We propose a neural network approach to price EU call options that significantly outperforms some existing pricing models and comes with guarantees that its predictions are economically reasonable. To achieve this, we introduce a class of gated neural networks that automatically learn to divide-and-conquer the problem space for robust and accurate pricing. We then derive instantiations of these networks that are 'rational by design' in terms of naturally encoding a valid call option surface that enforces no arbitrage principles. This integration of human insight within data-driven learning provides significantly better generalisation in pricing performance due to the encoded inductive bias in the learning, guarantees sanity in the model's predictions, and provides econometrically useful byproduct such as risk neutral density.
We construct realistic equity option market simulators based on generative adversarial networks (GANs). We consider recurrent and temporal convolutional architectures, and assess the impact of state compression. Option market simulators are highly relevant because they allow us to extend the limited real-world data sets available for the training and evaluation of option trading strategies. We show that network-based generators outperform classical methods on a range of benchmark metrics, and adversarial training achieves the best performance. Our work demonstrates for the first time that GANs can be successfully applied to the task of generating multivariate financial time series.
This work provides a review of this literature. The motivation for this summary arose from our companion paper Ruf and W ang . There we continue th e discussions of this note; in particular, of potentially problematic data leakage when training ANNs to historic financial data. This paper is organised in the following way. Section 2 featu res Table 1, a summary of the literature that concerns the use of ANNs for nonparametric pricing (and hedging) of options. Section 3 provides a list of recommended papers from Table 1. Section 4 provides a n overview of related work where ANNs are applied in the context of option pricing and hedging, but not necessarily as nonparametric estimation tools. Section 5 briefly discusses various regularisation techniq ues used in the reviewed literature.
We study neural networks as nonparametric estimation tools for the hedging of options. To this end, we design a network, named HedgeNet, that directly outputs a hedging strategy. This network is trained to minimise the hedging error instead of the pricing error. Applied to end-of-day and tick prices of S&P 500 and Euro Stoxx 50 options, the network is able to reduce the mean squared hedging error of the Black-Scholes benchmark significantly. We illustrate, however, that a similar benefit arises by simple linear regressions that incorporate the leverage effect. Finally, we show how a faulty training/test data split, possibly along with an additional 'tagging' of data, leads to a significant overestimation of the outperformance of neural networks.
Mathematical modelling is ubiquitous in the financial industry and drives key decision processes. Any given model provides only a crude approximation to reality and the risk of using an inadequate model is hard to detect and quantify. By contrast, modern data science techniques are opening the door to more robust and data-driven model selection mechanisms. However, most machine learning models are "black-boxes" as individual parameters do not have meaningful interpretation. The aim of this paper is to combine the above approaches achieving the best of both worlds. Combining neural networks with risk models based on classical stochastic differential equations (SDEs), we find robust bounds for prices of derivatives and the corresponding hedging strategies while incorporating relevant market data. The resulting model called neural SDE is an instantiation of generative models and is closely linked with the theory of causal optimal transport. Neural SDEs allow consistent calibration under both the risk-neutral and the real-world measures. Thus the model can be used to simulate market scenarios needed for assessing risk profiles and hedging strategies. We develop and analyse novel algorithms needed for efficient use of neural SDEs. We validate our approach with numerical experiments using both local and stochastic volatility models.