Fair pricing, and pricing paradoxes

Abstract

The St Petersburg Paradox revolves round the determination of a fair price for playing the St Petersburg Game. According to the original formulation, the price for the game is infinite, and, therefore, paradoxical. Although the St Petersburg Paradox can be seen as concerning merely a game, Paul Samuelson (1977) calls it a “fascinating chapter in the history of ideas”, a chapter that gave rise to a considerable number of papers over more than 200 years involving fields such as probability theory and economics. In a paper in this journal, Vivian (2013) undertook a numerical investigation of the St Petersburg Game. In this paper, the central issue of the paradox is identified as that of fair (risk-neutral) pricing, which is fundamental in economics and finance and involves important concepts such as no arbitrage, discounting, and risk-neutral measures. The model for the St Petersburg Game as set out in this paper is new and analytical and resolves the so-called pricing paradox by applying a discounting procedure. In this framework, it is shown that there is in fact no infinite price paradox, and simple formulas for obtaining a finite price for the game are also provided.