Tensor train representations of Greeks for Fourier-based pricing of multi-asset options

ArXiv ID: 2507.08482 “View on arXiv”

Authors: Rihito Sakurai, Koichi Miyamoto, Tsuyoshi Okubo

Abstract

Efficient computation of Greeks for multi-asset options remains a key challenge in quantitative finance. While Monte Carlo (MC) simulation is widely used, it suffers from the large sample complexity for high accuracy. We propose a framework to compute Greeks in a single evaluation of a tensor train (TT), which is obtained by compressing the Fourier transform (FT)-based pricing function via TT learning using tensor cross interpolation. Based on this TT representation, we introduce two approaches to compute Greeks: a numerical differentiation (ND) approach that applies a numerical differential operator to one tensor core and an analytical (AN) approach that constructs the TT of closed-form differentiation expressions of FT-based pricing. Numerical experiments on a five-asset min-call option in the Black-Sholes model show significant speed-ups of up to about $10^{“5”} \times$ over MC while maintaining comparable accuracy. The ND approach matches or exceeds the accuracy of the AN approach and requires lower computational complexity for constructing the TT representation, making it the preferred choice.

Keywords: Greeks, Tensor Train, Monte Carlo, Multi-Asset Options, Fourier Transform, Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper utilizes advanced tensor network mathematics (TT decompositions, TCI algorithms) and Fourier transform theory, indicative of high math complexity. It presents concrete empirical results with specific speed-up metrics (10^5x) and comparative accuracy assessments on a defined 5-asset option model, demonstrating implementation-heavy validation.

  flowchart TD
    A["Research Goal"] --> B["Methodology"]
    B --> C["Computation"]
    C --> D["Outcomes"]

    A --> A1["Efficient Greeks for<br>Multi-Asset Options"]

    B --> B1["FT-Based Pricing"]
    B --> B2["TT Compression"]
    B1 --> B3["Data: Black-Scholes<br>5-Asset Min-Call"]
    B2 --> B3

    C --> C1["ND: Numerical Differentiation"]
    C --> C2["AN: Analytical Differentiation"]
    C1 --> C3["Single TT Evaluation"]
    C2 --> C3

    D --> D1["Speed-up: ~10<sup>5</sup>x<br>vs Monte Carlo"]
    D --> D2["Accuracy: MC Comparable"]
    D --> D3["Preferred: ND Approach"]