Volatility dynamics

Abstract : We establish asymptotic links between two classes of stochastic volatility models describing the same derivative market : - a generic stochastic instantaneous volatility (SInsV) model, whose SDE system is a formal Wiener chaos without any specified state variable. - a sliding stochastic implied volatility (SImpV) class, another market model describing explicitly the joint dynamics of the underlying and of the associated European option surface. Each of these connections is achieved by layer, between a group of SInsV coefficients and set of (static and dynamic) SImpV differentials. The asymptotic approach leads to these cross-differentials being taken at the zero-expiry, At-The-Money point. We progress from a simple single-underlying and bi-dimensional setup, first to a multi-dimensional configuration, and then to a term-structure framework. We expose the structural modelling constraints and the asymmetry between the direct problem (from SInsV to SImpV) and the inverse one. We show that this Asymptotic Chaos Expansion (ACE) methodology is a powerful tool for model design and analysis. Focusing on local volatility models and their extensions, we compare ACE with the literature and exhibit a systematic bias in Gatheral's heuristics. In the multi-dimensional context we focus on stochastic-weights baskets, for which ACE provides intuitive results underlining the embedded induction. In the interest rates environment, we derive the first layer of smile descriptors for caplets, swaptions and bond options, within both a SV-HJM and a SV-LMM framework. Also, we prove that ACE can be automated for generic models, at any order, without formal calculus. The interest this algorithm is demonstrated by computing manually the 2nd and 3rd layers, in a generic bi-dimensional SInsV model. We present the applicative potential of ACE for calibration, pricing, hedging or relative value purposes, illustrated with numerical tests on the CEV-SABR model.
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Computational Finance [q-fin.CP]. Ecole Polytechnique X, 2011. English
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Submitted on : Monday, July 11, 2011 - 4:57:11 PM
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  • HAL Id : pastel-00600106, version 1



David Nicolay. Volatility dynamics. Computational Finance [q-fin.CP]. Ecole Polytechnique X, 2011. English. 〈pastel-00600106〉



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