I do not know to which swaption volatility matrix I have to calibrate the LMM in order to price back correctly the swaptions corresponding to the underlying swaps of a Bermudan Swaption.
My problem: For the LMM, I use the simple correlation form $\rho_{i,j}=e^{-\beta|i-j|}$ with beta fixed. This parameter is not taken into account in the calibration. For the volatility I use the form $$\sigma_i\left(t\right) = \phi_i \left(\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right)$$ Initially I take $\phi_i=1$ for all $i$ and calibrate the parameters $a,b,c,d$ to the ATM swaption market volatilities matrix by minimizing the MSE between these volatilities and the ones determined by the Rebonato swaption volatility approximation formula. We obtain: $$V_{ATM swaptions}^{Reb} = V_{ATM swaptions}^{market}$$ Afterwards we determine the parameters $\phi_i$ such that the diagonal of a fixed strike $K$ market volatility matrix is fitted exactly: $$V_{K swaptions}^{Reb} = V_{K swaptions}^{market}$$ It is important that the diagonal swaption volatilities (=swaption vols of swaptions corresponding to the underlying swaps of a Bermudan Swaption) of a fixed strike $K$ swaption vol matrix are priced correctly when we want to price a Bermudan Swaption with strike $K$.
However the Rebonato approximation formula is only accurate for ATM strikes. Hence, comparing the Rebonato approximated price with strike $K$ and the price obtained by a Monte Carlo routine with these calibrated parameters, we would in general see that: $$V_{K swaptions}^{Monte Carlo} \neq V_{K swaptions}^{Reb} = V_{K swaptions}^{market}$$ Hence the swaptions corresponding to the underlying swaps of a Bermudan Swaptions are not priced back correctly.
Does anybody know how I can improve my calibration routine in order to price back correctly the swaptions corresponding to a fixed strike?
Any help is appreciated. Thanks in advance.