Heston model can be calibrated by call and put option market data separately. The correlation from call option is very close to 1, while the one from put option is very close to -1. Does it mean people trading call option anticipate volatility goes up, when market goes up; and people trading put option also anticipate volatility goes up, when market goes down? From historical data the correlation between underlying and its correlation is close to 0. So I think there may be some trading strategy that can make profit from this discrepancy.

This sounds totally messed up to me. Assuming the underlying is reasonably appropriate for the model, I suggest you re-do using -only OTM options (puts *and* calls both included, at mid-point quotes, and excluding options with no bids) -the VIX white paper method to fix the option-implied forward (or, equivalently, the option-implied dividend yield)-a good range of maturities, say 1 month to 2 years, all included in a single calibration.If, after all that, you are still getting something from bizarro world, I would say post some charts of IV's (market and the re-done calibrated fits).The market IV's should be *smooth* as you move through the at-the-money point if you have carried out my suggestions. Of course, the Heston model fits will also be smooth.BTW, what is the underlying?

Thank you Alan.The underlying is an ETF mutual fund in a developing country.The BS implied vols for call option are not smooth even only OTM options are included. The BS implied vols for put option are just going down from low strike to high strike, which is much "smoother". There are great different between the call and put options' implied vol.Actually, the calibration errors of Heston model look small, which is around 10% of the option price on average. But I am not sure why the correlation is so close to 1 or -1

I see. In the US, great differences between (midpoint) put/call implied volatility is typically a symptom of being "hard to borrow",an issue which is significantly (but probably not completely) cured by the procedure I suggested. Yours may simply be due to extremely large bid-ask spreads or something else peculiar to that market, or the ETF structure. One thought: maybe the ETF trades substantially away from its NAV or is in danger of doing so? I would resolve that issue first before even thinking about a model. If there are other (perhaps illusionary) arbitrage opportunities in your options data, say negative butterfly spreads, you may want to first do an arbitrage-free fit, like Fengler's and thencalibrate your model to that fit.

Are you using your own implementation of Heston pricer or something that has been thoroughly checked by a third party? If it's your own you may want to check it works correctly for known cases. Which forward price are you using? Are options American or European?

Thank you all for your suggestions.The short sell rate is indeed very high, roughly 8% p.a. in this market. May I know if there is any industrial standard method to calculate implied volatility in such a case?If no, is there any problem if I do the following?Still assume the put-cal parity holds. I fixed the risk-free interest rate,and then use the market price of call and put to calculate the implied dividend by the put-call parity equation. Use this implied dividend to caluculate BS implied volatility, in such a case the put-call implied vol must equal.To calibrate Heston model, I take the average of implied dividend along the striks and then calibrate to match the BS implied vol.

I don't like the idea of an implied dividend at each strike. After all, it is supposed to be a proxy for the cost to borrow -- astrike-independent, but perhaps maturity-dependent concept. Instead, for US equities, I prefer the option-implied dividend given by the method I suggested in my first response, which differs by maturity but is fixed across strikes. Whether or not this procedure will produce something sensible for your market -- I have no way of knowing.

Thank you Alan. The method in white paper "The CBOE Volatility Index - VIX"generate a better result on BS implied vol. It's not a very smooth curve, but I can see some oscillations of volatility between 25% and 27% for options with long expiration days.1.However, I can not understand well the logics behind this method. 1.1 Why only OTM options are considered? Is it because they are traded more actively?1.2 Why do they calculate foward price," by identifying the strike price at which the absolute difference between the call and put prices is smallest"? Is it because the long and short side has the smallest disagreement on this strike?I calculate implied dividend like this Forward_imp=S0*exp[-(r-q)*T],where Forward_imp is calcluated from the white paper. S0 is the spot price, r is the fixed risk-free interest rate, T is time to expiration, q is the implied dividend to calculate. 2.On this BS surface, I try to calibrate it with Heston model, but I find the Heston model is very sensitive to my initial guess. By different initial guess, one parameter can be abnormal, say kappa reaches 1.0e-5, or sigma reaches 1.0e-5, or rho equal to -0.9999. Is this because of some option prices are arbitragable (I can obsever arbitrage on the quote indeed)? I think a fast cure is to add constraints to the parameters.

1.1 Why only OTM options are considered? Is it because they are traded more actively?... yes, and the spreads are smaller, and oft-times the theory replication is written using them.1.2 Why do they calculate foward price," by identifying the strike price at which the absolute difference between the call and put prices is smallest"? Is it because the long and short side has the smallest disagreement on this strike?... In a perfect market, selling c(K) and buying p(K), with c=p would replicate entering a forward position with K = forward price.Is this because of some option prices are arbitragable (I can obsever arbitrage on the quote indeed)? ... Perhaps, or perhaps the objective function has multiple local and shallow minima.

QuoteOriginally posted by: EdisonCruiseThank you Alan.The underlying is an ETF mutual fund in a developing country.The BS implied vols for call option are not smooth even only OTM options are included. The BS implied vols for put option are just going down from low strike to high strike, which is much "smoother". There are great different between the call and put options' implied vol.Actually, the calibration errors of Heston model look small, which is around 10% of the option price on average. But I am not sure why the correlation is so close to 1 or -1Basis point difference is small and within an estimation error, 10 basis points is large and should make your palms sweat but 10% difference between market and model is huge.Before going on with the complicated things make sure that- your model fits mostly traded strikes. And by that I mean from 75% to 115%. These are the strikes that trade in equities assuming maturities are from 2y to 5y.- use multiple maturities for each calibration. including only one skew per day makes no sense since you are supposed to come up with a model that can price all vanillas properly- use different div yields for forwards with different maturities- use different risk-free rates for different maturities. this is especially important if you have quite fresh data and work with US equitiesAlso. You should check this out https://www.math.nyu.edu/faculty/avella ... _Zhang.pdf. Personally, I have not read the option pricing section but it must have something to do with adjustments to the short gamma run by ETFs.