[Paper Review] No simple arbitrage for fractional Brownian motion

Rémi Peyre|arXiv (Cornell University)|Aug 3, 2015

Stochastic processes and financial applications1 references3 citations

TL;DR

This paper proves that fractional Brownian motion (fBm) with any Hurst parameter $ H \in (0,1) $, $ H \neq 1/2 $, admits no stopping time that is a local minimum at right with positive probability. The result establishes a strong no-arbitrage property for fBm, showing that no simple trading strategy based on local minima can yield risk-free profit, even in the presence of long-range dependence. The proof relies on pathwise regularity estimates, Gaussian vector analysis, and a law of iterated logarithm for a variant of fBm to control continuous unions of events.

ABSTRACT

We prove the following result: For $(Z_t)_{t \in \mathbf{R}}$ a fractional Brownian motion with arbitrary Hurst parameter, there does not exist any stopping time $τ$ adapted to the natural filtration of the increments of $Z$ such that, with positive probability, $τ$ a local minimum at right of the trajectory of $Z$.

Motivation & Objective

• To establish that no stopping time adapted to the natural filtration of fractional Brownian motion (fBm) can be a local minimum at right with positive probability.
• To demonstrate that fBm does not allow for 'simple arbitrage' strategies based on anticipating local minima in the trajectory.
• To extend the no-arbitrage property of standard Brownian motion to the long-range dependent case of fBm with $ H \neq 1/2 $.
• To provide a rigorous pathwise analysis of fBm trajectories using regularity estimates and Gaussian vector techniques to rule out the existence of such stopping times.

Proposed method

• Derives a pathwise representation of fBm as a stochastic integral with respect to a bilateral Brownian motion, using deterministic integration by parts.
• Introduces a variant of fBm to apply a law of iterated logarithm, capturing local oscillatory behavior near potential local minima.
• Uses regularity estimates on fBm to discretize continuous unions of events over time into finite unions, enabling probabilistic control.
• Applies Gaussian vector analysis to bound the determinant and inverse of covariance matrices arising from conditional distributions at stopping times.
• Employs operator norm estimates and matrix perturbation theory to control the behavior of covariance matrices under conditioning.
• Relies on technical lemmas involving binomial coefficients and geometric series to derive exponential bounds on matrix entries and determinants.

Experimental results

Research questions

• RQ1 Can a stopping time for fractional Brownian motion be a local minimum at right with positive probability?
• RQ2 Does the long-range dependence in fBm with $ H \neq 1/2 $ allow for simple arbitrage strategies based on local minima?
• RQ3 Is there a pathwise obstruction to detecting local minima in fBm trajectories using adapted stopping times?
• RQ4 How does the local behavior of fBm trajectories prevent the existence of such stopping times?

Key findings

• There does not exist any stopping time $ \tau $ adapted to the natural filtration of fBm that is a local minimum at right with positive probability.
• The result holds for all Hurst parameters $ H \in (0,1) $, including $ H \neq 1/2 $, and extends trivially to $ H = 1/2 $.
• The key obstruction arises from the joint Gaussian structure and Hölder regularity of fBm, which prevent the existence of predictable local minima.
• A law of iterated logarithm for a modified fBm variant is established to analyze local path behavior near candidate stopping times.
• Matrix estimates on the determinant and inverse of covariance matrices are derived, showing exponential decay in distance between indices, which controls the probability of rare events.
• The proof shows that the probability of a stopping time being a local minimum at right is zero, even when conditioning on such events, due to the intrinsic path regularity and dependence structure.