White noise and random walks — the null model every strategy must beat

White noise: the sound of no structure

White noise is the simplest time series: a sequence of shocks $\varepsilon_1, \varepsilon_2, \ldots$ — $\varepsilon$ is the Greek letter epsilon, the standard symbol for a shock, a fresh dose of pure random noise arriving at each time step. The shocks are iid — independent, all drawn from the same distribution — with mean 0 and constant variance $\sigma^2$ ("sigma squared"). Its ACF (D1-01) is zero at every lag — a flat line inside the noise band. Why care about the most boring series imaginable? Because it is the null model — the baseline claim that "there is no linear structure here at all." Every claim of predictability in returns is implicitly a claim that returns are not white noise. And D1-01's Fact 1 says that for the direction of returns, white noise is uncomfortably close to the truth.

Random walk: white noise, accumulated

A random walk is the running sum of white noise:

$P_t = P_0 + \sum_{i=1}^{t} \varepsilon_i, \qquad \text{equivalently} \qquad P_t = P_{t-1} + \varepsilon_t$

In words: the price at time $t$ ($P_t$, "P sub t") equals the starting price $P_0$ plus the sum of every shock so far — the $\Sigma$ is the for-loop from A1-00, adding up $\varepsilon_1$ through $\varepsilon_t$. Equivalently: today's price is yesterday's price plus one fresh shock.

Concrete: start at $P_0 = 0$ and flip coins for shocks of $+1$ or $-1$. Shocks of $+1, +1, -1$ give the walk $0, 1, 2, 1$.

The two models are one differencing apart. Differencing — replacing each value with its change, this value minus the previous one (np.diff) — undoes the summing. So: summing white noise gives a random walk; taking first differences of a random walk gives back the white noise. Map this onto markets and you get the working approximation of this whole track: prices behave like a random walk; returns behave like white noise. (For prices the sum is really of log returns, so the walk lives in log-price space — the compounding detail from B1.) The approximation is not exactly true — the deviations are the interesting part below — but it is shockingly close for liquid markets.

Why a random walk looks trendy

Here is the trap the pretest set. A walk is a running sum, so a lucky streak of positive shocks does not average away. It gets baked into the level and stays there until an equally lucky negative streak undoes it. Long excursions — extended stretches spent far above or below the start — are the norm, not the exception. Your eye reads those excursions as trends and the local extremes as support and resistance. The pattern-hungry brain does the rest. None of it predicts the next step, because the next step is a fresh coin flip by construction.

This is the statistical backbone of chart-pattern skepticism: the burden of proof is on the pattern, and "it looks like it works on this chart" cannot meet that burden, because random walks look like that too.

Variance grows with time

Independent shocks add in variance (the A1-04 rule, with every covariance term zero because the shocks are independent):

$\text{Var}(P_t - P_0) = t\,\sigma^2, \qquad \text{SD}(P_t - P_0) = \sigma\sqrt{t}$

In words: the variance of the total move after $t$ steps ("Var of P-t minus P-zero") equals $t$ times the variance of a single step. So the typical size of the move — the standard deviation, SD — grows like the square root of time: sigma times root t. Concrete: if one day's shock has standard deviation 1%, the typical move over 4 days is $1\% \times \sqrt{4} = 2\%$, not 4%.

Three things fall out of this one line. First, the $\sqrt{t}$ is the same square-root-of-time scaling you used to annualize volatility in A2-02. That convention silently assumes returns are uncorrelated — i.e. white noise. Second, forecast uncertainty grows without bound. A random walk has no anchor to revert to, so a 1-year-ahead price forecast is roughly $\sqrt{252}$ times as uncertain as a 1-day-ahead one. Third, since $P_t$ is a sum of many iid shocks, the CLT (A1-07) says its distribution is approximately normal even if the individual shocks are not. (Caveat: fat-tailed shocks converge slowly, so short-horizon returns stay visibly fat-tailed.)

The testable implication — and where edges live

If prices were exactly a random walk, then no function of past prices — no moving average, no pattern, no indicator — could have positive expected profit before costs. That is a falsifiable claim, and researchers have tested it. Lo and MacKinlay's variance-ratio tests — tests of whether variance really grows linearly in time, as a strict random walk requires — rejected the strict random walk for US stocks. (Historically: small positive autocorrelation in index returns, and more structure in the cross-section — the differences between individual stocks.) So the honest summary is: prices are approximately a random walk, and the deviations are small, unstable, and fight against costs.

That sentence is the job description of a quant. Volatility clustering (D1-01 Fact 2), weak momentum, mean reversion in spreads — every systematic edge is a specific, measured deviation from the random walk null. So the first question for any strategy pitch, including your own: which deviation from the random walk does this exploit, and what is the evidence that deviation exists? If there is no answer, the strategy is trading the patterns a coin flip makes.

Try it

Build a walk from its steps — the deterministic core of every simulation you will write in this track.

Explain it

Explain to a chart-reading friend, in plain words and no formulas, why the patterns on a cumulated-coin-flip chart look identical to the patterns on a stock chart — and what kind of evidence would actually distinguish a tradeable pattern from a coincidence. Say it out loud or write five sentences.

Predict before the next lesson

Next: stationarity and unit roots (D1-03). Before then, predict: you generate two independent random walks and regress one on the other with the tools from A2-06. What R-squared do you expect — and why might the answer be disturbing?