Local Volatility vs. Stochastic Volatility

Question

Are there any empirical observations or practices when to prefer Local Volatility Model for pricing over Stochastic Model or vice versa?

Answer 1

There is another reason why Stoc Vol Models should be usually preferred to Local Vol Models, this reason is explained in the Hagan et al. paper "Managing Smile Risk" about SABR process and is in simple terms the fact that "smile dynamics" is poorly predicted by local vol models leading to bad Hedging of exotic options.

Anyway Local Vol models have the good feature to be "arbitrage free" (at the begining) and I think that some link between both approach can be achieved by Markovian Projection Method.for this you can have a look at V. Piterbarg's paper on the subject and the references therein.

Regards

Answer 2

For pricing and hedging a portfolio of vanilla options, stochastic volatility is almost always preferable to local volatility since empirically it more accurately captures the evolution of the smile.

Answer 3

For pricing, there are a few products whose prices are sensitive to the forward smile and when you compute that with just local vol, it is not realistic.

So if you are a seller, you go to the next church and find something that looks kindof reasonable, and that kind of can reconstruct a reasonnable forward smile structure.

The game in pricing is to not sell for 0 something that you know can't be 0. Especially if 90% of the product price comes from this particular effect.

For hedging one could try to add this and that dependency, but the chance are you are off big time on 2 other things.

Answer 4

Where you have a strong view on lots of points in your Black-implied volatility surface, both in time and strike, then some of the popular stochastic volatility models introduce calibration error, so you might want to stick the honest local volatility model (non-parametric, calibrate-able to whatever you know about the Black-implied vol surface). If you still want to directly manipulate the shape of the forward skew (that is, the prices of forward starting options), or keep other good parts of stochastic volatility mentioned in answers above, then you need to look up stochastic local volatility models.

Hope this helps.

Answer 5

A similar question posted after this one has a very good answer. Especially the section that explains that the calibrated leverage surface is typically observed to flatten with maturity (a shortcoming of LV). Therefore the forward volatility smile will be less convex than on the initial pricing date and you will not be pricing deals properly which are primarily sensitive to forward volatility skew and smile. For example options on forward vol, cliquets, etc.

Generally, I think Stochastic Local Vol (SLV) is mainly used nowadays (at least for equity and certainly FX). Once calibrated to the vanilla market, LV and SV offer no extra flexibility in matching the dynamics of implied volatility. For example, prices for barriers and touches tend to be undervalued by LV but overvalued by SV. In SLV, mostly vol of vol and correlation control the mixing of LV and SV.

Hence, appropriate calibration of the mixing parameters will allow you to closely match market quotes. LV and stochastic SV are simply degenerate cases where the mixing fraction is such that only one or the other is used.

I would like to add that not only the model, but also the implementation matters (e.g. do I use a functional form to calibrate my LV, or a non-parametric grid of strikes and spot values). To price a wide range of structures, it may be worth considering both, a finite-difference solver of the PDE or MC simulation of the SDE for more exotic path-dependent structures. Ultimately, MC should converge to PDE but it is simply more computationally intensive to always use MC.