Volga
Volga (also called Vomma or DvegaDvol) is the second-order sensitivity of an option's price to changes in implied volatility—that is, the rate of change of vega with respect to volatility. It measures the convexity of the option's value with respect to implied volatility.
Key takeaways
- Volga is the second derivative of option price with respect to implied volatility: ∂²V/∂σ².
- It is the volatility analog of gamma: just as gamma measures how delta changes with price, volga measures how vega changes with volatility.
- Deeply out-of-the-money and in-the-money options have higher volga than at-the-money options, which explains the volatility smile phenomenon.
- Positive volga is generally beneficial for option buyers, as it means vega increases when volatility rises.
- Volga is a key input in volatility surface models and is used in variance swap pricing and stochastic volatility models.
Explanation
Within the framework of the Black-Scholes model, option sensitivities (Greeks) beyond the first order become critical for sophisticated options books and volatility arbitrage strategies. Volga occupies a central place among second-order Greeks, alongside vanna (the cross-derivative of delta with respect to volatility) and gamma. The term 'volga' is a contraction of 'volatility gamma,' underscoring its role as the curvature of option value in the volatility dimension.
Mathematically, under Black-Scholes, volga equals vega multiplied by d1 times d2 divided by σ: Volga = Vega × (d1 × d2) / σ, where d1 and d2 are the standard Black-Scholes distance-to-default terms. This formulation reveals an important structural property: volga is positive for standard vanilla options and attains its maximum away from at-the-money strikes. At the money, d1 and d2 straddle zero symmetrically and their product is negative but small in absolute value; deep OTM or ITM options exhibit large |d1 × d2| and correspondingly high volga.
This behavior is directly related to the volatility smile. Options with high volga are more sensitive to vol-of-vol (the volatility of implied volatility itself), and market participants demand premium for bearing this risk. In stochastic volatility models such as Heston and SABR, volga drives the curvature of the implied volatility surface—higher vol-of-vol produces more pronounced smile curvature. Practitioners use volga alongside vanna to construct 'vega-vanna-volga' (VVV) pricing methods, particularly in foreign exchange options markets where the smile is expressed in terms of at-the-money volatility, risk reversals (capturing vanna), and butterfly spreads (capturing volga).
Volatility traders and market makers hedge their books not only against movements in the underlying (delta, gamma) and the level of implied volatility (vega), but also against changes in the shape of the volatility surface. A book that is vega-flat but has significant residual volga exposure will profit or lose as volatility becomes more or less convex—analogous to a gamma-neutral book that still has curvature risk if the speed of gamma is non-zero. Managing volga typically involves trading butterfly structures (long OTM call and OTM put, short two ATM options), which are naturally long volga.
Formula
Volga = \frac{\partial^2 V}{\partial \sigma^2} = \text{Vega} \cdot \frac{d_1 \cdot d_2}{\sigma}
Example
A derivatives desk holds a large position in 1-month EUR/USD at-the-money straddles, making the book approximately vega-neutral after hedging with options. However, the desk is short a strip of deep OTM EUR/USD puts that have high volga. When realized volatility spikes during a central bank announcement and the implied vol surface simultaneously 'smiles' wider (vol-of-vol increases), the short OTM put position suffers outsized losses because its vega increases rapidly—the volga effect. The desk estimates its volga exposure at $250,000 per 1-vol-point change in vol-of-vol. To hedge, traders buy EUR/USD one-month 25-delta butterfly spreads (paying 0.3 vol in premium), which are inherently long volga, reducing the net volga risk to approximately $30,000 per vol-of-vol point.
Related terms
Arbitrage At The Money Average Rate Option Black Scholes Model Central Bank Convexity Default Delta Exchange Gamma Greeks Hedging