Position Sizing, Kelly & Risk of Ruin

EV says whether a marginal trade is attractive under a model. Position sizing says how much loss the bankroll can survive when the model is wrong or variance arrives in an unfriendly order. An all-in positive-EV bet can still end the strategy in one resolution.

Three sizing policies #

Fixed shares

Simple, but risk changes dramatically as bankroll and price change.

Fixed bankroll percentage

Scales exposure with available capital, but still needs an evidence-based percentage.

Kelly fraction

Maximizes expected logarithmic growth under an exact repeated model; highly sensitive to wrong probabilities and dependence.

Kelly for a 0/1 contract #

Let $f$ be the fraction of bankroll spent buying YES at $p$. Wealth multiplies by

$$W_{YES}=1+f\frac{1-p}{p}, \qquad W_{NO}=1-f.$$

Maximize

$$g(f)=q\log W_{YES}+(1-q)\log W_{NO}.$$

With no costs or shorting, the full-Kelly capital fraction is

$$f^*=\frac{q-p}{1-p}$$

when $q>p$; otherwise do not buy. At $q=0.65$, $p=0.60$, $f^*=0.05/0.40=0.125$: 12.5% of bankroll spent, not 65% and not the raw five-cent edge.

Why full Kelly is fragile #

If the true $q$ is 0.58 rather than 0.65, the trade at 0.60 has negative EV. Costs lower the optimal fraction. Correlated positions reuse the same bankroll risk. Fractional Kelly, hard event limits, and explicit model-error buffers deliberately sacrifice theoretical growth to reduce drawdown and misspecification risk.

Life example #

A business with a favorable project still does not mortgage everything on one launch. Survival preserves the option to take future opportunities.

Application to no-hedge #

Size no-hedge positions after aggregating every fill that shares the same resolution, reference, or model error. Use net edge, not gross $\delta$, and reduce further for uncertain calibration, latency, and contract mismatch risk.