Paper: arXiv 2601.00009

Authors: Lucas Arenstein, Michael Kastoryano

Abstract

Pricing multi-asset options via the Black-Scholes PDE is limited by the curse of dimensionality: classical full-grid solvers scale exponentially in the number of underlyings and are effectively restricted to three assets. Practitioners typically rely on Monte Carlo methods for computing complex instrument involving multiple correlated underlyings. We show that quantized tensor trains (QTT) turn the d-asset Black-Scholes PDE into a tractable high-dimensional problem on a personal computer. We construct QTT representations of the operator, payoffs, and boundary conditions with ranks that scale polynomially in d and polylogarithmically in the grid size, and build two solvers: a time-stepping algorithm for European and American options and a space-time algorithm for European options. We compute full-grid prices and Greeks for correlated basket and max-min options in three to five dimensions with high accuracy. The methods introduced can comfortably be pushed to full-grid solutions on 10-15 underlyings, with further algorithmic optimization and more compute power.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces advanced tensor network mathematics (QTT) to solve high-dimensional PDEs, demonstrating rigorous implementation on a laptop with specific numerical results for 3-5 assets. It focuses on algorithmic innovation and accuracy validation rather than live market deployment, fitting the high-math, high-rigor profile.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Scalable Multi-Asset Option Pricing<br>overcoming Curse of Dimensionality"] --> B["Methodology: Quantized Tensor Train QTT"]
  B --> C["Data Inputs<br>Black-Scholes PDE<br>Correlated Basket & Max-Min Options"]
  C --> D{"Computational Solvers"}
  D --> E["Time-Stepping Solver<br>American & European Options"]
  D --> F["Space-Time Solver<br>European Options"]
  E & F --> G["Key Outcomes<br>Full-Grid Pricing for 3-5 Assets<br>High Accuracy on PC"]
  G --> H["Future Potential<br>Scalable to 10-15 Assets"]