Paper: arXiv 2302.06682

Authors: Arun Kumar Polala, Bernhard Hientzsch

Abstract

Differential machine learning (DML) is a recently proposed technique that uses samplewise state derivatives to regularize least square fits to learn conditional expectations of functionals of stochastic processes as functions of state variables. Exploiting the derivative information leads to fewer samples than a vanilla ML approach for the same level of precision. This paper extends the methodology to parametric problems where the processes and functionals also depend on model and contract parameters, respectively. In addition, we propose adaptive parameter sampling to improve relative accuracy when the functionals have different magnitudes for different parameter sets. For calibration, we construct pricing surrogates for calibration instruments and optimize over them globally. We discuss strategies for robust calibration. We demonstrate the usefulness of our methodology on one-factor Cheyette models with benchmark rate volatility specification with an extra stochastic volatility factor on (two-curve) caplet prices at different strikes and maturities, first for parametric pricing, and then by calibrating to a given caplet volatility surface. To allow convenient and efficient simulation of processes and functionals and in particular the corresponding computation of samplewise derivatives, we propose to specify the processes and functionals in a low-code way close to mathematical notation which is then used to generate efficient computation of the functionals and derivatives in TensorFlow.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper extends Differential Machine Learning (DML) to parametric pricing and global calibration using one-factor Cheyette models with stochastic volatility. It combines solid mathematical extensions of DML with practical algorithmic innovations like adaptive parameter sampling and low-code automatic differentiation.

Research Flowchart

  flowchart TD
    A["Research Goal: Parametric DML for Efficient Pricing & Robust Calibration"] --> B["Inputs & Low-Code Setup: Models (e.g., Cheyette), Parameters & TensorFlow Auto-Diff"]
    B --> C["Methodology: Parametric Differential ML with Adaptive Parameter Sampling"]
    C --> D["Computational Process: Train Neural Network Surrogates with Samplewise State Derivatives"]
    D --> E["Calibration Process: Global Optimization over Surrogate Pricing Models"]
    E --> F["Key Outcomes: Fast Pricing, Highly Accurate Calibration & Sample Efficiency"]