Numeraire Methods for Anytime-Valid Inference in Finance

Financial practitioners routinely face statistical challenges involving data that is gathered or observed sequentially over time, rather than as a fixed and complete dataset. Examples include risk assessment and monitoring, evaluation of hedging and trading strategies, and estimation of price impact from trades. This project uses recent advances from the field of anytime-valid statistics to address these issues. This is a fast-growing field at the intersection of statistics, online learning, information theory, and game theory. Using ideas from mathematical finance, this project will make meaningful contributions to the theoretical foundations and broader goals of anytime-valid statistics, in particular toward mitigating what is known as the replication crisis in science. The project revolves around the concept of the numeraire e-variable, an optimal test statistic for general statistical hypotheses that is based on the testing-by-betting framework. This approach creates strong links to mathematical finance and offers powerful methodologies in complex, non-parametric, and temporally dependent settings-- scenarios common in financial applications. The primary objectives of this project are to (i) develop a general theory of the numeraire process, a dynamic analog of the numeraire e-variable; (ii) establish a formal framework for the effective null hypothesis, a fundamental object associated with the numeraire e-variable and e-process; and (iii) develop a particular application in finance, namely anytime-valid price impact estimation. Estimating price impact is an important and difficult inference problem which is amenable to the approaches developed here. This project will showcase the power and usefulness of anytime-valid statistics in finance and pave the way for its use in a broad range of financial applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.