Option pricing models without probability: a rough paths approach

TL;DR

This paper introduces a probability-free approach to option pricing and hedging using rough paths, providing a pathwise framework that enhances robustness and generalizes fundamental derivative trading principles.

Contribution

It presents a novel probability-free method for option pricing and hedging using rough paths, extending classical delta hedging with volatility swaps for improved robustness.

Findings

Pathwise option replication without probability

Small model misspecification leads to minimal profit/loss

Enhanced delta hedging with volatility swaps improves robustness

Abstract

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generalise the so-called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second order terms arising in rough-path integrals, resulting in improved robustness.