Deeply Learning Derivatives
Deeply Learning Derivatives
This paper demonstrates the application of deep neural networks (DNNs) to approximate derivative valuation functions, using a European call option on a worst-of basket of six stocks as a case study. The authors generate training data via Monte Carlo simulation, randomly sampling input parameters (stock prices, volatilities, maturities, correlations) and computing option values. They explore the impact of network architecture (number of layers and nodes), training set size, and Monte Carlo path count on accuracy and speed. Key findings include that DNNs can achieve accuracy comparable to Monte Carlo with 100,000 paths while being about a million times faster (6 milliseconds vs. over 10 minutes on a CPU). Moreover, using a larger training set with fewer Monte Carlo paths per sample (higher noise) leads to better generalization, as the network learns to average out the noise. The paper also highlights that DNNs do not suffer from the curse of dimensionality, making them suitable for high-dimensional problems. The methodology is broadly applicable to other derivative models and can be parallelized on GPUs for further speed gains.
Highlights
- 1Demonstrates deep neural networks as fast and accurate approximations for derivative valuation, using a basket option example.
- 2Develops a training methodology where training data is generated via Monte Carlo simulation, enabling scalability.
- 3Shows that DNNs can average out Monte Carlo noise, achieving lower error than the training data's inherent noise.
- 4Explores the impact of hyperparameters (layer width, depth, training data quality/quantity) on model accuracy and speed.
- 5Achieves approximately 1 million times faster inference compared to traditional Monte Carlo methods.
Methods
- MDeep Neural Networks (DNNs) with ReLU activation and linear output layer, trained via mini-batch gradient descent with Adam optimizer.
- MMonte Carlo simulation to generate training data for a European call option on a worst-of basket of six stocks.
- MHyperparameter tuning by varying number of hidden layers, nodes per layer, training set size, and Monte Carlo path count.
- MC-vine method for generating random correlation matrices for the basket's underlying assets.
Results
- RDNN models achieve accuracy comparable to Monte Carlo with 100k paths, but inference takes <6 ms vs. >10 minutes for MC.
- RUsing more training data with fewer Monte Carlo paths (higher noise) yields better generalization than fewer high-precision samples.
- RLarger models (more nodes per layer) improve accuracy but risk overfitting; optimal size depends on training data quantity.
- RThe neural network learns to reduce Monte Carlo noise, with test error lower than training error for noisy datasets.
- RTraining data generation took ~1 week on a 24-core server; model training took 3 hours to 1 week; inference is extremely fast.
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