Deep Learning and the Heston Model:Calibration & Hedging

Abstract

The computational speedup of computers has been one of the de ning characteristics of the 21st century. This has enabled very complex numerical methods for solving existing problems. As a result, one area that has seen an extraordinary rise in popularity over the last decade is what is called deep learning. Conceptually, deep learning is a numerical method that can be "taught" to perform certain numerical tasks, without explicit instructions, and learns in a similar way to us humans, i.e. by trial and error. This is made possible by what is called arti cial neural networks, which is the digital analogue to biological neural networks, like our brain. It uses interconnected layers of neurons that activates in a certain way when given some input data, and the objective of training a arti cial neural network is then to let the neural network learn how to activate its neurons when given vast amounts of training examples in order to make as accurate conclusions or predictions as possible. In this thesis we focus on deep learning in the context of nancial modelling. One very central concept in the nancial industry is pricing and risk management of nancial securi- ties. We will analyse one speci c type of security, namely the option. Options are nancial contracts struck on an underlying asset, such as a stock or a bond, which endows the buyer with the optionality to buy or sell the asset at some pre-speci ed price and time. Thereby, options are what is called a nancial derivative, since it derives its value from the under- lying asset. As it turns out, the concept of nding a fair price of this type of derivative is closely linked to the process of eliminating or reducing its risk, which is called hedging. Traditionally, pricing and hedging is achieved by methods from probability theory, where one imposes a certain model in order to describe how the underlying asset price evolves, and by extension price and hedge the option. This type of model needs to be calibrated to real data. Calibration is the task of nding parameters for the stochastic model, such that the resulting model prices coincide with their corresponding market prices. However, traditional calibration methods are often too slow for real time usage, which poses a practical problem since these models needs to be re-calibrated very often. The hedging problem on the other hand has been very di cult to automate in a realistic market setting and su ers from the simplistic nature of the classical stochastic models. The objective of this thesis is thus twofold. Firstly, we seek to calibrate a speci c prob- abilistic model, called the Heston model, introduced by Heston (1993) by applying neural networks as described by the deep calibration algorithm from Horvath et al. (2019) to a major U.S. equity index, the S&P-500. Deep calibration, amongst other things, addresses the calibration problem by being signi cantly faster, and also more universal, such that it applies to most option pricing models, than traditional methods. Secondly, we implement arti cial neural networks to address the hedging problem by a completely data driven approach, dubbed deep hedging and introduced by Buehler et al. (2019), that allows hedging under more realistic conditions, such as the inclusion of costs associated to trading. Furthermore, the deep hedging method has the potential to provide a broader framework in which hedging can be achieved, without the need for the classical probabilistic models. Our results show that the deep calibration algorithm is very accurate, and the deep hedging method, applied to simulations from the calibrated Heston model, nds hedging strategies that are very similar to the traditional hedging methods from classical pricing models, but deviates more when we introduce transaction costs. Our results also indicate that di erent ways of specifying the deep hedging algorithm returns hedging strategies that are di erent in distribution but on a pathwise basis, look similar.