With the emergence of more complex option pricing models, the demand for fast and accurate numerical pricing techniques is increasing. Due to a growing amount of accessible computational power, neural networks have become a feasible numerical method for approximating solutions to these pricing models. This work concentrates on analysing various neural network architectures on option pricing optimisation problems in a supervised and semi-supervised learning setting. We compare the mean-squared error (MSE) and computational training time of a multilayer perceptron (MLP), highway architecture and a recently developed DGM network (Sirignano et al., 2018) along with slight variations on the Black-Scholes and Heston European call option pricing problem as well as the implied volatility problem. We find that on nearly all the supervised learning problems, the generalised highway architecture outperforms its counterparts in terms of MSE relative to computation time. On the Black-Scholes problem, we noticed a reduction of 9.8% in MSE for the generalised highway network while containing 96.2% fewer parameters compared to the MLP considered in (Liu et al., 2019).

On the semi-supervised learning problem, where we directly optimise the neural network to fit the partial differential equation (PDE) and boundary/initial conditions, we concluded that the network architecture of the DGM allows for optimisation of both the interior condition as well as the non-smooth terminal condition. As this was not the case for the MLP and highway networks, the DGM network turned out to be the best performing network architecture on the semi-supervised learning problems. Additionally, we found indications that on the semi-supervised learning problem the performance of the DGM network remained consistent when increasing the dimensionality of the problem.

In this thesis we classify samples of music according to the genre that the music belongs to using neural networks. We divide this task into four parts. In the first part, we prepare the audio files to be used as input to a neural network. Specifically, we examine ways to create spectrograms. We then optimise the spectrograms by reducing and normalising them. The second part consists of theoretical information regarding neural networks. We initially look at the perceptron, the building blocks of any neural network, and then extend this notion to various networks, such as the multilayer perceptron, the recurrent network and the convolutional network. In the third part, we apply the theoretical knowledge that we gained in part two and implement a standard neural network, a recurrent neural network, a convolutional neural network, and a combination of
recurrent and convolutional neural networks. We examine various network structures and we evaluate them based on what the networks can learn, how fast they can learn it and how accurate its classifications are. Simultaneously, we focus on creating an efficient network, using the fewest amounts of computational resources possible. We train each of the networks with data that we created in part one, and compare the performance of the networks with each other. In terms of accuracy and loss measures, we find that the best performing network is the combination of the recurrent and convolutional neural network. This network is able to determine which of six considered genres a 3 second sound sample belongs to with an accuracy of 90%. However, in terms of computational resources required to train the models, the convolutional neural network with many kernels during training converges using least computational cost. We then experiment with the amount of kernels in the convolutional layers, and find that a layer with many kernels learns faster, but does not necessarily yield better results. This is because networks with fewer kernels eventually learn the same kernels that are significant. Finally, we consider the impact of varying sound sample lengths on the performance of the networks. For a 1 second sound sample, we see that the recurrent network outperforms the other networks in terms of accuracy of the predictions. However, the larger we make the sound samples between 1 and 3 seconds, the better each network performs. Part four consists of an explanation of each component in the system, and presents a complete system built with Google’s TensorFlow [15], in which all the components work together to create an end-to-end classification system.