A Machine-learning Approach to Parametric Option Pricing

This work explores a deep-learning approach to the problem of Parametric Option Pricing. In a first phase, neural networks are used to learn a pricing function starting from a set of prices computed by means of a suitable benchmark method. In a second phase, the method is coupled with a complexity reduction technique in order for it to be scaled up to higher dimensions. The contributions of this work are multiple. On the one hand, it shows the applicability of the neural-network approach to the parametric pricing problem. While few recent works have tackled a similar problem, this thesis shows that solutions can be found to more complex financial products, such as American and basket options. The pricer resulting from this method is fast and accurate, thus comparing favourably against the traditional Monte Carlo or PDE approaches. In addition, it provides results which are comparable to those obtained by recent developments in the use of Chebychev polynomial approximations, but without the constraints imposed by the need of having a fixed grid of Chebychev points. On the other hand, this work shows how to make the neural-net approach scalable to higher-dimensional problems. Indeed, the high number of parameters which can enter the pricing function (model and option parameters and underlying assets) can lead to problems which are not tractable by standard machines due to memory constraints. This thesis proposes to exploit a tensor-train (TT) decomposition which significantly compresses the tensor of prices used to train the neural network. As well as ensuring an accurate representation of the tensor entries, the TT-decomposition allows to retrieve the entries by means of simple products of three-dimensional ten- sors. For this reason, one does not need to store the whole training tensor, but can easily compute only the few entries of the small batch of samples which are needed for stochastic gradient-based methods used in the training of the neural net. The proposed methodology is tested on two practical cases: basket of call options, with up to 25 underlying assets, and American put options.