Why your delta moves when implied vol changes (vanna)

Question

You are delta-hedged on a book of out-of-the-money calls. The stock has not moved at all, but implied volatility just jumped. You check your risk and your delta is now off.

Why did your delta change with no move in the stock, and which way did it go for the out-of-the-money calls?

Accepted Answer

The cross-greek responsible is vanna, the sensitivity of delta to volatility (equivalently, of vega to spot): 	ext{Vanna} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial\,	ext{Vega}}{\partial S}. Delta depends on how likely the option is to finish in the money, and volatility changes those odds even when the stock is still. For an out-of-the-money call, raising volatility increases the chance it finishes in the money, so its delta rises toward 0.50. (Formally, for a call vanna = -\varphi(d1)\,d2/\sigma; an out-of-the-money call has d2 < 0, making vanna positive, delta goes up as …

Why your delta moves when implied vol changes (vanna)

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