Parametric Differential Machine Learning for Pricing and Calibration

TL;DR

This paper extends differential machine learning to parametric problems, enabling efficient pricing and calibration of financial models by leveraging derivatives and adaptive sampling, demonstrated on interest rate models with caplet data.

Contribution

The paper introduces an extension of differential machine learning to handle parametric dependencies and proposes adaptive sampling for improved accuracy across parameters.

Findings

Efficient pricing surrogates for calibration instruments.

Successful calibration to caplet volatility surface.

Demonstrated on one-factor Cheyette models with stochastic volatility.

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.