Pathwise derivatives valuation as a Cauchy problem on the space of Monte Carlo paths

Abstract

It is well-known that, under classical assumptions, the arbitrage-free value of European options contracts in complete continuous Markovian models is given by the solution to a Cauchy problem on Rd. Recent research has shown that similar results hold in a path-dependent context, whereby values solve an analogous Cauchy problem on the non-separable path space Dd of càdlàg paths, in an almost sure sense, that is – on an implicit subset with model probability one. This presents difficulties for numerical solution, as practitioners must work with real data time series and at most countably many operations. This thesis resolves this in a wide class of continuous path dependent market models, by showing that in this context derivatives’ valuation is equivalent to solving a Cauchy problem for any path in an explicit subset of a new Banach space M2
P .Rd/ of Monte Carlo paths, that is naturally adapted to practitioner intuition and numerical methods.

First, we develop a new framework for the pathwise analysis of market risk models. Based on a new notion of pathwise variance that generalises existing notions of quadratic variation, we construct the new Banach space M2 P .Rd/ and show how its geometry captures practitioner intuition about risk models’ volatility. The paths of a wide class of risk models can then be constructed explicitly as a family of integral expressions parameterised by noise paths. Second, we show how the Dupire (2009) differential operators for path functionals fit within the general theory of differentiation, and have a rich calculus. We characterise vertical differentials as the generators of strongly continuous groups on certain spaces of path functionals, and show how this allows for chain and product rules, and analogues of Taylor’s Theorem and smooth approximation. Similarly, the horizontal differential is characterised as the generator of a strongly continuous semigroup on spaces of functional processes. We prove a Fundamental Theorem of Calculus type result for this operator, and show how this framework can be applied to derivatives’ risk sensitivities. Third, we develop a theory of pathwise no arbitrage valuation on general Monte Carlo paths. We prove a new Itô formula for functional processes generalising that of Föllmer (1981) and Cont and Fournié (2010) to continuous paths with possibly discontinuous pathwise variance, and models which are not necessarily semimartingales. We then derive a corresponding valuation equation and robustness property for hedging error.

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Identifiers

ORCID for Enrico Gerding: orcid.org/0000-0001-7200-552X

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