1) × d2/σ; Volga = ∂Vega/∂σ = Vega × d1 × d2/σ
Example
A market maker holds a short strangle position: short 1,000 one-month calls at $105 strike and short 1,000 one-month puts at $95 strike (stock at $100, total net delta ≈ 0). The position has: theta = +$5,000/day (collecting $5,000 in time decay daily), gamma = -$200 per $1 stock move (delta changes by -$200 per $1 price change, accelerating losses if the stock moves sharply). On a day when the stock rallies $3 from $100 to $103, the delta shifts: Δdelta ≈ -$200 × $3 = -600 contracts equivalent. The market maker now has a net short delta of -600, requiring purchase of 600 equivalent shares to remain delta-neutral—a transaction that costs money and represents the gamma P&L drain. If this $3 move happens in a day where theta earned $5,000, the gamma loss = (0.5 × Gamma × Price Move²) = 0.5 × 200 × 9 = $900 per dollar of notional—but scaled to the full position this could be $9,000 or more, exceeding the daily theta earned and creating a net P&L loss for the market maker despite collecting premium.
Related terms
Backwardation Call Option Convexity Correlation Delta Equity Exotic Options Gamma Greeks Hedging Implied Volatility Market Maker