[Paper Review] Market Graph Clustering Via QUBO and Digital Annealing

Seo Woo Hong, Pierre Miasnikof+2 more|arXiv (Cornell University)|Jun 18, 2020

Game Theory and Applications23 references4 citations

TL;DR

This paper proposes a QUBO-based K-medoid clustering approach on a financial market graph to construct low-cardinality tracking portfolios for the S&P 500 index. By leveraging Fujitsu's Digital Annealer hardware, the method efficiently solves the NP-hard optimization problem, achieving a tracking error below 0.5% and high beta alignment using only 10 representative stocks annually.

ABSTRACT

Our goal is to find representative nodes of a market graph that best replicate the returns of a broader market graph (index), a common task in the financial industry. We model our reference index as a market graph and express the index tracking problem in a quadratic K-medoids form. We take advantage of a purpose built hardware architecture, the Fujitsu Digital Annealer, to circumvent the NP-hard nature of the problem and solve our formulation efficiently. In this article, we combine three separate areas of the literature, market graph models, K-medoid clustering and quadratic binary optimization modeling, to formulate the index-tracking problem as a quadratic K-medoid graph-clustering problem. Our initial results show we accurately replicate the returns of a broad market index, using only a small subset of its constituent assets. Moreover, our quadratic formulation allows us to take advantage of recent hardware advances, to overcome the NP-hard nature of the problem.

Motivation & Objective

• To develop an efficient method for constructing low-cardinality tracking portfolios that replicate broad market index returns.
• To address the NP-hard nature of K-medoid clustering in financial market graphs through quadratic binary optimization (QUBO).
• To leverage purpose-built quantum-inspired hardware (Fujitsu Digital Annealer) to solve large-scale QUBO formulations efficiently.
• To evaluate the performance of the proposed method using standard financial metrics like tracking error and portfolio beta.

Proposed method

• Model the S&P 500 constituent stocks (n=453) as a complete weighted market graph, with edge weights derived from Pearson correlation of daily log-returns.
• Transform the distance matrix into a robust similarity matrix Δ using δ_ij = 1 - exp(-0.5 × d_ij), where d_ij = √(2(1 - ρ_ij)).
• Formulate the K-medoid clustering problem as a QUBO: minimize zᵀ(γ11ᵀ - αΔ/2)z + zᵀ(βΔ1 - 2γk1), with z_i ∈ {0,1} indicating exemplar selection.
• Set parameters α = 1/k, β = 1/n, γ = 2, with k=10 exemplars selected per year.
• Use the Fujitsu Digital Annealer to solve the QUBO formulation, exploiting its hardware-optimized architecture for combinatorial optimization.
• Recompute the market graph annually using prior year’s returns and re-optimize the QUBO to generate a new tracking portfolio each year.

Experimental results

Research questions

• RQ1 Can a QUBO formulation of K-medoid clustering effectively identify a small subset of stocks that track the S&P 500 index with high accuracy?
• RQ2 To what extent can purpose-built hardware like the Digital Annealer overcome the computational intractability of NP-hard clustering in financial market graphs?
• RQ3 How does the performance of the QUBO-Digital Annealer approach compare to traditional index-tracking methods in terms of tracking error and beta alignment?
• RQ4 What is the impact of annual re-computation of the market graph on portfolio stability and tracking performance?

Key findings

• The method achieved a consistent annual tracking error of 0.005 (0.5%) across all four years (2016–2019), indicating strong replication of the S&P 500 daily returns.
• The tracking portfolio’s beta with respect to the S&P 500 varied between 0.691 and 1.167, with t-statistics exceeding 15, indicating statistically significant tracking performance.
• Despite using only 10 stocks per year, the method successfully replicated the S&P 500 index returns with high fidelity, demonstrating the effectiveness of the QUBO and Digital Annealer pipeline.
• The use of the Fujitsu Digital Annealer enabled efficient solution of the NP-hard QUBO problem, making large-scale index tracking feasible in practice.
• The robust similarity matrix Δ improved the stability of the market graph representation, enhancing clustering quality.
• The approach demonstrated strong empirical performance across multiple years, with low tracking error and high statistical significance, validating its practical applicability in portfolio construction.