Optimal Bayesian Betting & Favorable Games

Abstract

Trading has always been considered more of an art than a science. With the rise of quantitative finance, high-frequency trading, and the demoralized equity market, there is more than ever a need to understand why specific strategies make a profit and others do not. This work focuses on the why part and tries to distill down the art component of quantitative strategy development to more of a science discipline. At the same time, it tries to lay down the theoretical groundwork for further research. Bayesian statistics is well suited for such a problem because of the inherited uncertainty quantification and its synergy with typical decision science.To the author's best knowledge, most current works focus on a particular strategy and fail to realize that a strategy consists of various moving, intertwined, yet crucial components that require sophisticated statistical methods to disentangle and correctly attribute the "effect" to each element. Without first entangling, the analysis can quickly fail to uncover the valid underlying profit driver and get lost in the weeds.We used the first chapter to introduce the most crucial concept of gambling, Kelly's Criterion. We highlighted its connection with the well-known log utility function of money and the theory of utility maximization in optimal decision-making. Further theories were also developed and extended around the criterion to make it suited to the equity market. The second and third chapters each dive deeply into a particular area of quantitative finance. Even though people treat these two areas separately in practice, they follow the same underlying principle. The second chapter focuses on portfolio optimization. Starting with the classical mean-variance portfolio, we extend it with the Kelly Criterion and prove two things: the latter guarantees positive growth. At the same time, the former does not, and a trade-off exists between the Sharpe ratio and the growth rate. The third chapter leaps to pairs trading through the lens of Bayesian statistics and the generative stochastic model. Such formulation offers insight into the profit generation structure and the more statistically "correct" way to conduct such trades.