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Henry Chiu

Publications and source records attributed to Henry Chiu.

5 recordsLinked to original sources
arXiv2024

Model-free portfolio allocation in continuous-time

We present a non-probabilistic, path-by-path framework for studying path-dependent (i.e., where weight is a functional of time and historical time-series), long-only portfolio allocation in continuous-time based on [Chiu & Cont '23], where the fundamental concept of self-financing was introduced, independent of any integration theory. In this article, we extend this concept to a portfolio allocation strategy and characterize it by a path-dependent partial differential equation. We derive the general explicit solution that describes the evolution of wealth in generic markets, including price paths that may not evolve continuously or exhibit variation of any order. Explicit solution examples are provided. As an application of our continuous-time, path-dependent framework, we extend an aggregating algorithm of [Vovk '90] and the universal algorithm of [Cover '91] to continuous-time algorithms that combine multiple strategies into a single strategy. These continuous-time (meta) algorithms take multiple strategies as input (which may themselves be generated by other algorithms) and track the wealth generated by the best individual strategy and the best convex combination of strategies, with tracking error bounds in log wealth of order O(1) and O(ln t), respectively. This work extends Cover's theorem [Cover '91, Thm 6.1] to a continuous-time, model-free setting, recovering his celebrated error bound.

q-fin.MF↗
arXiv2022

A model-free approach to continuous-time finance

We present a non-probabilistic, pathwise approach to continuous-time finance based on causal functional calculus. We introduce a definition of self-financing, free from any integration concept and show that the value of a self-financing portfolio is a pathwise integral (every self-financing strategy is a gradient) and that generic domain of functional calculus is inherently arbitrage-free. We then consider the problem of hedging a path-dependent payoff across a generic set of scenarios. We apply the transition principle of Isaacs in differential games and obtain a verification theorem for the optimal solution, which is characterised by a fully non-linear path-dependent equation. For the Asian option, we obtain explicit solution.

q-fin.MF↗
arXiv2019

Causal functional calculus

We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulas which extend the results of \follmer\ (1981) and Cont \& Fourni\'e (2010) to a larger class of functionals, including \follmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends F\"ollmer-Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1-forms.

math.PR↗
arXiv2018

On pathwise quadratic variation for cadlag functions

We revisit H. Foellmer's concept of quadratic variation of a cadlag function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of cadlag processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition of quadratic variation which implies the Lebesgue decomposition as a result, rather than requiring it as an extra condition.

math.PR↗