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Dissertation writing help: On the Calibration of the SABR-Libor Market Model Correlations

SABR-LMM model - Custom Dissertation Writing Service

This work is concerned with the SABR-LMM model. This is a term structure model of interest forward rates with stochastic volatility that is a natural extension of both, the LIBOR market model (Brace-Gatarek-Musiela [1997]) and the SABR stochastic volatility model of Hagan et al. [2002]. While the seminal approximation formula (developed by Hagan et al. [2002]) to implied Black volatility using the SABR model parameters allows for a successful calibration of each forward rate dynamics to the volatility smile of the respective caplets/floorlets, an adequate calibration of the rich correlation structure of SABR-LMM (correlations among the forward rates, the volatilities and the cross correlations) is a challenging topic and of great interest in practice. Although widely used for calibration, it is well known that swaptions' volatilities carry only little information about correlations among the forward rates. As practically successful for the classical LMM, desirable would be to take the market swap rate correlations into account for the model calibration.

In this study we develop a new approach of calibrating the model correlations, aiming at incorporating the market information about the forward rate correlations implied from more correlation-sensitive products such as CMS spread derivatives, in which also swap rate correlations are involved. To this end we derive a displaced-diffusion model for the swap rate spreads with a SABR stochastic volatility. This we achieve by applying the Markovian projection technique which approximates the dynamics of the basket of forward rates, in terms of the terminal distribution, by a univariate displaced-diffusion. The CMS spread derivatives can then be priced using the SABR formulas for the implied volatility, taking the whole market smile of CMS spread options into consideration. For the ATM values in the payoff measure of the projected SDE we use a standard smile-consistent replication of the necessary convexity adjustment with swaptions. Numerical simulations conclude the work, giving a comparison between this method and the classical one of calibrating the model correlations to swaption volatilities. Furthermore, we study the performance of different parameterizations of the correlation (sub-)matrices.