Financial Time Series for Traders

What is a financial time series?

A financial time series is simply data indexed by time: a sequence of observations recorded at successive moments — every second, minute, hour or day. Markets never stop generating it. Each trade, quote and tick is stamped with a time, so a price chart is just a time series drawn on a screen.

Markets generate continuous streams of this data. A quant's first job is to treat that stream as a dataset — something to measure, transform and model — rather than a flickering chart to react to emotionally.

Prices vs returns

Raw prices are awkward to compare. A \$5 move means something very different for a \$10 stock than for a \$1,000 stock. Returns fix this by expressing each move as a fraction of the starting price, so movements are normalized and assets become directly comparable.

Simple return

The simple (arithmetic) return is the change in price divided by the old price:

Log return

Log (continuously compounded) returns take the natural log of the price ratio. They are convenient because they add over time instead of compounding multiplicatively, and they are nearly identical to simple returns for small moves.

Cumulative return

To combine many periods of simple returns into one total figure, chain the growth factors $(1 + r_t)$ and subtract 1:

Understanding volatility

Volatility measures how much returns fluctuate around their average. It is the statistical fingerprint of uncertainty: higher volatility means wider swings, less predictability and more risk.

We first need the mean (average) return over the sample:

Then volatility is the sample standard deviation of returns — using $n-1$ in the denominator (Bessel's correction) because we estimated the mean from the same data:

Daily volatility is scaled to an annual figure by multiplying by $\sqrt{252}$ — the square-root-of-time rule, where 252 is the number of trading days in a year:

Moving averages: smoothing noisy prices

Raw price series are jagged with noise. A moving average smooths them by averaging the last $n$ observations, revealing the underlying trend. The simple moving average (SMA) over a window of length $n$ is:

Shorter windows (e.g. 20-day) react fast but stay noisy; longer windows (e.g. 50-day) are smoother but lag. Comparing the two is the basis of many trend signals: when the fast average crosses above the slow one, momentum is turning up.

Rolling windows: statistics over a sliding window

Both the SMA and rolling volatility are rolling statistics: you fix a window width, compute a statistic, then slide the window one step at a time across the series. Each step produces one new output value, so a single price series spawns a smooth derived series.

flowchart LR
  subgraph SERIES[Price series P₁ … P₄₋]
    P1[P₁]:::s --> P2[P₂]:::s --> P3[P₃]:::s --> P4[P₄]:::s --> P5[P₅]:::s
  end
  W1["Window {P₁,P₂,P₃}"]:::d --> S1[SMA / σ #1]:::d
  W2["Window {P₂,P₃,P₄}"]:::d --> S2[SMA / σ #2]:::d
  W3["Window {P₃,P₄,P₅}"]:::d --> S3[SMA / σ #3]:::d
  classDef s fill:#eef2ff,stroke:#4f46e5,stroke-width:1px,color:#3730a3;
  classDef d fill:#ecfdf5,stroke:#0d9488,color:#0d9488;
      

From prices to returns: the core transformation

Almost every analysis starts by converting the price series into a return series, then deriving statistics from the returns. This pipeline is the backbone of the lab session.

flowchart LR
  A([Raw price series P_t]):::s --> B([Compute returns r_t]):::s
  B --> C([Mean return r̄]):::d
  B --> D([Rolling volatility σ]):::d
  A --> E([Moving averages SMA]):::d
  C --> F([Signal vs noise:
trend or mean reversion?]):::s
  D --> F
  E --> F
  classDef s fill:#eef2ff,stroke:#4f46e5,stroke-width:1px,color:#3730a3;
  classDef d fill:#ecfdf5,stroke:#0d9488,color:#0d9488;
      

Trend following vs mean reversion

Two opposing intuitions explain most price behaviour, and a quant must decide which regime an asset is in.

Randomness and noise

Not every wiggle means something. Much of short-term price movement is noise — random fluctuation with no predictive content. The researcher's hardest job is distinguishing genuine signal from noise.

Lab, assignment & mini project

Key takeaways