Pricing Barrier Options under Lévy Processes using Dimension-reduced Cosine Expansion

Abstract

Barrier options are fundamental financial tools that give rise to pricing challenges, particularly when embedded within stochastic models. This study directs its focus towards Lévy processes as a strategic approach to navigate and resolve these intricate complexities. The model assumption adopted in this thesis is that the underlying log-asset price follows a Levy process. This assumption, in combination with the COS method, enables us to reduce the dimensionality of the pricing problem for barrier options. To be more precise, the strike price and the log-asset price are both factored out from the calculation formula. The traditional COS method, known for rapid European option valuation, depends on the knowledge of the ch.f.. The COS method has been extended to pricing barrier options in [1] and [2]. However, both methods depend on a recursive calculation formula whereby the number of recursion equal to the number of monitoring dates, and thus, the calculation speed lags behind pricing European options , especially with quite a number of monitoring dates. Our key insight lies in the potential of using the traditional COS method for pricing barrier options. As the first step, we integrate the barrier hitting probability into the ch.f., thereby transforming it into a survival ch.f. Although the exact function of the survival ch.f. is unknown, its values on the grid used by the COS method can be accurately solved via Singular Value Decomposition (SVD). Testing results robustly reinforces our findings. Building on this, we work out two techniques to approximate the characteristic function through supervised learning. The model takes Lévy process model parameters and yields approximation function of the ch.f.. The first method, COS-GPR, combines Gaussian Process Regression (GPR) with the traditional COS method for ch.f. estimation. Assuming mutual independence among multivariate model outputs, COS-GPR addresses GPR’s limitations at boundaries with an additional Fourier expansion. Notably, COS-GPR demonstrates significant significantly faster calculation than the existing COS Barrier by seven times while maintaining heightened accuracy, with occasional outliers. The second method, the COS Fourier CPD (CFC) method, replaces the ch.f. in the COS method by a Fourier-series expansion of the ch.f., after which the dimension is strategically reduced using Canonical Polyadic Decomposition (CPD). CFC achieves superior overall accuracy compared to COS Barrier, with a speed increase of 180 times and minimal instances of extreme errors. In comparison to COS-GPR, CFC’s accuracy slightly decreases but exhibits a smaller maximum error. The CFC method has a 180- fold increase in speed compared to the COS Barrier method and maintains linear time complexity as training points per dimension increase. In conclusion, this study introduces efficient methods based on supervised machine learning for barrier option pricing under Levy processes. The fusion of conventional iii iv methods with advanced approaches leads to significant improvements in both speed and precision. These innovations hold the potential to transform financial derivative valuation, enhancing accuracy and efficiency in the process.