# Libor Market Model Calibration

Currently I am doing a research on the plain vanilla multi-curve framework Libor Market Model meaning that no stochastic volatility is involved. I had the idea to calibrate to the swaption market. In the volatility calibration procedure I minimize the error between the market quoted swaption volatilities and the volatilities obtained by Rebonato's swaption volatility approximation formula. Which minimization method would be convenient for this particular problem?

I have the following parametrizations for the instantaneous correlation and instantaneous volatility, respectively: $\rho_{i,j} = e^{-\beta |T_i - T_j |}$

and

$\sigma_i(t) = (a+b(T_i - t))e^{-c(T_i-t)}+d$

I would like to include the correlation parameter $\beta$ in the calibration.

Does anyone have a suggestion on how I can calibrate the correlation? I suppose this has to be done before calibrating the volatility parameters $a,b,c,d$ since the Rebonato's approximation formula requires the instataneous correlation as input. To which target value do I have to compare it to since no instantaneous correlation can directly be deduced from the market quoted volatilities and which error do I have to minimize?

Furthermore, after having calibrated the model I can use the calibrated parameters in a Monte Carlo routing where the forward Libor rates are simulated under the spot measure to price a swaption. Can anyone advise me on how I can test the accuracy of the calibrated parameters by comparing the swaption prices obtained by Monte Carlo, by Rebonato's approximation formula and the market quoted volatilities? (side question: Do I retrieve the price of swaption obtained throught Monte Carlo in terms of volatility by taking the volatility inserted in the simulation?)

Would it make sense to compare these results in a graph where different strikes are plotted (ATM + x) against the three different types of volatilities described above for a given set of calibrated parameters? Or is this only relevant when the volatility in the model is stochastic?

When I calibrate the Libor Market Model using Rebonato's approach, I use Rebonato's closed form approximation formula which allows the calibration of correlation parameter along with the other vol paramaters. 