Fast At-the-Money Calibration of the LIBOR Market Model through Lagrange Multipliers

Abstract. We claim to have developed the optimal methodology for non-parametric cal-ibration of market model to the prices of at-the-money (ATM) caps/floors and swaptions, and to the historic correlations of the LIBOR rates. We take the approach of divide-and-conquer: first fit the model to historic correlations, then to the implied Black volatilities of the input options. Regularization is adopted and the calibration is cast into minimization-maximization problems by the method of Lagrange multiplier. By utilizing the quadratic functional form of both objective function and constraints, we solve the inner maximiza-tion problems with a single matrix eigenvalue decomposition, which renders the efficiency of our method. The outer minimization problems, meanwhile, are nicely subdued by gradient-based descending methods due to the convexity of the objective functions. The well-posedness of the Lagrange multiplier problems and the convergence of the descending methods are rigorously justified. Numerical results show that we have achieved very quality calibration. We have also developed a technique to calculate the hedging ratios of a de-rivative security with respect to the benchmark derivative instruments, using the auxiliary results of the calibration. Key words: LIBOR market model, non-parametric calibration, constrained optimiza-tion, Lagrange multiplier method