Asymptotic Chaos Expansions in Finance

Stochastic instantaneous volatility models such as Heston, SABR or SV-LMM have mostly been developed to control the shape and joint dynamics of the implied volatility surface. In principle, they are well suited for pricing and hedging vanilla and exotic options, for relative value strategies or for risk management. In practice however, most SV models lack a closed form valuation for European options. This book presents the recently developed Asymptotic Chaos Expansions methodology (ACE) which addresses that issue. Indeed its generic algorithm provides, for any regular SV model, the pure asymptotes at any order for both the static and dynamic maps of the implied volatility surface. Furthermore, ACE is programmable and can complement other approximation methods. Hence it allows a systematic approach to designing, parameterising, calibrating and exploiting SV models, typically for Vega hedging or American Monte-Carlo.

Asymptotic Chaos Expansions in Finance illustrates the ACE approach for single underlyings (such as a stock price or FX rate), baskets (indexes, spreads) and term structure models (especially SV-HJM and SV-LMM). It also establishes fundamental links between the Wiener chaos of the instantaneous volatility and the small-time asymptotic structure of the stochastic implied volatility framework. It is addressed primarily to financial mathematics researchers and graduate students, interested in stochastic volatility, asymptotics or market models. Moreover, as it contains many self-contained approximation results, it will be useful to practitioners modelling the shape of the smile and its evolution.

Exposes some structural links, both static and dynamic, between classic stochastic instantaneous volatility models and the more recent stochastic implied volatility model class Provides a programmable methodology to compute the small-time asymptotics, at any order, of the smile associated to any regular stochastic volatility model Presents simple but powerful illustrations of the methodology, in particular some applications to Local Volatility models which expose the systematic bias of the 'most probable path' method Includes self-contained, high-order generic approximations for single-underlying SV models (such as Heston or SABR) to improve calibration and Vega-hedging Extends the ACE approach progressively, first to multi-dimensional frameworks and baskets, then to term structure models. In particular, derives the asymptotic smiles of generic SV-HJM and SV-LMM models Includes supplementary material: sn.pub/extras

Autorentext
David Nicolay received his Ph.D. degree in financial mathematics from Ecole Polytechnique, France. Currently he is a front office quantitative researcher for a financial institution in London. His research interests include the modelling of interest rates and hybrid derivatives, Monte-Carlo methods and asymptotic approaches.

Inhalt
Introduction.- Volatility dynamics for a single underlying: foundations.- Volatility dynamics for a single underlying: advanced methods.- Practical applications and testing.- Volatility dynamics in a term structure.- Implied Dynamics in the SV-HJM framework.- Implied Dynamics in the SV-LMM framework.- Conclusion.