Differential Machine Learning for 0DTE Options with Stochastic Volatility and Jumps

TL;DR

A differential machine learning method under a stochastic-volatility jump-diffusion model that computes prices and Greeks in a single network evaluation that improves jump-term approximation relative to one-stage baselines, keeps price errors close to one-stage alternatives while improving Greeks accuracy, and produces stable one-day delta hedges is presented.

Abstract

We present a differential machine learning method for zero-days-to-expiry (0DTE) options under a stochastic-volatility jump-diffusion model that computes prices and Greeks in a single network evaluation. To handle the ultra-short-maturity regime, we represent the price in Black--Scholes form with a maturity-gated variance correction, and combine supervision on prices and Greeks with a PIDE-residual penalty. To make the jump contribution identifiable, we introduce a separate jump-operator network and train it with a three-stage procedure. In Bates-model simulations, the method improves jump-term approximation relative to one-stage baselines, keeps price errors close to one-stage alternatives while improving Greeks accuracy, produces stable one-day delta hedges, and is substantially faster than a Fourier-based pricing benchmark.

Figures (6)

• Figure 1: Architecture and training procedure. Stages 1--3 describe the three-stage training scheme. A variance-correction network returns $\Delta V_\phi(\mathbf{x})$, multiplied by a deterministic maturity function $g(\tau)$ so that the correction vanishes as $\tau\to 0$. Prices are produced by substituting the resulting effective variance into the Black--Scholes call formula, yielding $u_\phi(\mathbf{x})$ (not the plain BS price unless $\Delta V_\phi\equiv 0$). A separate network outputs the compensated jump contribution $J_\psi(\mathbf{x})$. Greeks are obtained by automatic differentiation of $u_\phi$. The jump-PIDE residual $R(\mathbf{x})$ is computed and penalized at randomly sampled points. Stages 1--3 depict the three-stage training scheme used for jump-term identifiability.
• Figure 2: Three-stage model: true vs predicted (price, delta, gamma, vega) on validation.
• Figure 3: Error distributions for the three-stage model.
• Figure 4: Jump-term check in the data domain ($|x|\le 0.5$): predicted compensated jump term vs numeric integral proxy.
• Figure 5: One-day $\Delta$-hedging P&L (True vs DML delta), stressed Bates SVJD parameter regime.