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I am currently studying Chapter 6 of the book Stochastic Volatility Modelling by Lorenzo Bergomi which is about the Heston model.

After presenting the Heston Model he makes some claims about why the Heston Model is unsuitable for handling exotic options:

  • the Heston model lacks flexibility,
  • the Heston model has some peculiar idiosyncrasies.

At the end he says that the fundamental problem is its usage because the SDE $$ \begin{cases}dS_{t} & = \sqrt{\xi^{t}_{t}} S_{t} dW_{t} \\ d \xi^{T}_{t} & = \nu e^{-k(T-t)} \sqrt{\xi^{t}_{t}}dZ_{t}. \end{cases} $$

is useless.

Here is my questions:

  1. What does it mean that a model lacks flexibility?
  2. What features of the model are peculiar?
  3. Why is the SDE useless?
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  • $\begingroup$ Aren't the claims at all supported in the book? $\endgroup$ Commented Feb 18 at 14:44

2 Answers 2

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I have not read Bergomi's book, but I have a lot of respect for his work and am in accord with the observations you have distilled from it. So, the points below might not quite agree with the reasons Bergomi himself had for making those statements:

Flexibility

When pricing exotics, we want a calibrated model that agrees with the volatility surface of non-exotic options, or at least comes close to no-arbitrage agreement. The Heston model does not have a lot of parameters, and there are many observable volatility surface shapes that can't be reproduced particularly well by any choice from the Heston parameter space.

I'm always torn by this kind of criticism, personally. Ultimately we fit the volatility surface in order to choose hedge parameters. A common "flexible" solution is local volatility models, and I think of those as mathematical artifice. They provide comfort via an illusion of exactitude, and I'm unconvinced their hedge parameters are very good.

Peculiarity

The forward surfaces of the Heston model do not look anything like volatility surfaces we observe in the wild. That is to say, if you take some future $(\tau; S, \nu, \xi)$ point, generate prices to some longer $T>t$, and look at the BSM-skew implied from there, it does not resemble a typical volatility skew.

Usefulness

Here my preference for a model with jumps comes into play. Even if we have the common case of an equity index underlying, you just need those jumps to get good skews in the short term. For this reason, I find Heston too limited.

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In addition to Brian's answer, which already covers much of it, I'd like to add a few more things:

  • I don't agree with Bergomi that the (Heston / 1 factor) SDE is useless (did he really write that?), in fact Heston is a good testing ground as Heston has closed form solutions for various derivatives.

  • One peculiarity of the Heston model is that the vol-of-vol decreases as vol increases. It's a peculiarity because this gives downward sloping implied vol curves for vol derivatives, which is not observed in practice. Hence SABR or 3/2 model might be a better option for vol derivatives.

  • In terms of flexibility, where Bergomi is coming from is I believe the joint calibration to index options and varswaps or vix futures and options. Heston doesn't fare well in joint calibrations, which is one of the reasons multi-factor models for vol have been introduced by Buehler and also Bergomi of course.

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  • $\begingroup$ He does not literally write "useless" however he writes (and I quote) "which practical pricing or hedging issue naturally calls for that SDE?". You can give a better interpretation, I interpreted as "useless". $\endgroup$ Commented Feb 18 at 17:14
  • $\begingroup$ Can you point me to the multi factor models you are talking about? $\endgroup$ Commented Feb 18 at 18:21
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    $\begingroup$ @Pedro These two to begin with papers.ssrn.com/sol3/papers.cfm?abstract_id=1493294, papers.ssrn.com/sol3/papers.cfm?abstract_id=1493308 $\endgroup$ Commented Feb 18 at 19:22

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