It seems that Heston model is the standard stochastic vol model used in practice due to itbeing "solvable", i.e. one has "closed" (up to a Laplace transform) form solution for European optionsand the volatility process can be integrated to give something like V_t ~ Z where Z is a non-centralchi-squared distribution and stuff like integral V_s ds are known.However, Heston model only fits well mid-long term vol surface and underskews short term and worstof all does not give correct market dynamics (forward skew becomes a smile). It seems pricing cliquets, Napoleons, options on variance, etc. seem very dangerous in this framework. Some people seem to introduce jumps but it only alleviates the forward skew problem but not eliminate it.Can someone guide me to any literature on other types of stochastic models? e.g. the classdV = a(b-V)dt + c V^d dW (d=0.5 being Heston). I did some study on realized vol time seriesof SPX and found that d=1. seems to fit the market very well. But the vol reversion rate I get from the time series implies that a ~ 0.8 days^-1 ~ 200 years^-1 which seemsridiculous large considering that most Heston fits to implied vol surface I've seen gives a ~ 2 years^-1. Has anyone run into similar results?Basically I would like to find a stochastic vol model (perhaps with jumps) that fits skew of all maturitiesreasonably well (better than Heston) and gives the correct market dynamics, i.e. gives forward vol surface resemblesspot vol surface. Would playing with "d" above be of any help?Thanks.

Keep d=1 and add Merton's lognormal jumps. Theresult is a model that can fit the SPX quite well in my experience.regards,

The parameter d has little effect on the volatility smile around at the money, but it does affect the asymptotics of the smile for large strikes. Moreover, some choices of d will give you models with peculiar and undesirable properties. See this for details.

Schoutens (SST) discusses stochastic volatility models with with jumps, maybe it's worth having a look. A.

see below [sorry].

In an article recently published in Wilmott, we discuss how the smile problem should generally be solved, and the questions of exotic option pricing and hedging (barrier options, cliquets, etc.) particularly addressed.

Very nice paper by Wim Schoutens !

thanks for the paper, numbersix

Thanks for the references. I found all the papers very helpful. I've decided to implementd=1/2 and d=1 models since there seems to be several issues with d<1/2 and d>3/2 (momentsblow up, becomes only local martingale, vol explosion, etc.). I basically have Monte-Carlodone and hope to implement a 2D finite-difference scheme for these models. However,I am curious what the correct boundary conditions to put in the volatility, especially forthe lower barrier. Due to mean reversion, I don't think it is as simple as putting theintrinsic value of the option. Is FD too ambitious? I guess adding jumps will be just solvinga 2D PIDE but the boundary condition seems hard to overcome. Any advice? Thanks again.Also, no one addressed the issue of forward skew. Can this be solved?

Actually, from the moment explosion prospective, the safest choice is 0<d<1/2. For d<1/2, all moments of X (the spot) exist for all times. For d=1/2, high-order moments of X blow up in finite time. For 1 >= d > 1/2 and zero (or positive) correlation, none of the moments of X of order greater than 1 exist. Also, for 0<d<=1/2, X is always a proper martingale

QuoteOriginally posted by: pleeAlso, no one addressed the issue of forward skew.I believe we did.This is how I see the problem. As you have pointed out earlier, Heston is fashionable because it is solvable. It is always the same old story of people picking up models they can more or less analytically solve for vanilla option pricing. The solvability constraint usually imposes some parametric form on the model. As a consequence, people end up discussing the value of some parameter (for instance, your "d") that has no meaning other than being a parameter in the particular parametric process, and that may have totally unrealistic or artificial consequences (and by that I mean, unrelated to the practical problem at hand, which is the fitting, the pricing and the hedging of some options in the real world) such as moments blowing up, forward volatility surfaces not being consistent with the market prices or market experience of cliquets, artificial boundary conditions to impose on the volatility grid, etc.I am of the opinion of building up a practical and useful model going from the task you intend it for to the mathematical expression of the model, and not the other way round.As discussed in our paper, there is no way you can guarantee that the forward implied volatility surface (or in other words, smile dynamics which also affect the barrier option prices) will conform with your expectations unless you build this conformity into your model. There is no way to guarantee all this but to calibrate your model to the given prices of the forward starting options (or barrier options). Think how lucky you must be for some particular choice of a particular parameter of the particular parametric solvable model you have chosen to exactly fit the real market dynamics, and fit the right prices and the right hedges of the cliquets or barrier options!This is why we have proposed a model (or a family of models we have called "Nobody" - my favourite word is "representation") that you can exactly adapt to your given problem, and calibrate from the start to the market prices of forward starting options or barrier options. For instance, our model will predict forward volatility surfaces that remain similar to the spot volatility surface, if you calibrate it today to the corresponding prices of the corresponding forward starting options. Also our model incorporates jumps in asset, stochastic volatility, and jumps in volatility correlated with asset jumps (as these features seem to be required in order to realistically explain the shapes of smiles both on the short term and the long term).There is no hope, of course, that such a model might be analytically solvable. You have to integrate it numerically. It is parametric, but not exactly in the sense that Heston, or Merton, or Bates, or Pan Duffie Singleton, etc., are. In other words, our representation does not produce fancy single-valued parameters such as "mean-reversion of volatility," "volatility of volatility", etc., that you will have to worry about calibrating or observing (when they are unobservable) or making sure they don't blow up. Although the concepts behind the parameters are certainly relevant (and our model certainly takes them into account), their reflection into mathematical words or letters such as your "d" or "a" or "b", and the mathematical problems they will automatically generate (fear of blowing up, etc.) is only the consequence of the imperative of solvability and parametrization in the "bad sense."The representation that we found was most suited to answering all the above requirements (calibration to all kinds of options, speed, numerical tractability, flexibility, down-to-earth parametrization) is the regime-switching model. I attach a paper which further discusses the practical (even philosophical) implications of such a representation.

QuoteOriginally posted by: SPAAGGVery nice paper by Wim Schoutens !Shoutens calibrate on call prices. Is this reasonable? Obvisously your error for OTM calls will "weight" less in the rmse. And high-strike-implied-vols will be in general fitted worse than low-strike-implied-vols. I don't find in the paper a table with observed and fitted implied vol.