Hi everybodyWhich is the market standard in joint modeling different equities under stochastic volatility?These are my ideas, but they are not still tested:case 1: one marketIf equities X, Y, Z belong to the same market I, one could try to calibrate the parameters of the stock and vol processes V on the options written on I. Then X, Y and Z will be governed by two brownians, one is the market brownian multiplied by the market stochastic vol, the other is an independent brownian multiplied by a const vol coefficient.[the same for Y, Z, each with its rho and sigma coefficient]Yes, I suppose that playing just with rho and sigma I will not get a superb fit. But I believe it's too expensive to have stoch vol processes for each stock of the market.case 2: different marketsIn this case I would calibrate a stoch vol model for each market. Here the problem is to identify correlation among the brownian driving stocks and those driving volatilities. Are volatilities directly correlated or it's just a side effect of stocks correlation?Stefano

I don't think there is a market standard because I don'tbelieve there has been a lot of work in multivariate stochastic vol.However, I agree that your case I model is a natural first thingto try, since it's (roughly speaking) a CAPM-type model where weonly let the market vol. be stochastic. It should lead naturallyto individual names having less pronounced 'skews' than themarket, which I believe is true. To the extent that indivudual nameshave more symmetrical smiles than the index, you may need toadd some jumps. Please post any interesting results that you find.You might want to search on the multivariate GARCH literature,which is perhaps more extensive on this topic.regards,