Abstract

This chapter examines investments within the Cox Ross Rubinstein binomial asset pricing model, and describes simple mathematical parallels between long and short positions in the underlying asset and conventional bets made at risk neutral betting odds. The concepts of log optimal investment and Kelly betting are reconciled, and a formula provided for replicating Kelly bets with a portfolio of riskless bonds and derivatives of the underlying asset. At a practical level, the chapter defines a new derivative called a $1 Kelly bet . This is an easily implementable betting instrument that enables traders in sportsbetting and other prediction markets to automatically manage an arbitrary sum of capital over repeated bets in the way that exploits their abilities to estimate probabilities to the greatest possible long run monetary effect.

12.1 Introduction

Betting on horse races or football games and investment in stocks are often viewed as essentially the same activity (e.g. Thorp, 1971, 2000; Hausch et al., 1981; Asch et al., 1982; Ziemba and Hausch, 1985; Thaler and Ziemba, 1988; Pope and Peel, 1989; Hausch and Ziemba, 1990; Golec and Tamarkin, 1991; Shin, 1993; Hausch et al., 1994; Woodland and Woodland, 1994; Gray and Gray, 1997; Vaughan Williams and Paton, 1997; Gandar et al., 1998; Sauer, 1998; Vaughan Williams 1999, 2005; Ozgit 2005; Smith et al., 2006).

There are many parallels between trading in financial markets and sports wagering. in both settings, investors with heterogeneous beliefs and information seek to profit through trading as uncertainty...