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Why does the market assign a vol skew in the presence of a spot vs vol correlation. Both heuristic/fundamental answers as well as mathematical explanations welcomed

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Suppose you were to price 2 instruments: a strongly OTM put and a strongly OTM Call.

In the standard BS settings, instantaneous volatility is assumed to be constant. Consequently, the implied volatility of these 2 instruments will be the same, resulting in an absence of implied volatility skew.

Now, assume a negative spot/instantaneous volatility correlation. Make it very negative ($\rho \approx -1$) to make things clearer.

  • When you price the OTM put, it is the strongly bearish paths that will matter to you (think Monte Carlo if you like, only the paths that finish below the strike $ K << S_0 $ will contribute to the option price), that is, the paths where the spot is expected to strongly decrease. Because of the negative spot/vol correlation, along the latter "bearish" paths, instantaneous volatility is expected to increase, thus making "effective" volatility (i.e. the implied volatility) higher compared to the pure BS case.
  • When you price the OTM call, it is the strongly bullish paths that will matter to you (think Monte Carlo if you like, only the paths that finish above the strike $ K >> S_0 $ will contribute to the option price), that is, the paths where the spot is expected to strongly increase. Because of the negative spot/vol correlation, along the latter "bullish" paths, instantaneous volatility is expected to decrease, thus making "effective" volatility (i.e. the implied volatility) lower compared to the pure BS case.

This illustrates how a negative spot/(instantaneous) volatility correlation leads to a negative implied volatility skew.

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