This may seem silly, but I need to price S&P index derivatives with maturities that range from 5-17 years and I want to use stochastic volatility. I tried to use the Heston model, but the parameters that come out of my calibration cause the volatility process to hit zero too much and introduce a large bias in the results even for a large number of timesteps and second order discretization methods (I have to price with Monte Carlo methods due to the highly exotic nature of the derivatives). I even tried exact simulation (Brodie and Kaya - 2004), but this was way too slow for the amount of timesteps that I need. Does anyone have a suggestion for a better model for pricing these extremely long term derivatives?-FYI I have tried SABR also, which seems to be good on the vanillas but bad on the exotics.

You might want to try the GARCH diffusion,which is the process dV = k (b - V) dt + c V dB(t).Of course, there are two processes, the statistical and therisk-neutral; c should be the same in both, but feelfree to play around with k and b to fit the market.For the risk-neutral, b is usually not too far from what you think the implied vol^2 would be at very long maturities. You mustalready have a value in mind for b from your Heston model work.A fourth parameter is the equity-vol correlation (more on that below). For starters, you might want to see if the equilibrium stationary distribution can be made plausiblefor your application, given your best parameterestimates. I like this model because:(i) it's problem-free: the process certainly won't hit zero,like the Heston model, or explode to +infinity aslong as b > 0 && k > 0. (ii) It has empirical justification from a -lot- of GARCH work. With such long-dated securities, perhaps you cantake zero equity-volatility correlation (I don't know,just suggesting something). With that, vanilla pricing isrelatively simple, just average the BS vanilla priceover the vol. distribution p(t,Vbar), where Vbar is thetime-average volatility from 0 to t. That distribution is easily simulated. It should get pretty close to the stationary vol. distribution p(V) for k t >> 1, so this may meanthat at 17 years or 10 years, etc., you can just use the stationary dist.without much error. If the model looks plausible after all these checks, Iwould bring back in the correlation and fit the thingto the 1-3 year vanilla spx options. Don't worry about veryshort maturiities as the model lacks the very wide tailsfor that, but the cures for that won't help you. Then, you could start to investigate your exotics and whether or not dropping the correlation works for you ornot. (Price your stuff both with and without). A note on pricing. With or without correlation,vanilla europeans are straightforward -- you need simulate -only- the volatility process. (see the wilmottthread "Fast Monte Carlo for Vanillas ..." for more on that).If you have to do a 2 factor simulation for the exotics to get a rough price, so be it. I think the task should be to first see if you like the model.Once you like a model, then it will seem worth the trouble touse/create a pde solver or related method for your final implementation. It would be interesting to hear progress reports;if it's not proprietary, what exactly is your payoff structure? regards,

I agree that GARCH methods have considerable potential here.However, for long-maturity options it is then crucial to take a view about the character of the volatility process - is it characterized by short memory (SM, as in Alan's notes) or by a long memory (LM)? SM processes typically have volatility half-lives of a few months, so that today's volatility level has relatively little impact on the implied for a 5-year option. But LM processes allow far-away volatilities to depend on many years of historical returns - and there is compelling evidence that volatility behaves "like" a LM process.Good luck!

Good point. I see a lot of competing ideas about this, including apaper of your own. Some propose fractionally integrated models,and some suggest it is all an artifact of structural breaks, regimeswitching, etc. How should the original poster decide amongthe various alternatives?regards,

You ask a very difficult question!Best wishes.

Svetlana and Alan, thanks for your input. I have come across a paper by Christopher Jones in which he studies the dynamics of the general volatility model, dV = k(b - V)dt + c (V^g) dB(t). I discretized and calibrated this process to my data set, and the values returned were interesting. I got a g-value of around .7 and the rest of the parameters looked like my old Heston ones. When I went to price the exotic derivatives the prices were spot on, and the process never hits zero for my calibration. I believe that I agree with Svetlana in that volatility behaves like a LM process, because of these results and also the fact that in Jones' paper he calibrates his process from years of historical data to estimate the parameters. With regards to the correlation, my calibration returned a very high negative correlation, but I believe that this was mainly to reduce the error in pricing the shorter maturity options (1-5 y). The fit is pretty remarkable, I still have to see what it does with different exotic structures but it is giving me great promise in my initial tests. I will let you know of my progress. The structure of the option I ultimatley want to price is proprietary, but the few more common exotics I am pricing are 5,7,10 year lookbacks (365 days between lookback samples) and 3.5-15 and 5-15 quarterly averaged asian options. -a