(Trading)RISK_FORMULAS_FOR_PROPORTIONAL_BETTING(William_Chin,2004,)_[pdf].pdf

VISIT…
RISK FORMULAS FOR PROPORTIONAL
BETTING
William Chin, .
Department of Mathematical Sciences
DePaul University
Chicago, IL
Marc Ingenoso, .
Conger Asset Management, .
Chicago, IL
Email: marcingenoso@
Introduction
A canon of the theory of betting is that the optimal procedure is to bet
proportionally to one's advantage, adjusted by variance (see [Ep, Th, ]
for discussion and more references). This is the well-known "Kelly Criterion". It results
in maximum expected geometric rate of bankroll growth, but entails wild swings, which
are not for the faint of heart. A more risk-averse strategy used by many is to scale things
back and bet a fraction of the Kelly bet. This is monly by blackjack teams (see
) and futures traders, . [Vi], where the Kelly fraction is referred to as
“optimal f”.
In this article we examine what happens when we bet a fraction of Kelly in terms
of the risk of losing specified proportions of one's bank. We employ a diffusion model,
which is a continuous approximation of discrete reality. This model is appropriate when
the bets made are "small" in relation to the bankroll. The resulting formulae are limiting
versions of discrete analogs and are often much simpler and more elegant. This is the
theoretical set-up used for the Kelly theory.
The main result presented gives the probability that one will win a specified
multiple of one’s bankroll before losing to specified fraction as a function of the fraction
of Kelly bet. This formula () was reported in [Go]. There it is derived from a more
complicated blackjack-specific stochastic model. See also [Th] for related results. Our
approach results in the same formula, but more assumes from the outset a standard
“continuous random walk with drift” model. We do not have a historical citation, but it is
certainly true our results here are almost as old as Stochastic Calculus itself, and predate
1
any mathematical analyses of blackjack. It i