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It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this?

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For diffusion models (i.e. no jumps):

  1. Local volatility models:

    1. match vanilla options market prices;
    2. give unrealistic volatility dynamic (smile flattens when we move forward in time);
  2. Stochastic volatility models:

    1. don't match vanilla options market prices (not enough skew for short dated expiries);
    2. give realistic volatility dynamics.

You might want to have the best of both worlds, and in this case you have to combine both (local stochastic volatility models), or include correlated stock and volatility jumps (but it's not easy to calibrate this kind of models).

For the doc supporting point 2.1. above, you can refer to chapters 3 and 7 of Gatheral's the Volatility Surface. It's not that recent (uses SPX data of Sep 15th, 2005) but it does illustrate the point and the results are still valid.

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    $\begingroup$ On a more technical note, stochastic volatility models do NOT always give realistic dynamics in practice. Take the popular Heston model for example, when calibrated to real market data, the Feller condition is typically violated, which implies a volatility process that many times has zero as its most probably value. $\endgroup$ – Yian Pap Jun 14 '18 at 17:19

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