A novel cosine network for pricing European and barrier options under stochastic volatility models

Abstract

This thesis presents a comprehensive exploration of the rough Heston model as a means to enhance financial derivative pricing and calibration in the context of the complex behavior of market volatility. Recognizing the limitations of classical models, such as the Black-Scholes and the standard Heston model, which assume constant or mean reverting volatility, this research delves into the application of rough volatility models that account for the empirical ’memory’ effect observed in financial markets. These models, inspired by the fractional Brownian motion with a Hurst parameter less than 0.5, offer a more accurate representation of the volatility surface.
A significant portion of the thesis is dedicated to the development of a novel cosine tensor network that expedites the supervised learning of the characteristic function of the lifted Heston model. This advancement is pivotal for the rapid pricing and calibration of European options under the rough Heston framework. The cosine tensor network, leveraging the characteristic function’s availability, enables the efficient application of Fourier-based methods, such as the COS method, for option pricing. This approach is further extended to the pricing of path-dependent options like barrier and Bermudan options through the 2-dimensional COS method.
The thesis is methodically structured, beginning with a foundational overview of option pricing and volatility, followed by a literature review that situates rough volatility within the broader context of quantitative finance. Subsequent chapters detail the mathematical framework of the rough Heston model, its derivation from market microstructures, and the introduction of the lifted Heston model as a multi-factor approximation.
Empirical analysis is provided to validate the lifted Heston model against traditional methods, demonstrating its superior performance and accuracy. The thesis culminates in the presentation of a new calibration method, supported by real market data, and a novel benchmark for pricing complex derivatives under the rough Heston model.
The research encapsulated in this thesis not only sets a new standard for computational speed and precision in the pricing and calibration of financial derivatives under the rough Heston model but also opens avenues for future research, particularly in the application of rough volatility models to other areas of financial mathematics.