I know when pricing a Bermudan swaption by Hull-White, people usually calibrate mean reversion to auto correlation of core swap rates. Let's say for a time grid like T1, T2, ... Tn, we have core swap rate S(T1, Tn), S(T2, Tn),...S(Tn-1, Tn). Can someone tell me what is the auto correlation used in calibration? The correlation between S(T1, Tn), S(T2, Tn),...S(Tn-1, Tn), the different swap rate? or the same swap rate like S(T1, Tn) ate different fixing time t, s where t<s<T1?Another thing confusing me is that according to the following equation we can approximate the auto correlation of swap rate like this, but the right hand side is only a function of fixing time t, s, and mean reversion parameter, it has nothing to do with what swap rate we are using, which should be identified by index k, n, l ,p. Then what auto correlation should we use?

What is K and u in your expression on the RHS?Thanks,

k is the mean reversion of Hull-White, if it's a constant, we have RHS = √(e^2kt-1) / √(e^2ks-1)On the LHS, k,n,l,p are index of time for swap rate, i.e. swap rate starting from time k, ending at time n, swap rate starting from time l, ending at time p. I'm really confused, don't know what core swap rate to use since RHS has nothing to do with what swap rate are concerned.

Hi,It's a little bit strange when you compute Expectation of Variance volatility of x(k,n)(t) and x(l,p)(s) doesn't appear.With this correction , we will through the volatility of OU process the dependency of Swap rate

First of all, which swap rate correlation should be used?The correlation between S(T1, Tn), S(T2, Tn),...S(Tn-1, Tn), the different swap rate? or the same swap rate like S(T1, Tn)(t) S(T1, Tn)(s) at different fixing time t, s where t<s<T1?

why the auto correlation is so important in pricing Bermudan option?