I am trying to simulate the Libor Market Model and the Heston Model together. I have got 30 forward rates to be simulated using LMM and 20 equities using the Heston model.
For the forward rate correlation, I am using a parametrized form, corr(t,T) = \beta_0 + (1 - \beta_0) * exp(-\beta_1 * |T - t|), this is the correlation structure I have assumed in the calibration part as well.
On the other hand, I do not have any predefined structure of correlation among equities.
How do I link the parametric form of correlation that I have used in the calibration of LMM with the historical data of equities? Should I consider building a hybrid model?
If I try to find a correlation matrix of 30 fwd rates + 20 equities, then I lose the parametric assumption on the correlation among fwd rates.
Am I missing something obvious here?
thank you in advance,