Variance, Losing Streaks & Sample Size

An edge is usually small compared with the binary swing of one contract. Consequently, a profitable model can look terrible for a while, and an unprofitable model can look brilliant over a lucky sample.

Variance of one binary purchase #

For $X=Y-p$ with $Y\sim Bernoulli(q)$,

$$E[X]=q-p, \qquad Var(X)=q(1-q).$$

At $q=0.65$, standard deviation per one-share trade is

$$\sqrt{0.65(0.35)}\approx0.477,$$

while buying at 0.60 has edge only 0.05. Noise is almost ten times the expected profit per trade.

Losing streaks are normal #

The probability that a particular block of five independent 65% trades all loses is

$$(1-q)^5=0.35^5\approx0.00525.$$

That sounds rare, but a long trading history contains many overlapping five-trade blocks, so eventually seeing such a streak is much more likely. A streak does not by itself prove the edge vanished.

Law of large numbers, carefully #

For independent comparable trades, average PnL tends toward EV as sample size grows. The standard error shrinks at roughly $1/\sqrt n$. It does not help when $q$ changes, costs are omitted, fills are selected, or trades share outcomes.

Life example #

A skilled basketball player can miss several free throws in a row. Skill affects the distribution, not a deterministic schedule of makes and misses.

Application to no-hedge #

Estimate effective sample size at the resolution-event level. Ten fills in one market are mostly one binary experiment. Review streaks against calibrated simulations or exact clustered models, while also checking for real regime change and reference degradation.