A model-free approach to continuous-time finance

TL;DR

This paper introduces a non-probabilistic, pathwise framework for continuous-time finance using causal functional calculus, providing new insights into self-financing portfolios, arbitrage-free domains, and explicit solutions for path-dependent options.

Contribution

It develops a model-free, pathwise approach to continuous-time finance, defining self-financing portfolios without integration and solving for optimal hedging of path-dependent payoffs.

Findings

Self-financing portfolios are pathwise integrals and gradients.

The domain of functional calculus is inherently arbitrage-free.

Explicit solution obtained for Asian options.

Abstract

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.