Learning outcomes

After this module you can:

1. 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)).
2. 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.
3. 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 $f^* = \Sigma^{-1}(\mu - r\mathbf 1)$.
4. 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 $x^{2/c - 1}$.
5. Quantify how estimation error in $\mu$ reduces realised growth and derive the optimal shrinkage $t^2/(1+t^2)$ of the estimated Kelly fraction.
6. 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.
7. Compute drawdown probabilities for Brownian log-wealth and simulate maximum-drawdown distributions for a trade list.
8. 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.