The Libor Market Model And Its Calibration To The South African Market

The South African interest rate market has mainly been focused on vanilla interest rate products and hence can be seen as underdeveloped in this regard when compared, for instance, to the associated equity market. Market participants subscribe this aspect to a lack of demand and sophistication of investors within the market. This is, however, expected to change given the influx of international banks into the South African market over the past couple of years. The current market methodology, for the pricing of vanilla interest rate options in the South African market, is the standard Black model with some mechanism to incorporate interest rate smiles. This mechanism is typically in the form of the SABR model. The most signi cant drawback of this approach is the fact that it models each forward rate in isolation. Hence, there is no way to incorporate the joint dynamics between different forward rates and consequently cannot be used for the pricing of exotic interest rate options. In anticipation of these new market developments, we explore the possibility of calibrating the LIBOR market model to the South African market. This dissertation follows a bottom up approach and hence considers all aspects associated with such an implementation. The work mainly focuses on the calibration to at-the-money interest rate options. A possible extension to the SABR model, while remaining within the LMM framework, is considered in the final chapter. Copyright.

The Libor Market Model (LMM) is a mathematical model for pricing and risk management of interest rate derivatives and has been built on the framework of modelling forward rates. For the conceptual understanding of the model a strong background in the fields of mathematics, statistics, finance and especially for implementation, computer science is necessary. The book provides the ne cessary groundwork to understand the LMM and delivers a framework to implement a working model where possible calibration and parameterization methods for volatility and correlation are explained. Special emphasis lies also on the trade off of speed and correctness where differences in choosing random number generators and the advantages of factor reduction are shown.

The LIBOR Market Model (LMM) is the first model of interest rates dynamics consistent with the market practice of pricing interest rate derivatives and therefore it is widely used by financial institution for valuation of interest rate derivatives. This book provides a full practitioner's approach to the LIBOR Market Model. It adopts the specific language of a quantitative analyst to the largest possible level and is one of first books on the subject written entirely by quants. The book is divided into three parts - theory, calibration and simulation. New and important issues are covered, such as various drift approximations, various parametric and nonparametric calibrations, and the uncertain volatility approach to smile modelling; a version of the HJM model based on market observables and the duality between BGM and HJM models. Co-authored by Dariusz Gatarek, the 'G' in the BGM model who is internationally known for his work on LIBOR market models, this book offers an essential perspective on the global benchmark for short-term interest rates.

The Libor Market Model is a financial model used to price and hedge exotic interest rate derivatives. The model is accepted and used widely due to its consistence with the standard market formula, Black's cap (floor) formula. This compatibility simplifies the calibration because the Black's quoted prices for standard interest rate derivatives can be directly used as an input for the model. The goal of this book is to examine the Libor Market Model theoretically and apply it practically to the pricing of standard caps, discrete barriers, European swaptions and ratchets. The dynamic of the Libor Market Model will be derived and all steps of its implementation using Monte Carlo simulation will be explained. Implementation is fulfilled using different volatility and correlation structuring. Certain care should be taken when calibrating the Libor Market Model and structuring the forward rate volatilities and correlations as they may affect prices of interest rate derivatives considerably. The book is aimed at graduate students of finance and practitioners implementing this model in practice. C source code, used for pricing interest rate derivatives in this book, may be ordered at the following web site: http: //www.irina-goetsch.com/libor-market-model/

This thesis presents a study of LIBOR market model calibration. In particular, the study builds on the prevailing calibration methodologies in an attempt to find a method that simultaneously recovers implied volatility and forward rate correlations structures from market prices of plain vanilla options. In order to ensure that complex derivative pricing and hedging requirements are jointly addressed, the study extends the performance analysis of calibration methods from a static level of goodness-of-fit with market prices test, to a dynamic level of approximation to next period's LIBOR (London Interbank Offer Rate) dynamics when tested on a series of market prices. Among the methodologies considered, the results show that for caplets, full calibration results in least pricing error when tested on an intra-day pricing prediction, and generates a stable evolution of day-to-day implied volatility. For swaptions, analytic approximation provides better estimate on an intra-day pricing but Monte Carlo simulation with parametrized correlations matrix provides a stable evolution of volatility and correlation (or covariance). This approach for swaptions calibration outperforms the other methods used despite the modifications made in volatility and initial thetas specifications. All together, the results suggest that the Monte Carlo method with parametrized correlations appear to be superior as it provides smooth evolution of covariance of forward rates that is desired in complex derivative pricing and hedging.

The main objective of this work is to construct and implement a LIBOR market model and a Swaptions market model for the South African market.In his Thesis, Victor Gumbo starts by recapitulating the basic theory of arbitrage pricing, forward measures and term structure models for zero-coupon bonds. He goes on to describe and analyze the LIBOR market models. Apart from the standard models, he goes on to discusses market practice and provides numerous formulae for pricing as well as terminal measure existence. In Chapter 3, he gives a similar outline for Swap Market models. It should be emphasized that these models are quite complicated from a theoretical point of view but Victor manages to give an extremely pedagical account of this difficult theory.

Interest rate traders have been using the SABR model to price vanilla products for more than a decade. However this model suffers however from a severe limitation: its inability to value exotic products. A term structure model à la LIBOR Market Model (LMM) is often employed to value these more complex derivatives, however the LMM is unable to capture the volatility smile. A joint SABR LIBOR Market Model is the natural evolution towards a consistent pricing of vanilla and exotic products. Knowledge of these models is essential to all aspiring interest rate quants, traders and risk managers, as well an understanding of their failings and alternatives. SABR and SABR Libor Market Models in Practice is an accessible guide to modern interest rate modelling. Rather than covering an array of models which are seldom used in practice, it focuses on the SABR model, the market standard for vanilla products, the LIBOR Market Model, the most commonly used model for exotic products and the extended SABR LIBOR Market Model. The book takes a hands-on approach, demonstrating simply how to implement and work with these models in a market setting. It bridges the gap between the understanding of the models from a conceptual and mathematical perspective and the actual implementation by supplementing the interest rate theory with modelling specific, practical code examples written in Python.

The Libor Market Models arise from the general multi-factor Heath-Jarrow-Morton interest rate model. The Libor Market Models assume that, say, 3 months simple rates are log-normal. With pricing formulae for caps/floors and swaptions this makes the model easy to calibrate for a specific choice of volatility function. We describe how to calibrate the model using a non-parametric volatility function. We apply a smoothness criteria to the quality of fit used in calibration as erratic volatilities otherwise result from the calibration. We perform numerical studies using real market data from several markets to check the robustness of the implementation towards changes in model/calibration parameters. The implementation is indeed very robust and market quotes are matched within bid-offer spread.