Shapes of Implied Volatility with Positive Mass at Zero


Book Description

We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can -- theoretically -- distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the CEV process, and in jump-to-default models. Note that while Lee's moment formula tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics do not apply in this setting -- essentially due to the breakdown of Put-Call duality.




Large Deviations and Asymptotic Methods in Finance


Book Description

Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts. Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour. Graduate students, researchers and practitioners will find this book very useful, and the diversity of topics will appeal to people from mathematical finance, probability theory and differential geometry.




Modern SABR Analytics


Book Description

Focusing on recent advances in option pricing under the SABR model, this book shows how to price options under this model in an arbitrage-free, theoretically consistent manner. It extends SABR to a negative rates environment, and shows how to generalize it to a similar model with additional degrees of freedom, allowing simultaneous model calibration to swaptions and CMSs. Since the SABR model is used on practically every trading floor to construct interest rate options volatility cubes in an arbitrage-free manner, a careful treatment of it is extremely important. The book will be of interest to experienced industry practitioners, as well as to students and professors in academia. Aimed mainly at financial industry practitioners (for example quants and former physicists) this book will also be interesting to mathematicians who seek intuition in the mathematical finance.







From Implied to Spot Volatilities


Book Description

This paper is concerned with the relation between spot and implied volatilities. The main result is the derivation of a new equation which gives the dynamics of the spot volatility in terms of the shape and the dynamics of the implied volatility surface. This equation is a consequence of no-arbitrage constraints on the implied volatility surface right before expiry. We first observe that the spot volatility can be recovered from the limit as the expiry tends to zero of at-the-money implied volatilities. Then, we derive the semimartingale decomposition of implied volatilities at any expiry and strike from the no-arbitrage condition. Finally the spot volatility dynamics is found by performing an asymptotic analysis of these dynamics as the expiry tends to zero. As a consequence of this equation, we give general formulas to compute the shape of the implied volatility surface around the at-the-money strike and for short expiries in general spot volatilitymodels.







Implied Volatility


Book Description

Perpetual motion is absolute zero thus implied volatility (universally).




A Structural Framework for the Pricing of Corporate Securities


Book Description

A treatment of structural credit risk models for simultaneous and consistent pricing of corporate securities. This book takes us from the economic principles of firm value models to the empirical implementation, through the development of an economic framework. It provides exposition of corporate securities pricing for academics and practitioners.




Volatility Trading, + website


Book Description

In Volatility Trading, Sinclair offers you a quantitative model for measuring volatility in order to gain an edge in your everyday option trading endeavors. With an accessible, straightforward approach. He guides traders through the basics of option pricing, volatility measurement, hedging, money management, and trade evaluation. In addition, Sinclair explains the often-overlooked psychological aspects of trading, revealing both how behavioral psychology can create market conditions traders can take advantage of-and how it can lead them astray. Psychological biases, he asserts, are probably the drivers behind most sources of edge available to a volatility trader. Your goal, Sinclair explains, must be clearly defined and easily expressed-if you cannot explain it in one sentence, you probably aren't completely clear about what it is. The same applies to your statistical edge. If you do not know exactly what your edge is, you shouldn't trade. He shows how, in addition to the numerical evaluation of a potential trade, you should be able to identify and evaluate the reason why implied volatility is priced where it is, that is, why an edge exists. This means it is also necessary to be on top of recent news stories, sector trends, and behavioral psychology. Finally, Sinclair underscores why trades need to be sized correctly, which means that each trade is evaluated according to its projected return and risk in the overall context of your goals. As the author concludes, while we also need to pay attention to seemingly mundane things like having good execution software, a comfortable office, and getting enough sleep, it is knowledge that is the ultimate source of edge. So, all else being equal, the trader with the greater knowledge will be the more successful. This book, and its companion CD-ROM, will provide that knowledge. The CD-ROM includes spreadsheets designed to help you forecast volatility and evaluate trades together with simulation engines.




Fuel Hedging and Risk Management


Book Description

A hands-on guide to navigating the new fuel markets Fuel Hedging and Risk Management: Strategies for Airlines, Shippers and Other Consumers provides a clear and practical understanding of commodity price dynamics, key fuel hedging techniques, and risk management strategies for the corporate fuel consumer. It covers the commodity markets and derivative instruments in a manner accessible to corporate treasurers, financial officers, risk managers, commodity traders, structurers, as well as quantitative professionals dealing in the energy markets. The book includes a wide variety of key topics related to commodities and derivatives markets, financial risk analysis of commodity consumers, hedge program design and implementation, vanilla derivatives and exotic hedging products. The book is unique in providing intuitive guidance on understanding the dynamics of forward curves and volatility term structure for commodities, fuel derivatives valuation and counterparty risk concepts such as CVA, DVA and FVA. Fully up-to-date and relevant, this book includes comprehensive case studies that illustrate the hedging process from conception to execution and monitoring of hedges in diverse situations. This practical guide will help the reader: Gain expert insight into all aspects of fuel hedging, price and volatility drivers and dynamics. Develop a framework for financial risk analysis and hedge programs. Navigate volatile energy markets by employing effective risk management techniques. Manage unwanted risks associated with commodity derivatives by understanding liquidity and credit risk calculations, exposure optimization techniques, credit charges such as CVA, DVA, FVA, etc.