You are long a 25-delta risk reversal in an index, long the call and short the put, delta hedged. Give the sign of your vanna and volga, and say what happens to the position if the index falls 5% while implied volatility rises 5 points.

You are long a 25-delta risk reversal in an index, long the call and short the put, delta hedged. Give the sign of your vanna and volga, and say what happens to the position if the index falls 5% while implied volatility rises 5 points.

Approach: Vanna is the change of vega with spot, volga the change of vega with volatility. Work out each leg's contribution and then apply the two moves together, remembering they are correlated in an index.

The position is long vanna and roughly volga neutral, and the stated move loses on both the direction and the volatility. Vanna is d(vega)/dS: an out-of-the-money call gains vega as spot rises toward it and an out-of-the-money put gains vega as spot falls toward it, so being long the call and short the put makes vega increase with spot and decrease as spot falls, which is positive vanna, and it means the position becomes short volatility exactly when the index sells off. Volga, the convexity of value in volatility, is positive for both wings and the two legs largely cancel, so a risk reversal is the market's clean vanna trade while a butterfly is its volga trade. On a 5% fall with volatility up 5 points, the short put leg is now nearer the money with more vega, and it is short, so the volatility rise costs money on a leg whose vega just grew. That combination is why the index skew prices the risk reversal with puts over calls in the first place.

Follow-up: How would you construct a position that is long volga and flat vanna, and what does it cost in theta?

Key concepts: vanna, volga, risk reversal, skew, vega.