In this paper we claim to have developed the optimal methodology for non-parametric calibration of market model for LIBOR term rates to a set of benchmark at-the-money (ATM) cap/floor and swaption prices, as well as the historic correlation of the LIBOR rates. It is well-known that the calibration to correlation matrices can be decoupled from the calibration to the input prices. Yet both remain to be middle-scale optimization problems. The former typically requires a rank-reduction pre-processing on the input matrices, which has been very costly to achieve. The latter naturally arises as a constraint minimization problem with quadratic objective and constraint functions (of local volatility coefficients), which however does not render itself to any standard methods. We superpose a convex function to the objective function, and turn the calibration into two ``mini-max" problems along the approach of Lagrange multiplier. The key to the efficiency of our new methodology is that the inner ``max" problems are solved as linear eigenvalue problem, which requires no iteration and thus is extremely fast. We have justified the convergence of the Lagrange multiplier approach, and achieved numerically very good quality of calibrations, in the sense of stability of the volatility surface with respect to the changes in the number of driving factors or input price values. With some extra costs, we can also obtained the sensitivities of the volatility surface with respect to the input prices. Such sensitivities can be used to calculate the hedging ratio of a portfolio by the benchmark instruments.
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