#### P/L best correlates to fixed strike performance since positions are discrete

You cannot tell if a long option position made money just from seeing that ATM vol increased. It is possible ATM vol increased but your strike vol decreased.

#### Example of sticky local vol

As a sticky local volatility causes a negative correlation between spot and Black-Scholes volatility (shown below), this re-mark is profitable for long skew positions. As the value of this re-mark is exactly equal to the cost of skew theta, skew trades break even in a sticky local volatility regime. If volatility surfaces move as predicted by sticky local volatility, then skew is priced fairly (as skew trades do not make a loss or profit).
Local volatility is the name given for the instantaneous volatility of an underlying (ie, the exact volatility it has at a certain point). The Black-Scholes volatility of an option with strike K is equal to the average local (or instantaneous) volatility of all possible paths of the underlying from spot to strike K. This can be approximated by the average of the local volatility at spot and the local volatility at strike K.

Black-Scholes skew is half the local volatility skew (due to averaging).
As local volatility skew is twice the Black-Scholes skew, and ATM volatilities are the same, a sticky local volatility surface implies a negative correlation between spot and implied volatility. This can be seen by the ATM Black-Scholes volatility resetting higher if spot declines
Example of local volatility skew = 2x Black-Scholes skew
Assume the local volatility for the 90% strike is 22% and the ATM local volatility is 20%. The 90%-100% local volatility skew is therefore 2%. As the Black-Scholes 90% strike option will have an implied volatility of 21% (the average of 22% and 20%), it has a 90%-100% skew of 1% (as the ATM Black-Scholes volatility is equal to the 20% ATM local volatility).

[Me: The skew understates what vol would be because the BS skew is the average of all possible paths. If the local volatility at lower strike was lower than implied by 2 x the skew then a long skew position would not pay off its theta]

• Long Skew Should Win No Matter What The Market Does If There Is Neg Spot/Vol Correlation
• When markets fall, the primary driver of the risk reversal’s value is the put (which is now more ATM than the call), and the put value will increase due to the rise in implied volatility (due to negative correlation with spot).

When the market rises, the position gets shorter vega as the call drives the RR's value and vol is falling. Since skew position always wins, it must have a cost and that is the cost of skew (skew theta) or vol premium to own that position.
The surface needs to outperform the skew by 2x to breakeven to the skew theta

• Vanna If spot/vol is negatively correlated, the vol outperforms the skew during a sell-off. The risk/reversal's negative delta will increase even faster than the what is simply implied by B-S gamma. This is the effect of vanna.
• Vanna = dDelta/dVol (and = dVega/dSpot)

• Spot decreases —> Vol increases —> Delta due to Vol Change increases It's second order of gamma because the gamma effect is change in delta due to spot, but vanna is the change in delta due to the change in vol derived from the change in spot.
• 1st order change in delta: change in spot
• 2nd order change in delta: change in delta due to change in vol [due to change in spot]
•
• The cost of this second order convexity is the additional theta associated with the premium skewed vol.