LIBOR and Swap Market Models

Libor and Swap Market Models for the Pricing of Interest Rate Derivatives: An Empirical Analysis

Bermudan Swaptions in the LIBOR Market Model

LIBOR Market Model with Stochastic Volatility

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 Libor Market Model with Default Risk

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 chapter we draw our attention to the market models. We start by briefly discussing short rate models, with the goal of seeing where they fit in the story of interest rate modelling. We then make a natural transition from the more general short rate models to the Heath-Jarrow-Morton framework, then to the definition of the LIBOR Market Model (LMM).

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.

Constant Maturity Swaps, Forward Measure and LIBOR Market Model

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

Calibration, Simulation and Hedging in a Heston Libor Market Model with Stochastic Basis

In the more classical models for equities and interest rates evolution, constant volatility is usually assumed. However, in practice the volatilities are not constant in financial markets and different models allowing a varying local or stochastic volatility also appear in the literature. Particularly, here we consider the SABR model that has been first introduced in a paper by Hagan and coworkers, where an asymptotic closed-form formula for the implied volatility of European plain-vanilla options with short maturities is proposed. More recently, different works (Mercurio and Morini, Modeling Interest Rates: Advances in Derivatives Pricing, Risk Books 2009; Hagan and Lesniewski, LIBOR market model with SABR style stochastic volatility. Working Paper. http://lesniewski.us/papers/working/SABRLMM.pdf, 2008; Rebonato, A time-homogeneous SABR-consistent extension of the LMM. Risk, 2008) have extended the use of SABR model in the context of LIBOR market models for the evolution of forward rates (SABR-LMM). One drawback of these models in practice comes from the increase of computational cost, mainly due to the growth of model parameters to be calibrated. Additionally, sometimes either it is not always possible to compute an analytical approximation for the implied volatility or its expression results to be very complex, so that numerical methods (for example, Monte Carlo in the calibration process) have to be used. In this work we mainly review some recently proposed global optimization techniques based on Simulated Annealing (SA) algorithms and its implementation on Graphics Processing Units (GPUs) in order to highly speed up the calibration and pricing of different kinds of options and interest rate derivatives. Finally, we present some examples corresponding to real market data.KeywordsSABR volatility modelsSABR/LIBOR market modelsParallel simulated annealingGPUs

Having in view the pricing of commodity derivatives in Libor Market Model (LMM) setting, we first analyze the set of basic rates we need to formulate the model by using the spanning tree concept taken from graph theory. Next, we present an efficient procedure for Monte Carlo simulation of the dynamics of the rates associated to LMM, avoiding the presence of the rates dependent drifts (drift‐free simulation) and the presence of negative deflated bond prices and negative forward rates. The method is based upon a new parameterization of the martingales introduced by Glasserman and Zhao and it is extended to a Cross‐Market Model for commodities. Finally, a particular example of commodity derivative (spread option) pricing problem is considered. Copyright © 2014 John Wiley & Sons, Ltd.

In the preceding chapters we considered different term structure models, for instance, the Gaussian short rate models or models for the instantaneous forward rate.