Option Vanna, Spot–Volatility Correlation, and Vanna P&L

# Vanna for call vs put intuition


# Vanna for call vs put intuition












In equity Vanna is negative. Hence it means that when the stock prices drop the implied volatility goes up.

I understand this intuitively in the sense that as the spot goes down, you have more chance of the downside put being ITM hence more demand for the downside puts which explains that IV goes up.

Yet what I don't get is why the IV of upside calls goes up? If spot goes down then upside calls are getting extremely OTM and hence their IV should drop which would imply a positive vanna right?

## Answer by Chris Taylor (score 7, accepted)

https://quant.stackexchange.com/a/82280

You are confusing a couple of different concepts (which have similar names, so the confusion is not surprising!)

The vanna of an option with price U is simply the sensitivity of delta to changes in implied volatility, or equivalently the sensitivity of of vega to changes in spot. It is the same for calls and puts:

$$ {\rm Vanna} = \frac{\partial^2 U}{\partial \sigma \partial S} = \frac{\partial}{\partial\sigma}\left( \frac{\partial U}{\partial S}\right) = \frac{\partial}{\partial S}\left( \frac{\partial U}{\partial \sigma}\right) $$

Generally speaking, vanna for an option is positive when strike > spot and negative when strike < spot. You can intuitively see this by plotting delta as a funciton of strike for a low implied vol vs. high implied vol option.

You then have the concept of Vanna P&L which is the P&L attributable to vanna, due to changes in both spot and implied volatility:

$$ {\rm Vanna PnL} = {\rm Vanna} \cdot \Delta S \cdot \Delta \sigma $$

For an option with strike > spot (typically this would be an out of the money call) the vanna P&L will be positive if spot and vol move in the same direction, and negative if they move in different directions. For an option with strike < spot (typically an out of the money put) the vanna P&L will be negative if spot and vol move in the same direction, and positive if they move in different directions.

Finally you have the spot-vol correlation which tells you whether spot and vol typically move in the same direction or not. This is sometimes confusingly called positive/negative skew, or positive/negative vanna. For equity index options, spot-vol correlation is negative, which means that spot and vol generally move in opposite directions (i.e price down/vol up or price up/vol down). Normally this correlation holds across the entire vol surface, i.e. all implied vols will be up if price is down, not just put vols (although put vols may rise more when spot is down ... this is because of spot-skew correlation which is a different phenomenon).

Since spot-vol correlation is negative for equity index options, we see that for calls (with strike > spot) we generally have negative vanna pnl (due to positive vanna, and negative spot-vol correlation) and for puts (with strike < spot) we generally have positive vanna pnl (due to negative vanna, and negative spot-vol correlation).

Of course, this isn't free money. The tendency of puts to have positive vanna pnl and calls to have negative vanna pnl needs to be compensated somehow, otherwise we could go long puts/short calls and delta hedge, and we would have a money-generating machine. The explanation is that puts tend to have higher implied volatility and hence higher theta than calls, so a long put/short call position (aka a risk reversal) has negative theta, the the theta decay exactly offsets (in a risk-neutral world) the expected vanna pnl.

## Answer by Jo&#227;o (score 1)

https://quant.stackexchange.com/a/82279

The Constant Elasticity of Variance (CEV) model defines volatility as: \begin{align*} \sigma(S, \tau) = \sigma S^{\zeta - 1} \end{align*}

\begin{align*} where, 0 < \zeta < 1 \end{align*}

This model from Cox 1975, ζ is designed to capture the leverage effect. Describing the behaviour of an increasing volatility when the stock price decreases.

So, when S decreases, the volatility σ(S,τ) increases.

This is consistent with the empirical behavior of equity markets, where downside moves come with higher implied volatilities.

On Vanna,

When the entire IV surface shifts up due to increased volatility, even far OTM calls can experience an IV increase.

The volatility skew readjusts as spot moves lower, keeping the overall shape but lifting IV levels across strikes.

I think the error may be:

You assume that only OTM calls should always lose IV, but in reality, the entire skew structure moves, leading to IV increases even for upside calls, the entire vol skew gets repriced higher.