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In some environments, like now, fixed strike vol is reliably rising when markets rally and falling when they sell off The terms vanna and volga are commonly used by practitioners to denote the partial derivatives ∂vega/∂spot and ∂vega/∂vol of an option’s vega with respect to the underlying asset and its volatility, respectively. So even if skew was literally flat, a long skew position (long downside, short upside) would be losing money on vanna exposure!!
Zero rates for notational clarity One brownian motion zt drives the whole implied volatility surface. Di usion stock price dynamics
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