Volatility of Volatility

Vol of vol directly determines the fatness of the tails in the risk-neutral distribution. Higher values amplify the probability of extreme market moves.

• A vol of vol of zero collapses a stochastic volatility model to the Black-Scholes framework with constant volatility.
• Increasing vol of vol generates leptokurtosis, producing the smile curvature observed in equity index options.
• This parameter is the primary driver of the price of out-of-the-money options relative to at-the-money options.

In models like the Heston model, vol of vol acts as the diffusion coefficient on the variance process. It scales the random shocks to variance.

• The variance process follows: dV = κ(θ - V)dt + ξ√V dW where ξ (xi) is the vol of vol.
• A higher ξ increases the speed at which variance can wander away from its long-term mean θ.
• The Feller condition (2κθ > ξ²) ensures the variance process remains strictly positive, preventing zero-variance traps.

Vol of vol governs the curvature of the volatility smile. It is the primary parameter traders adjust to fit short-dated, deep out-of-the-money options.

• In the SABR model, the vol of vol (ν) controls the backbone of the smile's convexity.
• A flat smile implies near-zero vol of vol; a steep smile implies high vol of vol.
• Calibrating vol of vol to market data reveals the market's expectation for variance of variance risk premium.

Vol of vol risk manifests as Volga (vomma), the second-order sensitivity of an option's price to changes in implied volatility.

• Volga = ∂²C/∂σ², measuring the convexity of the option price in volatility space.
• Exotic derivatives with discontinuous payoffs, such as digital options, exhibit extreme volga sensitivity.
• The Vanna-Volga method constructs a hedging portfolio of vanilla options to neutralize both vanna and volga exposures.

Tradable instruments now exist that directly reference the expected volatility of volatility.

• The VVIX Index (CBOE) measures the 30-day implied volatility of the VIX index itself, representing the vol of vol for the S&P 500.
• VVIX typically spikes during market crises, reflecting regime-switching uncertainty about future volatility levels.
• A VVIX futures curve in contango signals a stable vol-of-vol environment; backwardation signals immediate stress.

Vol of vol is notoriously difficult to calibrate stably from sparse market data. It is highly sensitive to the term structure of the smile.

• Short-dated options provide the strongest signal for vol of vol; long-dated options are dominated by mean-reversion speed.
• Regularization techniques (Tikhonov, Bayesian priors) are essential to prevent vol of vol from exploding during optimization.
• Market practitioners often bootstrap vol of vol from variance swap term structures rather than individual option prices.