A dynamic pythagorean exponent: Applications to baseball and structural modeling in risk management

Abstract

The famous Pythagorean theorem relating a team's winning percentage over a season to it's total runs scored and surrendered has been the subject of much attention, in both popular culture and academia. In academia, a main focus has been to find the optimal value for the exponent in the formula, originally taken to be 2 in Bill James' work. The optimal value has been obtained in many published works via regression or other statistical estimation techniques, and calibrated to various sports to reflect the type of play in each sport. In this short article, we borrow from ideas in quantitative finance and risk management to estimate the distribution of these Pythagorean exponents, viewing them as random variables that can evolve over time. We carry out our analysis in a manner similar to that of implied volatility from the Black‐Scholes formula. We present the result of our analysis of 150 years of team data via the Lahman package in R, and also provide dynamic models for an evolving measure of the Pythagorean exponent. This includes suggestions for modeling credit default as well as contract valuation in sports via macro variables in connection with historically implied parameters.