Valuation of CMS range notes in a multifactor LIBOR market model

A Review of CMS Swap Pricing Approaches

Valuation of Swaps and Options on Constant Maturity CDS Spreads

A Unified Market Model for Swaptions and Constant Maturity Swaps

In this paper an efficient calibration method for the multi-factor LIBOR Market Model (LMM) is proposed and is applied for the Japanese interest rate market. At first the joint calibration method in the cap and swaption market is demonstrated using a new parameterization for the correlation matrix in the LMM. Then we implement the proposed methodology for calibrating the Japanese cap and swaption markets, where the computational procedure is shown to be tractable and provides a practical estimation for the implied correlation matrix in the LMM. The empirical analysis also illustrates that Black's swaption volatilities through our calibration fit the market data almost exactly and that the estimated implied correlation matrix is smooth and stable.

In this paper, we propose a new exogenous model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model with a perfect fit to the observed term-structure. We use the difference between two independent CIR processes and apply the deterministic-shift extension technique. To allow for a fast calibration to the market swaption surface, we apply the Gram–Charlier expansion to calculate the swaption prices in our model. We run several numerical tests to demonstrate the strengths of this model by using Monte-Carlo techniques. In particular, the model produces close Bermudan swaption prices compared to Bloomberg’s Hull–White one-factor model. Moreover, it finds constant maturity swap (CMS) rates very close to Bloomberg’s CMS rates.

Bermudan Swaptions in the LIBOR Market Model

Constant Maturity Swaps, Forward Measure and LIBOR Market Model

A Libor Market Model with Default Risk

A new variant of the LIBOR market model is implemented and calibrated simultaneously to both at-the-money and out-of-the-money caps and swaptions. This model is a two-factor version of a new class of the almost Markovian LIBOR market models with properties long sought after: (i) the almost Markovian parameterization of the LIBOR market model volatility functions is unique and asymptotically exact in the limit of a short time horizon up to a few years, (ii) only minimum plausible assumptions are required to derive the implemented volatility parameterization, (iii) the calibration yields very good results, (iv) the calibration is almost immediate, (v) the implemented LIBOR market model has a related short-rate model. Numerical results for the two-factor case show that the volatility functions for the LIBOR market model can be imported into its short-rate model cousin without adjustment.

A number of models have been successfully developed wherein the theoretical prices for caps, floors, and swaptions produced by the model are of the Black-76 form. In these models, discrete market rates are modelled like LIBOR rates in the LIBOR market models or forward swap rates in the swap market models; and under a suitable choice of numeraires, market rates can be modelled log normally. LIBOR caps and the market practice for pricing and quoting these instruments are discussed. It is shown that given a swap market model, the LIBOR rates will not be lognormal; thus, LIBOR market models and swap models are generally incompatible. Practice exercises are included.

We present a parallel algorithm and its implementation that computes lower and upper bounds for prices of Bermudan swaptions. The evolving of the underlying forward rates is assumed to follow the extended multi-factor LIBOR market model. We follow the Longstaff-Schwartz least-squares approach in computing a lower bound and the Andersen-Broadie duality-based procedure in computing an upper bound. Parallelisation in the implementation is achieved through POSIX threading. High-performance Intel MKL functions are used for regression and linear algebra operations. The parallel implementation was tested using Bermudan swaptions with different parameters on Intel multi-core machines. In all the tests the parallel program produced close results to those reported in the previous studies. Significant speedups were observed against an efficient sequential implementation built for comparison.

On the Information in the Interest Rate Term Structure and Option Prices

LIBOR Market Model with Stochastic Volatility

The Libor market model has established itself as the benchmark model for interest rate derivatives. If the observed correlation and volatility surfaces cannot be reproduced by a model, we cannot hope to get meaningful prices; therefore the crucial task, before it comes to pricing and hedging, is to calibrate the model to given market data. An overview of the Libor market model is given and it is shown how to obtain a robust calibration. We show how to extend the model to include the market smile by making use of stochastic volatility. For these stochastic volatility Libor market models a new time homogeneous skew parametrization is introduced. Furthermore a new approximative terminal correlation formula based on the same parameter averaging technique used for pricing in a stochastic volatility Libor market model is presented. We look at a very robust initial calibration procedure; here we introduce some new approximations making it possible to use global optimizers. The optimal choice of local and global optimizers for each step of the calibration will be dealt with and an analysis of the robustness of the calibrated parameters is performed. Two extensions to the cross currency Libor market model introduced by Schlögl are developed. These make use of displacements and stochastic volatility to fit the observed skew and smile respectively. The calculation of the Greeks for exotic interest rate derivatives is crucial for hedging and therefore for trading these products. For a Libor market model this task can only be fulfilled by means of nontrivial Monte Carlo based methods. A new method for the calculation of the weights for the proxy simulation scheme method is developed making use of the conditional independence of the underlying from the stochastic volatility process, which can be used for any model for which this independence holds. The developed estimators are fast enough to be used in everyday practice.

The main objective of the present thesis is to provide efficient methods and tools for calibrating Libor Market Models (LMM) to market-prices of caps, swaptions and CMS spread options, and in this way to extract the volatility and correlation information implicitly contained in these products.