Vanna Walls vs Gamma Walls: How Implied Volatility Moves Dealer Hedges

Summary. Gamma measures how dealer hedges change when spot moves. Vanna measures how they change when implied volatility moves. This article derives vanna, shows its sign across strikes and how it varies with tenor, builds a vanna exposure (VEX) profile for a hypothetical chain, and compares the size of vanna-driven hedging with gamma-driven hedging. It also corrects two common misconceptions: that vanna is a "proxy" for vega, and that out-of-the-money puts have positive vanna.

1. Definition

Vanna is the sensitivity of an option's delta to implied volatility, or equivalently the sensitivity of vega to spot [1, 2]. In the Black-Scholes model with zero rates and dividends,

The expression is identical for a call and a put at the same strike, because their deltas differ by a constant. Vega ($\mathcal{V}$) is a different quantity: the sensitivity of the option's price to volatility. A vega exposure chart and a vanna exposure chart answer different questions and should not be used interchangeably.

2. Sign across strikes

The sign of vanna follows the sign of $-d_2$. For strikes above spot ($d_2<0$) vanna is positive: higher volatility raises the delta of an out-of-the-money call. For strikes below spot ($d_2>0$) vanna is negative: higher volatility makes the delta of an out-of-the-money put more negative. Figure 1 shows the profile for a 21-day option. A 6,300 call has vanna of about +0.016 delta per volatility point; a 5,700 put has about −0.014.

3. Dealer hedging from vanna

Under the convention that dealers are long customer-sold calls and short customer-bought puts [3], the change in dealer delta for a change in implied volatility $\Delta \sigma$ (in volatility points) is

• Short out-of-the-money puts: negative vanna times a short position gives a positive contribution. When volatility rises, dealer delta increases and dealers sell to rebalance. When volatility falls, they buy.
• Long out-of-the-money calls: positive vanna times a long position also gives a positive contribution, with the same hedging direction.

Under this convention both wings push the same way: falling implied volatility leads to dealer buying and rising implied volatility to dealer selling. Figure 2 shows the per-strike profile for the hypothetical chain used in this series. Net VEX is about +$2.0 billion of delta per volatility point. A two-point fall in implied volatility would therefore require roughly $3.9 billion of buying, comparable to the gamma hedge for a move of about 1.1% at the current spot (net GEX of about $3.5 billion per 1%).

Because index implied volatility usually falls when the index rises [4, 5], vanna hedging and the volatility-spot relationship reinforce each other: a rally lowers implied volatility, which leads to dealer buying, which supports the rally. The same loop works in reverse in a sell-off. This is the mechanism behind the common observation that falling volatility after an event often coincides with a steady rally.

The spot-volatility relationship that drives this loop is strong and stable in the data. Figure 3 shows the 63-day rolling correlation between daily S&P 500 returns and daily changes in the VIX from October 2016 to October 2026 [6, 7]. It was negative throughout the sample, with a median of −0.82, and never rose above −0.45.

4. Tenor: where vanna lives

Vanna for a fixed percentage distance from spot peaks at a tenor that depends on that distance (Figure 4). For a call 2% out of the money, vanna is largest with about 4 trading days to expiry; for 5% out of the money, about 25 days; for 10%, about 99 days. Further out-of-the-money options therefore carry their vanna in longer-dated expiries. Positioning in longer-dated wings, which may contribute little gamma near spot, can still contribute a large share of the dealer book's sensitivity to implied volatility.

5. When vanna matters more than gamma

• Volatility moves without spot moves. After a scheduled event, implied volatility can drop several points while the index barely moves (see event volatility). Gamma hedging is small in that case; vanna hedging is not.
• Heavy longer-dated wing positioning. Large out-of-the-money put protection several months out carries substantial vanna even when it contributes little to the near-spot gamma map.
• Divergence between the gamma map and the vanna map. When the strikes with the largest VEX differ from the gamma walls, the response to a volatility shock is concentrated somewhere other than the gamma levels.

6. When vanna matters less

In a fast sell-off, spot moves dominate. Gamma hedging, and the dealers' short gamma in the put wing, are the first-order effect; vanna adds to the selling as volatility rises, but it is not the leading driver. Vanna analysis is most useful in calm or post-event conditions, when implied volatility is moving more than spot.

7. Practical reading

1. Compute or view the VEX profile alongside the GEX profile, using the same sign convention.
2. Translate net VEX into dollars per volatility point and compare it with GEX per 1% to judge which effect is larger.
3. Before scheduled events, estimate the likely post-event fall in implied volatility and the implied dealer buying.
4. Treat VEX levels as context for the size and direction of hedging flow, not as precise turning points.

DataFirst shows gamma, delta, vanna and charm exposure on the same strike axis, so the four profiles can be compared directly.

For educational and informational purposes only. Not financial advice. Options trading involves substantial risk of loss. Figures use a hypothetical option chain and the Black-Scholes model.

References

1. Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas (2nd ed.). McGraw-Hill.
2. Taleb, N. N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley.
3. SqueezeMetrics. (2016, revised 2017). Gamma Exposure (GEX). White paper. squeezemetrics.com/monitor/download/pdf/white_paper.pdf
4. Black, F. (1976). Studies of stock price volatility changes. Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economic Statistics Section, 177–181.
5. Bollen, N. P. B., & Whaley, R. E. (2004). Does net buying pressure affect the shape of implied volatility functions? Journal of Finance, 59(2), 711–753. doi.org/10.1111/j.1540-6261.2004.00647.x
6. S&P Dow Jones Indices LLC. S&P 500 [SP500]. Retrieved from FRED, Federal Reserve Bank of St. Louis, 9 October 2026. fred.stlouisfed.org/series/SP500
7. Cboe Global Markets. Cboe Volatility Index: VIX [VIXCLS]. Retrieved from FRED, Federal Reserve Bank of St. Louis, 9 October 2026. fred.stlouisfed.org/series/VIXCLS