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I would like to know if someone could provide a summarized view of the advantages and disadvantages of the approaches on the volatility surface issues, such as:

  1. Local vol
  2. Stochastic Vol (Heston/SVI)
  3. Parametrization (Carr and Wu approach)
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Please embed the links to the definitions (wiki) of the different models and avoid to much abbreviations... –  SRKX Jan 14 '13 at 10:54
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4 Answers

up vote 6 down vote accepted

The volatiltiy surface is just a representation of European option prices as a function of strike and maturity in a different "unit" - namely implied volatility (while the term implied volatility has to be made precise by the model used to convert prices (quotes) into implied volatilities - for example: we may consider log-normal vols and normal vols). Volatility is often preferred over prices, e.g., when considering interpolations of European option prices (although this may introduce difficulties like arbitrage violations, see, e.g., http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1964634 ).

A local volatility model can generate a perfect fit to the implied volatility surface via Dupire's formula as long as the given surface is arbitrage free. In other words: the model can calibrate to a surface of European option prices. Since this calibration is done by an analytic formula the calibration is exact and fast.

Parametric models, like stochastic volatility models, usually are more difficult to calibrate to Europen option price. The formulas that are derived for calibration are usually more complex and often the model does not produce an exact fit. Obviously, the reason to use a stochastic volatility model (or parametric model) is not given by the need to calibrate to European options. The reason is to capture other effects of the model. An important effect to be considered is the forward volatility. Let $t=0$ denote today. Given the model is calibrated to the implied volatitliy surface. How does the volatlity surface generated by the model look like in $t=t_1$ at state $S(t_1) = S_1$? The forward volatility will describe the option price conditional to a future point in time. It is important for "Options on Options" and "Forward Start Options". In other words: More exotic products depend on this feature. While European option only depend on the terminal distributions conditional to today, such a feature depends on the dynamics (conditional transition probabilities). In a local volatility model the forward volatility shows a possibly unrealistic behavior: it flattens out. The smile is vanishing. A stochastic volatility model can produce a more realistic forward volatility surface, where the smile is almost self similar..

Another aspect are sensitivities (hedge ratios): Using a local volatility model may imply a too rigid assumption on how the volatiltiy surface depends on the spot. This then has implications on the calculation of sensitivities (greeks). Afaik, this was the main motivatoin to introduce the SABR model (which is a stochastic volatility model used to interpolate the implied volatility surface): to have a more realistic behavior w.r.t. Greeks).

To summarize:

Local Volatility Model:

  • Advantage: Fast and exact calibration to the volatility surface.
  • Suitable for products which only depend on terminal distribution of the underlying (no "conditional properties").
  • Not suitable for more complex products which depend heavily on "conditional properties".

Stochastic Volatility Model:

  • Advantage: Can produce more realistic dynamics, e.g. forward volatility. Can produce more realistic hedge dynamics.
  • Disadvantage: For products which depend only on terminal distributions the fit of the volatility surface may be too poor.
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Thanks for the very detailed and summarized answer, truly appreciate your help! –  AZhu Jan 20 '13 at 3:45
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I do not have the time right now to write up a summary concise enough but at the same time trying to really touch on all the points that have to be made to delineate the above. Instead I point you to couple papers that are concise enough to skim over in a matter of minutes in order to understand the differences.

Jim Gatheral on Local vs Stoch Vols: http://www.math.ku.dk/~rolf/teaching/ctff03/Gatheral.1.pdf

Here a great paper that touches on pretty much all the models out there (the widely published ones obviously): http://wiredspace.wits.ac.za/bitstream/handle/10539/1495/lisa.pdf;jsessionid=CD3D69EBEFFD957D0B7BB5293E92C7DA?sequence=1

By the way, stochastic and parameterized models are often not clearly divided, there are a number of stochastic vol models that use parameterization and calibration techniques. What you may want to instead focus on is on what kind of volatility the model is based, for example, unobserved integrated volatility or instantaneous volatility. Also you want to really focus on the volatility dynamics and whether the dynamics of a certain model correspond with the observed market dynamics. This very reason was one of the major reasons the SABR model came about.

Just my two cents ;-)

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Thanks @Freddy! –  AZhu Jan 20 '13 at 3:55
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The best overview I have seen so far (in terms of theory AND practicality) is in
Paul Wilmott On Quantitative Finance Second Edition, pp. 826 - 830:

You can find the start: Here

...and the closing overview and summary: Here

(Unfortunately I haven't found the whole excerpt on the web but I would be happy to include a link in this answer [as long as it is referring to a legal copy] - just let me know in the comments).

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Thanks @vonjd, I will do some readings on the book then. –  AZhu Jan 20 '13 at 3:55
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I can comment since Jim Gatheral is actually my mentor.

The advantages and disadvantages of Stochastic Vol, Local Vol and Parameterized Vol really depends on what you are using it for. If you are using it to price exotic options, Stochastic vol is generally a more accepted principle than local vol. The reason because stochastic vol assumes time homogenous volatility surfaces whereas local vol assumes that the forward vol skew that you calculate today is what the vol skew will look like in the future + some stochastic term. Parameterized vol is generally recommended to use to price VIX options/variance swaps as well as for market makers.

One example to show the difference between local vol and stochastic vol is to hedge a one touch option which is the equivalent to a american binary option. The hedge for this would be to buy a call spread maturing shortly and keep buying the call spread until the one touch expires or it is exercised. Because we assume the atm skew in the future will be what we see the forward skew today in local vol, the price will be lower for the call spread. Whereas under stochastic vol, we assume the atm skew today is what we will see in the future, therefore, the price of the call spread will be more expensive under stochastic vol than local vol.

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Thanks a lot for your response, I think your response is definitely very practical and is very helpful from a practitioner stand of view. And could you kindly elaborate on why the parametrized vol is useful for VIX options? Is it because of the easiness in computation? However, isn't parametrization generate arbitrage-free surface only under certain conditions? Thanks for your clarification –  AZhu Jan 20 '13 at 3:54
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We can compute the price of vix options using spanning formula from Peter Carr using Ito's Lemma (I will elaborate on this a later further later as it is trading hours). There are arbitrage free parameterization of the vol surface under all conditions that Jim Gatheral derived in SVI right here papers.ssrn.com/sol3/Delivery.cfm/…. –  Andrew Jan 22 '13 at 17:52
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