Does the volatility parametrization that I have chosen give an underspecified model? Which volatility parametrization in the Libor Market Model would suit the best for the particular case described here under?

I have implemented the Libor Market Model in Matlab and am simulating paths of forward Libor Rates under the spot measure, whose dynamics are given by $$dL_n\left(t\right)=\sigma_n\left(t\right)L_n\left(t\right)\sum_{j=q\left(t\right)}^n \frac{\tau_j \rho_{j,n} \sigma_j\left(t\right)L_j\left(t\right)}{1+\tau_j L_j\left(t\right)}dt + \sigma_n\left(t\right)L_n\left(t\right)dW\left(t\right)$$ where $$L_n\left(t\right):=L\left(t;T_n,T_{n+1}\right),$$ $$\tau_n = T_{n+1}-T_n,$$ For the volatility parametrization I have chosen: $$\sigma_n\left(t\right) = \left(a+b\left(T_n-t\right)\right)e^{-c\left(T_n-t\right)}+d$$ where $a,b,c,d$ are constants and $T_n-t$ represents the time to maturity. For the correlation I have taken $$\rho_{i,j}\left(t\right) = e^{-\beta\left|i-j\right|}$$ with $\beta=0.05$ constant. I have calibrated the Libor Market Model to the co-terminal market swaption volatilities 1Y15Y,2Y14Y,...,14Y1Y,15Y1Y.

After calibration, I see that the market swaption volatilities are not fitted accurately with this volatility parametrization consisting of 4 degrees of freedom. My question is if it is possible that this model is underspecified with this parametrization and if it would be a wise choice to choose another parametrization with more degrees of freedom if we are calibrating to 15 swaption vols. If so, which volatility parametrization would you recommend?

Thanks in advance.



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