There have been various posts on this topic, but they don't really discuss the intuition behind the benefits of the stochastic local volatility (SLV) models over normal stochastic volatility (SV) models.

In other words:

  1. Why does including the leverage function $L(S_t,t)$ in the SV model capture the whole volatility surface? Why can't $\rho$ in the SV model be correctly calibrated to match the affect of $L(S_t,t)$?

  2. Why does not having $L(S_t,t)$ (So using an SV model) misprice deep ITM/OTM options and exotics?

  3. If SLV models have those benefits over the standard SV models, why even use SV models?

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    $\begingroup$ Does this answer your question? When to use a Local Vol model vs Stochastic Vol Model?. With regards to your third point, I don't think SV models are used much on its own. $\endgroup$
    – AKdemy
    Commented Nov 22, 2023 at 5:09
  • $\begingroup$ I had already looked at those responses and the skimmed the linked papers. I was more so wanting to understand the intuition of why SV models misprice barrier and exotics. Those papers (and other's elsewhere), use empirical data to show it's more accurate. I was hoping someone could give a mathematical reasoning for a leverage function rather than just calibrating $\rho$ in an SV model. $\endgroup$ Commented Nov 22, 2023 at 5:16
  • $\begingroup$ Did you ever try to for an SV? If so, you will realize it's hard to match market data, especially if surfaces look something like this. $\endgroup$
    – AKdemy
    Commented Nov 22, 2023 at 5:30
  • $\begingroup$ @AKdemy it's funny that you say that, because I am quite literally doing that right now. My long-term vol is always WAY smaller than it reasonably should be. For example, for option 'SPX250620C04600000' on yfinance, it requires a ridiculously small long-term vol and v0 to match this option. Unless traders' are pricing SPY yield as 3%? Or like you said, it doesn't match well $\endgroup$ Commented Nov 22, 2023 at 6:20
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    $\begingroup$ The leverage function is just an infinite parametrization trick to insure you can price back vanilla options - its basically forcing the marginal distributions to match the market ones. Not sure what type of mathematical reason you would get for why stochastic vol doesn't price back the market, I know heuristically the risk-reversal dynamic with spot is generally the wrong way around but in essence it is because it's inherently the wrong model. $\endgroup$
    – river_rat
    Commented Nov 22, 2023 at 9:44