What are the advantages/disadvantages of these approaches to deal with volatility surface?

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)

Answer

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.