Differential Deep Learning for Pricing Exotic Financial Derivatives

Sammanfattning: Calculating the value of a financial derivative is a central problem in quantitative finance. For many exotic derivatives there are no closed-form solutions for present values, instead, computationally expensive Monte Carlo methods are used for valuation. In addition to the present value, the sensitivities (derivatives with respect to the input parameters) are of interest. This study trains deep neural networks to approximate the present value of an autocallable, a highly path-dependent and customizable financial derivative in the form of a contract between two parties. This study pretrains deep neural networks on 2 million samples corresponding to one contract for an autocallable, and then retrains the same networks on 500 000 samples corresponding to a new, slightly different, contract. A differential method of training which includes the derivatives with respect to the inputs in the loss function, is compared to a baseline "vanilla" method in which the loss is merely the mean squared error of the predicted present value and the ground truth. The performance of the differential method is compared to the baseline method in terms of error, dependency on data, and number of epochs of training required to achieve a low error. The study finds that including derivatives (differentials) obtained by the finite difference method in the loss function yields several improvements when recalibrating a model to fit new data. The differential model achieves a significantly lower generalisation error. It is also less reliant on data, achieving lower errors than the non-differential model, despite using 10% of the data available for retraining. Additionally, the lowest error level achieved by the non-differential model is achieved within a few epochs by the differential model. This study shows that using derivatives obtained by the finite difference method is a suitable option when analytic derivatives cannot be obtained through automatic differentiation and can be used for differential machine learning.