Asymptotic Chaos Expansions in Finance : Theory and Practice 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 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