Multi-Dimensional Evolutionary Algorithms for Training Neural Networks with Heterogeneous Architectures in Pricing of American Style Options

MULTI-DIMENSIONAL EVOLUTIONARY ALGORITHMS FOR TRAINING NEURAL NETWORKS WITH HETEROGENEOUS ARCHITECTURES IN PRICING OF AMERICAN STYLE OPTIONS Andrew Clinton Sharp, PhD George Mason University, 2016 Dissertation Director: Dr. James Gentle A common problem in training artificial neural networks is determining the optimal network architecture for the particular problem. An ontology is presented that allows for the generalization of the multi-dimensional differential evolution (MD-DE) algorithm in training an ensemble of neural networks with heterogeneous architectures. This generalized algorithm is referred to as the multi-dimensional evolutionary algorithm (MD-EA) since it provides a framework for performing any of the evolutionary optimization algorithms such as differential evolution or genetic algorithms on neural networks with heterogeneous architectures. The original MD-DE algorithm and three variants are compared using a set of six representative functions while varying several parameters such as numbers of hidden nodes and size of the training population. These representative functions contain a wide array of potential function behaviors including polynomial, trigonometric, discontinuous, nondifferentiable, and a finite singularity. A random sample from a Poisson distribution is used to construct an ensemble of single and multi-layer neural networks with heterogeneous architectures that are then used to evaluate the proposed algorithms’ performance. An artificial neural network (ANN) training algorithm is presented that leverages any of the f-divergences as well as test statistics such as the Kolmogorov-Smirnov (K-S) or Anderson-Darling (A-D) as potential objective functions. These objective functions are chosen for online training of ANNs with a single observation at each time interval. The K-S statistic is further explored as a non-differentiable objective function for ANN training in online learning. This statistic is used to ensure convergence in the distributional sense for the learning of the conditional probability distribution of an input time series. Using the K-S statistic allows for the assumption of stationarity in the underlying model to be removed. Least squares Monte Carlo (LSM) is used to demonstrate the networks’ ability to learn the appropriate dynamics of simulated arithmetic and geometric Brownian motion (GBM) processes. The LSM algorithm estimates the American style option price of an asset following a known process. The known process is replaced by the learned conditional probability distribution output from the neural networks and is used to generate many sample asset price paths. It is shown that the variance in the estimated LSM price is due to the variance of the estimated parameters using a finite number of observations from a single price path realization of the underlying stochastic process. Chapter 1: Introduction Using artificial neural networks to learn a particular behavior of interest is typically accompanied by several issues related to the architecture of the networks, type and properties of the objective function, and determining if the underlying process to be modeled is constant or dynamic. A novel algorithm is presented that extends an existing multidimensional differential evolution algorithm to a larger class of optimization algorithms. The nature of this algorithm allows for the definition of an overall complexity parameter that sets the number of parameters to be used within a neural network. This parameter may remain constant or may be continuously controlled during run time. By controlling the expected value of this complexity parameter instead of setting a specific neural network architecture, the trained networks are allowed to add or remove nodes and layers implicitly in response to the problem. Combining this controllable parameter for model complexity with a suitable optimization algorithm along with a growing and pruning algorithm allows for a heterogeneous population of neural networks to train and adapt to a potentially changing underlying process. This allows the user to change error functions, merge in new hidden node types, add or remove new models to the ensemble, and continuously change the overall complexity of the problem without having to restart the models’ learning. A pre-run exploration phase to determine an optimal network architecture is also no longer necessary. The models are in particular in this dissertation used to learn the conditional probability distribution of an underlying asset price process in an online learning fashion. Data is presented to the model as it is observed and only used for a time window of a particular length (this time window need not be constant). Applications for this algorithm apply to any online learning problem where the underlying process may not be constant such as for financial assets, animal migrations, and real time controls for industrial processes. To 1 verify the learning of the ANNs, least squares Monte Carlo is used to estimate American style option prices based upon the final learned ANN models with respect to the underlying process.