Option Modelling by Deep Learning

Abstract

In this thesis we aim to provide a fully data driven approach for modelling financial derivatives, exclusively using deep learning. In order for a derivatives model to be plausible, it should adhere to the principle of no-arbitrage which has profound consequences on both pricing and risk management. As a consequence of the Black-Scholes model in Black & Scholes (1973), arbitrage theory was born. Arbitrage theory provides the necessary and sufficient formal conditions for a model to be free of arbitrage and the two most important results are the first and second fundamental theorems of arbitrage. Intuitively, under so called market completeness, the current price of any derivative/contingent claim in the model must reflect all available information and the price is unique, irrespective of risk-preferences. In order to arrive at an explicit arbitrage free price of any contingent claim, a choice must be made in order to simulate the distribution of the asset in the future. Traditionally this is achieved by the theory of random processes and martingales. However, the choice of random process introduces a type of model risk. In Buehler et al. (2019), a formal theory was provided under which hedging and consecutively pricing can be achieved irrespective of choice of model through deep learning. However, the challenge of choosing the right random process still remains. Recent developments in the area of generative modelling and in particular the successful implementation of generative adversarial networks (GAN) in Goodfellow et al. (2014) may provide a solution. Intuitively speaking, a GAN is a game theoretic learning based model in which two components, called the generator and discriminator, competes. The objective being to approximate the distribution of a given random variable. The objective of this thesis is to extend the deep hedging algorithm in Buehler et al. (2019) with a generative adversarial network. In particular we use the TimeGAN model developed by Yoon et al. (2019). We illustrate model performance in a simulation environment using geometric Brownian motion and Black-Scholes prices of options. Thus, the objective our model is to approximate the theoretically optimal hedge using only sample paths of the trained generator. Our results indicates that this objective is achieved, however in order to generalise to real market data, some tweaks to the algorithm should be considered.