[Paper Review] Robust Arbitrage Conditions for Financial Markets

Derek Singh, Shuzhong Zhang|arXiv (Cornell University)|Apr 20, 2020

Risk and Portfolio Optimization31 references4 citations

TL;DR

This paper introduces a robust framework for analyzing arbitrage in financial markets under distributional uncertainty using Wasserstein distance as a measure of ambiguity. It derives dual formulations for weak and strong arbitrage conditions, defines statistical arbitrage, and computes critical Wasserstein radii beyond which arbitrage becomes unavoidable, with computational results showing convergence to a stable minimum distance of approximately 130.07 to the nearest arbitrage-free measure.

ABSTRACT

This paper investigates arbitrage properties of financial markets under distributional uncertainty using Wasserstein distance as the ambiguity measure. The weak and strong forms of the classical arbitrage conditions are considered. A relaxation is introduced for which we coin the term statistical arbitrage. The simpler dual formulations of the robust arbitrage conditions are derived. A number of interesting questions arise in this context. One question is: can we compute a critical Wasserstein radius beyond which an arbitrage opportunity exists? What is the shape of the curve mapping the degree of ambiguity to statistical arbitrage levels? Other questions arise regarding the structure of best (worst) case distributions and optimal portfolios. Towards answering these questions, some theory is developed and computational experiments are conducted for specific problem instances. Finally some open questions and suggestions for future research are discussed.

Motivation & Objective

• To develop a robust framework for detecting arbitrage under distributional uncertainty in financial markets.
• To formalize the concept of statistical arbitrage as a relaxation of classical weak and strong arbitrage conditions.
• To derive dual formulations of robust arbitrage conditions using Lagrangian duality and Wasserstein ambiguity.
• To compute critical Wasserstein radii beyond which arbitrage becomes unavoidable, and to identify optimal portfolios and worst-case distributions.
• To apply the framework to real-world market data, including index baskets and pairs trading, and validate via computational experiments.

Proposed method

• Uses Wasserstein distance as a measure of ambiguity to quantify distributional uncertainty in asset return distributions.
• Applies Farkas' lemma and Lagrangian duality to derive simplified dual formulations of robust arbitrage conditions.
• Introduces statistical arbitrage as a relaxation of classical arbitrage, defined via a penalty relaxation problem (NSPR) with a parameter β.
• Solves the nearest arbitrage-free (NA) problem via nonlinear programming (NLP) using the Knitro solver on real market data.
• Employs a subgradient method to solve the tight relaxation problem (NSPRT), yielding the critical Wasserstein radius.
• Constructs worst-case and best-case distributions by solving the dual problem, with support derived from the optimal state prices q*.

Experimental results

Research questions

• RQ1 What is the critical Wasserstein radius beyond which an arbitrage opportunity must exist in a given market model?
• RQ2 How does the level of statistical arbitrage vary as a function of the Wasserstein ambiguity radius?
• RQ3 What is the structure of the worst-case and best-case probability distributions under robust arbitrage constraints?
• RQ4 How can optimal portfolios be computed under distributional uncertainty using Wasserstein-based robust optimization?
• RQ5 What is the computational complexity of determining the nearest arbitrage-free measure under Wasserstein ambiguity?

Key findings

• The critical Wasserstein radius for the 2019 index basket data converges to approximately 130.07, indicating the threshold beyond which arbitrage becomes unavoidable.
• The minimum distance to the nearest arbitrage-free measure, computed via the NSPR problem, stabilizes at 130.07 for β ≥ 64, indicating convergence of the relaxation.
• The optimal portfolio weights are derived from the dual solution, with the state price vector q* showing non-zero probabilities only on two scenarios (0.229 and 0.668), indicating concentration on key market states.
• The reconstructed payoff matrix ˜X* closely matches the original data, with a residual norm ∥p − ˜X*q*∥₂ ≈ 4.78×10⁻⁷, confirming high accuracy in the nearest NA approximation.
• The worst-case distribution is supported on a minimal set of scenarios, consistent with the dual solution, and the optimal portfolio achieves a near-zero cost with guaranteed positive payoff in high-probability states.
• The NP-hardness of the nearest arbitrage-free problem is formally established, confirming the computational challenge of exact solutions.