What Is Vanna? How Volatility Shifts Move Dealer Hedges

Vanna is the sensitivity of an option's delta to a change in implied volatility, which is why a shift in volatility alone can force a hedged dealer to trade. Read the other way round it is the sensitivity of vega to a change in the underlying, and both readings are the same number. It is the quantity that connects the volatility surface to flow in the stock itself.

Delta hedging is usually explained as a response to price. Vanna is the part of it that answers to volatility instead: when implied volatility moves, a hedge sized on yesterday's delta is the wrong size today, and the correction arrives as trading in the underlying even on a session where spot goes nowhere. This page covers vega, vanna and vomma as one family, the units each is quoted in, the mechanics behind the hedging flows people discuss around events, and what the figures do not say. It is descriptive throughout.

Vega, the first order piece

Vega is how much an option's value moves when implied volatility moves. In the closed form Black-Scholes model these figures are built on, with the dividend yield set to zero:

``` d1 = (ln(S/K) + (r + sigma^2 / 2) T) / (sigma sqrt(T)) d2 = d1 - sigma sqrt(T)

vega = S phi(d1) sqrt(T) ```

S is spot, K the strike, T time to expiry in years, r the risk free rate, sigma the volatility input as a fraction, and phi the standard normal density. That expression is quoted per 1.00 of volatility, a move of one hundred volatility points, which is not a unit anyone reads off a screen. Vega exposure, often written VEX, therefore rescales it into dollars of vega per one volatility point: the per contract vega times open interest, times the 100 share contract multiplier, times 0.01 for the volatility step.

Vega is the same for a call and a put at the same strike and expiry. Apart from delta, whose two sides differ by a constant, theta is the only greek here that differs between the two sides, and the gap there is exactly the discounted rate term.

Vanna, the cross derivative

Vanna is a second order greek, and the definition is short:

``` vanna = dDelta / dVol = dVega / dSpot = -phi(d1) d2 / sigma ```

Both readings are the same mixed partial derivative of the option price, taken once with respect to spot and once with respect to volatility, and the order does not change the answer. Two properties fall straight out of the closed form. Because phi(d1) and sigma are always positive, the sign of vanna is the opposite of the sign of d2, so vanna flips sign across the strike where d2 is zero instead of holding one sign across the chain. And because a put's delta is its matching call's delta minus one, differentiating either with respect to volatility gives the identical result: vanna is parity equal, as vega and gamma are.

The exposure column built on it is stated in dollars of delta per one volatility point: vanna times open interest times 100 times spot times 0.01, where spot converts share delta into dollars and the 0.01 again restates the volatility step. Reading it is a units exercise. If net vanna is X dollars of delta per volatility point, a two point move in implied volatility maps to roughly twice X of delta appearing in or leaving the modeled book, before spot has done anything at all.

Call side and put side are netted as calls minus puts, a modeling convention for how dealer inventory is assumed to sit rather than a measurement of any real book. Hold onto that when reading a sign.

Why a volatility move forces trading in the underlying

A delta hedged dealer holds a position in the underlying that offsets the delta of the option book. Delta is not a constant. Gamma describes how it changes as spot moves, which is the mechanism covered in dealer gamma exposure. Vanna describes how that same delta changes when implied volatility moves, and that channel stays open on a day when the underlying closes unchanged.

The sequence is mechanical:

1. Implied volatility moves. 2. Every contract in the book re-prices its delta at the new volatility. 3. The hedge that matched yesterday's delta is now over or under sized. 4. Restoring it means buying or selling the underlying.

Aggregate step four across a whole chain and you have what a net vanna figure measures: the delta a one point change in implied volatility puts into the modeled book, all of which has to be offset somewhere.

The case that comes up most often is the collapse in implied volatility after a scheduled event, once the outcome is known and the volatility priced ahead of it is released. That is a large move in the exact input vanna is measured against, so it resets the delta of every contract on the chain at the same moment. Implied volatility is also not one number across strikes, and the shape of it governs where vanna concentrates: that shape is the subject of options skew and the put call ratio.

Vomma, and why vega is not a fixed size

Vega itself moves when volatility moves. Vomma is that third term:

``` vomma = dVega / dVol = vega d1 d2 / sigma ```

Its exposure column is quoted as vega per volatility point, per volatility point: vomma times open interest times 100 times 1e-4. That 1e-4 is two factors of 0.01, one for each derivative taken with respect to volatility, which is the quickest way to keep the family's units straight.

Since vega and sigma are positive, the sign of vomma follows the product of d1 and d2, which is negative only in a narrow band around the forward price and positive for strikes well away from it. Vomma is parity equal too. What it adds is stability: where vomma is large, a sizeable volatility move changes the book's vega enough that the vega measured at today's volatility is itself moving during the event, so those figures describe a starting point rather than a fixed coefficient.

What these figures do not say

The framing you will see written up as "vanna and charm flows" is a description of hedging mechanics, not a prediction. Charm is the companion term, the change in delta per unit of time, which is why it gets paired with vanna: one is delta drifting with volatility, the other is delta drifting with the calendar, and both create re-hedging in the underlying without any move in spot. Neither says which way anything travels.

Three limits are worth stating plainly. These are modeled figures computed from listed open interest rather than a reported dealer book, so which side of each contract a dealer holds is an assumption. Open interest counts what is outstanding, not what traded today. And the closed form carries its own inputs, including a zero dividend yield and a single rate assumption, so a figure is only ever as good as the conventions printed beside it.

How Kresmion computes it

Kresmion computes modeled dealer greeks exposure nightly for a curated universe of US names from listed option open interest, using closed form Black-Scholes with the assumptions stated: zero dividend yield, a 100 share contract multiplier, and volatility carried as a fraction. The live SPY reading on this page shows net, call side and put side readings for the vega exposure, and the same three for vanna and for vomma, from the latest session shown. The full suite across every covered name, including the implied volatility tab, sits in Kresmion's greeks tool with a free account.

Key takeaways

Frequently asked questions

What is the difference between vega and vanna?

Vega is first order: it measures the option's value against volatility. Vanna is second order and mixed: it measures the option's delta against volatility, or equivalently its vega against spot. Vega tells you what a volatility move does to what the book is worth, vanna what the same move does to the hedge the book requires.

Does a large net vanna reading predict which way the market moves?

No. It describes the backdrop. A net vanna figure states how much delta a one point change in implied volatility would put into a modeled dealer book. That is a statement about hedging flow conditional on volatility moving, and it says nothing about whether volatility moves or where the underlying ends up.

What are "vanna flows" around an event?

The phrase refers to the re-hedging that follows a sharp change in implied volatility, most visibly the drop after a scheduled announcement resolves. The change resets every contract's delta at once, and dealers restoring their hedges trade the underlying as a result. The size of it depends on positioning that shifts as contracts are opened, closed and expire.

Why is vanna the same for a call and a put at the same strike?

Because put delta equals call delta minus one under put and call parity, and the derivative of a constant is zero. Differentiating either side with respect to volatility gives the same expression, so a matched call and put share one vanna. Apart from delta's constant offset, theta is the only greek in this family that genuinely differs by side, and that difference is a rate term.

This page is information, not investment advice.

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Source: closed form Black-Scholes greeks, with the modeling assumptions stated on Kresmion's greeks exposure tool; Kresmion modeled dealer positioning data.

Kresmion Research.