Given known call option prices, there is a unique local volatility function consistent with those prices.

So why use stochastic volatility models? We can use the market to find local volatility, and then that's our model, no?

Why do we need to complicate things by introducing a stochastic volatility model?

Doesn't that also mean that we need to find a model that produces the same local volatility given by Dupire's equation, since otherwise it would not match the market prices. How is there any guarantee it does that?


Well, what you find is that the introduction of stochastic vol changes the delta of your options. So what does this mean? If the new delta reduces the variance of your hedged portfolio versus the pure local vol model , then it means that the introduction of stochastic vol has resulted in a better description of market dynamics versus the pure local vol model.

Secondly, what you also find is that you can have different models all of which reprice the vanilla options, but that some exotic options have very different prices in the different models. For example , the introduction of stochastic vol can be done in a way that preserves the vanilla option prices , but it lowers the value of forward implied volatilities in the model versus a simple local vol model. Thus, exotics that depend on forward vols ( cliques, Bermudan etc) are priced very differently. Hence another reason to introduce stochastic vol is to improve the pricing of exotics, given the vanilla market.

  • 1
    $\begingroup$ I believe that stoch vol normally increases the value of cliques. $\endgroup$
    – will
    Jul 3 '19 at 7:57
  • $\begingroup$ I think for Cliquets that have caps and floors you may be right, since there is forward skew exposure. Maybe I should have used a different example , although the broader point remains. $\endgroup$
    – dm63
    Jul 3 '19 at 10:34

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