Abstract
Stochastic calculus without probability: pathwise stochastic integration and applications in mathematical finance
by
Rama Cont
Mathematical models in finance are formulated in probabilistic terms and use the tools of stochastic calculus to derive pricing and hedging formulas and risk measures. Yet the specification of such stochastic models is subject to great uncertainty and, at the end, market participants experience are concerned with their loss in a single scenario, the realized historical scenario, rather than an average across hypothetical scenarios. This calls for a pathwise framework for in which all quantities are defined scenario by scenario.
We show that this program may be realized using a notion of pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums, extending an idea of H. Follmer (1979) to path-dependent integrands. We show that this integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals and obtain a pathwise `signal plus noise' decomposition, which is a deterministic analog of the semimartingale decomposition, for regular functionals of such irregular paths.
We use this framework to develop a model-free framework for the pathwise analysis of hedging strategies for a general class of path-dependent derivatives. In particular, we derive a pathwise version of the hedging error for hedging strategies for path-dependent claims and extend the classical result of El Karoui, Jeanblanc and Shreve (1997) to path-dependent options, without any probabilistic assumptions on the underlying price process.
Joint work with Anna ANANOVA (Imperial College London) and Candia RIGA (University of Zurich).
References:
A Ananova, R Cont (2016)
Pathwise integration with respect to paths of finite quadratic variation,
_Journal de Mathematiques Pures et Appliquees_.
R Cont (2012)
Functional Ito calculus and functional Kolmogorov equations, in: Bally et al, Stochastic int??gration by parts and Functional Ito calculus (Barcelona Summer School on Stochastic Analysis 2012), Springer.