This book, first published in 2000, addresses pricing and hedging derivative securities in uncertain and changing market volatility.

This is one of the only books to describe uncertain volatility models in mathematical finance and their computer implementation for portfolios of vanilla, barrier and American options in equity and FX markets. Uncertain volatility models place subjective constraints on the volatility of the stochastic process of the underlying asset and evaluate option portfolios under worst- and best-case scenarios. This book, which is bundled with software, is aimed at graduate students, researchers and practitioners who wish to study advanced aspects of volatility risk in portfolios of vanilla and exotic options. The reader is assumed to be familiar with arbitrage pricing theory.

Along with the extraordinary growth in the derivatives market over the last decade, the impact of model choice, and model parameter usage, has become a major source of valuation uncertainty. This book concentrates on equity derivatives and charts, step by step, how key assumptions on the dynamics of stocks impact on the value of exotics. The presentation is technical, but maintains a strong focus on intuition and practical application./a

Packed with insights, Lorenzo Bergomi's Stochastic Volatility Modeling explains how stochastic volatility is used to address issues arising in the modeling of derivatives, including:Which trading issues do we tackle with stochastic volatility? How do we design models and assess their relevance? How do we tell which models are usable and when does c

The trading of financial derivatives and products in financial markets has influenced the development of the world economy. Over the last few decades, a rapid growth in complex financial systems, which can generate unstable conditions in financial markets, has been observed. Therefore models are being developed to study and examine the uncertainty surrounding these financial systems in different circumstances. The important milestone of this work can be traced to the Black-Scholes formula for option pricing which was published in 1973 and revolutionized the financial industry by introducing the no-arbitrage principle [8]. This model assumed that the average rates of return and volatility are constant, however, this is not realistic. Therefore, several models have been developed, based on pragmatic studies, which generalize the Black-Scholes formula to acquire more knowledge for these financial systems. In this project, we will focus on Stochastic Differential Equations (SDEs) models in finance which do not have explicit solutions so far. In particular, Lewis [47] developed the mean-reverting-theta processes which can not only model the volatility but also the asset price. Therefore, we will establish the Euler-Maruyama (EM) numerical schemes to approximate the solution to this model and show that the EM approximate solution will converge in probability to the true solution under certain conditions. The convergence property of the corresponding step process will be examined under the same conditions to determine its application in finance. In addition, the Markov-switching format of this model can be used to explain some erratic situations observed in financial data. Under the same conditions on parameters of mean-reverting-theta model, the Markov-switching model will be examined to show that the EM approximate solution to this model will converge in probability to the true solution. Although previous models fit to a certain type of financial data, they can not be used to explain behaviour of the unpredictable abrupt structural changes in financial markets. However, the mean-reverting-theta stochastic volatility model driven by a Poisson jump process explains some of this phenomenon. Therefore, we will examine the analytical properties of EM approximate solutions to this model for two conditions of the parameters theta and beta. Since it is possible to obtain a more generalized formula for this stochastic volatility jump process, by incorporating a hybrid concept into this SDE model, we will consider the mean-reverting-theta volatility model with Poisson jumps driven by two independent Markov processes. Existing financial instruments are not strong enough to examine the convergence property of the approximate solution to this model. Therefore, we will establish EM approximate solutions to this model and examine their convergence property, when we assume similar parameter conditions to the mean-reverting-theta model. Finally, we will show that these approximate solutions of the SDE models can be used to evaluate financial quantities, options and bonds for example.

The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced. The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. The fast-growing interest for hybrid products has led to a new chapter. A special focus here is devoted to the pricing of inflation-linked derivatives. The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.

A new edition of a successful, well-established book that provides the reader with a text focused on practical rather than theoretical aspects of financial modelling Includes a new chapter devoted to volatility risk The theme of stochastic volatility reappears systematically and has been revised fundamentally, presenting a much more detailed analyses of interest-rate models