Differential Machine Learning
Differential Machine Learning
Differential machine learning is a novel supervised learning framework that augments training data with differentials (derivatives) of labels with respect to inputs. In the context of financial derivatives pricing and risk management, it leverages automatic adjoint differentiation (AAD) to efficiently compute pathwise differentials of simulated payoffs. The method trains a twin neural network that simultaneously predicts prices and risk sensitivities (Greeks) by minimizing a combined loss function of value and derivative errors. This approach dramatically improves learning efficiency, especially with small datasets and high-dimensional inputs, as demonstrated on basket options and worst-of autocallables. Differential ML acts as a bias-free regularizer, reducing overfitting and enabling accurate online approximations for risk reports, capital calculations (XVA, FRTB, SIMM-MVA), and other applications where fast, disposable pricing functions are needed. The paper provides theoretical foundations, practical implementation details, and open-source TensorFlow code, establishing differential ML as a powerful tool for quantitative finance and beyond.
Highlights
- 1Introduces differential machine learning, combining automatic adjoint differentiation (AAD) with supervised learning to train on both labels and their differentials.
- 2Demonstrates vastly improved accuracy and convergence speed, especially with small datasets and high-dimensional inputs, compared to standard ML.
- 3Provides a practical solution for online pricing and risk approximation in derivatives risk management, enabling fast computation for regulations like XVA, FRTB, and SIMM-MVA.
- 4Shows that differential training acts as a bias-free regularization, reducing variance without introducing bias, akin to data augmentation.
- 5Includes TensorFlow implementation and numerical examples (basket options, worst-of autocallables) showing up to thousandfold improvement in error with small training sets.
Methods
- MTwin network architecture: combines feedforward network for value prediction with backpropagation for derivative prediction, sharing weights.
- MDifferential training: minimizes combined mean squared error (MSE) of value and derivative predictions, using pathwise differentials computed via AAD.
- MOnline learning from sampled payoffs: trains on simulated cash flows with unbiased but noisy labels and differentials, avoiding costly ground-truth pricing.
- MSmoothing of discontinuous payoffs (e.g., barriers, digitals) to ensure well-defined pathwise differentials for AAD.
Results
- RFor a 7-asset basket option, differential ML with 1k samples outperforms standard ML with 64k samples in value accuracy, and dramatically in derivative accuracy.
- RFor a worst-of-four autocallable in local volatility models, differential ML with 8k samples achieves value error 0.4 and delta error 2.5, vs. standard ML with 32k samples giving 1.2 and 32.5.
- RDifferential ML converges to the correct pricing function with much smaller datasets, reducing training time by orders of magnitude while maintaining accuracy comparable to Monte Carlo.
- RThe method effectively learns the true dimensionality of the problem (e.g., basket option reduces to one dimension), avoiding overfitting in high-dimensional spaces.
- RDifferential training provides bias-free regularization, improving generalization without the bias-variance tradeoff typical of L1/L2 regularization.
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