Part I · Foundations · M03 · week 2 · 14 h

The mathematics of betting: expectancy, Kelly, ruin

Expectancy and R-multiples, the Kelly criterion from log utility, overbetting, fractional Kelly, estimation risk, gambler's ruin, drawdown distributions and position sizing in practice.

Start reading · Learning outcomes

Builds on M01 Market structureM02 Statistics of returns

Prerequisites M01 (costs, lot sizes, margin), M02 (log returns, volatility drag 2.2, sampling error of the mean 2.4, fat tails 2.6, correlation 2.9). Probability: martingales and the optional stopping theorem, the strong law of large numbers, Brownian motion with drift (the facts used in 3.4 and 3.8 are collected in the Brownian toolkit, M02 · 2.11). Time: ~14 hours (notes 6 h, problems 5 h, code 3 h). Primary texts: Thorp, "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market" (2006); MacLean, Thorp & Ziemba (eds.), The Kelly Capital Growth Investment Criterion (World Scientific, 2011); Feller, An Introduction to Probability Theory and Its Applications, Vol. I (3rd ed., 1968), ch. XIV; Tharp, Trade Your Way to Financial Freedom (2nd ed., 2006), ch. 7 and 14.

Learning outcomes

Express any trading system's results in R-multiples, compute its expectancy and System Quality Number, state what the SQN does and does not prove, and compute how many trades are needed to distinguish an edge from noise ((3.2a)–(3.2b)).
Derive the Kelly fraction $f^* = p - q/b$ by maximising expected log wealth, prove that it maximises the almost-sure asymptotic growth rate, and prove that for even-money bets any $f \ge 2f^*$ gives negative growth.
Derive the continuous-time Kelly fraction $f^* = (\mu - r)/\sigma^2$ and growth rate $g(f) = r + f(\mu - r) - f^2\sigma^2/2$ , and its multi-asset form $f^* = \Sigma^{-1}(\mu - r\mathbf 1)$ .
Prove that betting $c$ times Kelly yields $2c - c^2$ of the maximal excess growth with $c$ times the volatility, and that the probability of ever losing to a fraction $x$ of wealth is $x^{2/c - 1}$ .
Quantify how estimation error in $\mu$ reduces realised growth and derive the optimal shrinkage $t^2/(1+t^2)$ of the estimated Kelly fraction.
Derive the gambler's-ruin probability by a difference equation and by optional stopping, and bound the probability of a given loss under fixed-fractional betting by $x^\theta$ with $\theta$ the Lundberg exponent.
Compute drawdown probabilities for Brownian log-wealth and simulate maximum-drawdown distributions for a trade list.
Size a position in shares or lots from a risk budget and a stop, and explain how gaps, correlation and estimation error modify the answer.

Sections ≈ 2 h 35 min of study

Foundation 4 sections · ≈ 45 min
Core 6 sections · ≈ 1 h 10 min
Advanced 4 sections · ≈ 40 min

Learning outcomes Foundation 5′
3.1 Expectancy, R-multiples and the System Quality Number Foundation 15′
3.2 The Kelly criterion for a binary bet Core 15′
3.3 Overbetting and the shape of $g$ Advanced 10′
Figure Kelly growth curve and the overbetting zone
3.4 Continuous-time Kelly and fractional Kelly Core 15′
Figure Fractional Kelly: growth given up, drawdowns avoided
3.5 Estimation risk and many bets Advanced 10′
3.6 Gambler's ruin Core 10′
Figure Gambler's ruin with fixed bets
3.7 Ruin under fixed-fractional betting Advanced 10′
3.8 Drawdowns Advanced 10′
Figure Drawdowns at a multiple of Kelly
Lab Nifty drawdowns at constant leverage
3.9 Position sizing in practice Foundation 10′
3.10 Case studies Core 20′
Summary of key results Foundation 15′
Readings and resources Core 5′
Facts that change Core 5′