What Is the Vol-of-Vol Greek (∂V/∂ξ)?
Last reviewed: by Options Analysis Suite Research.
Vol of Vol Greek (∂V/∂ξ) is the first derivative of option value with respect to the Heston vol-of-vol parameter (denoted xi or nu across notations). The Heston variance process is dv = kappa(theta - v)dt + nu sqrt(v) dW; nu controls the diffusion magnitude of variance itself - the "volatility of volatility." This Greek captures the structural sensitivity to vol-of-vol regime and drives smile-curvature pricing.
What Is the Heston Vol-of-Vol Greek?
The Vol of Vol Greek quantifies how option value changes when Heston's nu (vol-of-vol) parameter shifts. A higher nu means variance fluctuates more (volatility itself is volatile), which fattens both wings of the return distribution and produces more curvature in the IV smile. Long OTM options (puts and calls) gain value when nu rises because their payoff depends on extreme paths of the underlying.
Two intuitions. First, the Vol of Vol Greek is the smile-curvature sensitivity Greek - changes in nu map directly to changes in butterfly pricing. Second, this Greek is the Heston analog of vomma in some sense - both capture vol-convexity exposure - though they target different parameters (vomma is BS-style ∂²V/∂σ²; this Greek is ∂V/∂ξ in the Heston framework).
Why Does It Matter?
Three operational contexts. First, butterfly trades. Long ATM short OTM butterfly structures are explicit positive-vol-of-vol-Greek positions. They profit when vol-of-vol regime rises (smile curvature steepens).
Second, deep-OTM tail-risk hedging. Far-OTM puts on equity indices have value that is dominated by vol-of-vol regime - a low-nu environment makes deep OTM puts cheap; a high-nu environment makes them expensive. Tail-hedge programs explicitly track this Greek.
Third, regime detection. Aggregate Vol of Vol Greek exposure across a book signals whether the book is structurally long or short curvature. The VVIX index is the market-priced version of vol-of-vol; cross-checking aggregate Greek vs VVIX level is a sanity check on positioning.
How Heston Computes Vol of Vol Greek
Computed by central finite difference on the Heston pricer: bump nu by 1% (multiplicatively), re-price, take the difference. Closed-form differentiation of the characteristic function exists but is operationally rare.
Related Greeks
This Greek is one of four core Heston-parameter Greeks. Siblings: Kappa Der, Theta Param, Rho Der. The BS analog is vomma (vol convexity).
Related Concepts
Vol of Vol · Vomma · Heston Model · SABR · Volatility Smile · All 17 Greeks
References & Further Reading
- Heston, S. L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility." Review of Financial Studies, 6(2), 327-343.
- Gatheral, J. (2006). The Volatility Surface. Wiley.
- CBOE. VVIX White Paper. cboe.com.
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This page is part of the 17 Greeks reference covering every options Greek with formula, intuition, worked example, and how each pricing model computes it. Browse the full documentation.