The signature transform applied to Limit Order Book data in the cryptocurrency market

The path signature is a mathematical object that provides a universal, nonparametric feature representation of paths space in a structured linear space. Defined as a sequence of iterated integrals, the signature transform encodes data based on its arrival order, independently of the arrival time, which is a very desirable property in many applications. Moreover, due to their universal nonlinearity property, path signatures linearize the feature space, translating every forecasting problem into a simple linear regression. This thesis is based on a modeling framework proposed by Xin Guo et al. from the University of California, Berkeley, in which a two-step Lasso regression with adaptive weights is developed. In this framework, the adaptive weights are computed using the signature kernel, which offers a novel approach to measuring similarity between time series. The methodology is applied to high-frequency financial data from the Bitcoin perpetual futures limit order book data. Furthermore, to reduce the computational cost of high-dimensional signature features, we also explore the use of randomized signatures as a dimensionality reduction technique. It begins with a presentation of perpetual contracts and an introduction to the main concepts of market microstructure. This is followed by a mathematical overview of path signatures, focusing on their key properties and their application as feature maps in time series analysis. The modeling framework is then presented, including implementation strategies and parameter selection. Finally, the work concludes with the analysis of the forecasting results, highlighting both the strengths and limitations of the proposed methodology, and discussing its relevance for high-frequency financial data modeling.