Fractional Brownian Motion

Ordinary Brownian motion has no memory: whichever way the price just moved tells you nothing about which way it moves next. But real markets sometimes show streaks — a stock that's been climbing keeps finding buyers, or realized volatility that spikes tends to spike again in choppy little bursts rather than smoothing out. Fractional Brownian motion (fBM) is Brownian motion generalized with a single "memory dial" that can produce either kind of behavior, trending or choppier-than-random, while keeping everything else about the process's structure intact.

An analogy: a hiker with a stubborn or fickle stride

Ordinary Brownian motion is like a hiker who flips a coin at every step to decide left or right, with total amnesia about the last flip. Now imagine a hiker who is slightly stubborn — after stepping right, they're a bit more likely to step right again, producing longer, smoother-looking excursions in one direction (a trending hiker). Or imagine a hiker who is fickle — after stepping right, they're more likely to immediately correct back left, producing a jagged, jittery, constantly-reversing path that looks rougher than ordinary randomness. Fractional Brownian motion is a single family that covers stubborn, fickle, and perfectly amnesiac (ordinary Brownian motion) as special cases.

The mathematics, one symbol at a time

Fractional Brownian motion $B^H_t$ is characterized by the Hurst parameter $H \in (0,1)$, which controls memory via the covariance structure

$$\mathrm{Cov}(B^H_s, B^H_t) = \tfrac{1}{2}\left(s^{2H} + t^{2H} - |t-s|^{2H}\right).$$

You don't need to compute this directly to use the intuition: $H = 0.5$ recovers ordinary Brownian motion (no memory, independent increments). $H > 0.5$ gives persistent paths — an up-move makes a further up-move slightly more likely, producing smoother, trending-looking trajectories. $H < 0.5$ gives anti-persistent paths — an up-move makes a down-move slightly more likely, producing rougher, more jagged trajectories that reverse direction more often than pure randomness. The parameter also controls how path roughness scales: increments over a time step $\Delta t$ have standard deviation proportional to $(\Delta t)^H$, versus the familiar $(\Delta t)^{0.5}$ for ordinary Brownian motion — a smaller $H$ means increments shrink more slowly as you zoom in, which is exactly what "rougher" means mathematically.

Worked example 1: reading the Hurst exponent off real data

A common estimator regresses $\log(\text{range of a window})$ against $\log(\text{window length})$ and reads the slope as $H$. Suppose windows of length 10, 20, 40, 80 days give typical ranges of 2.0, 2.8, 3.9, 5.5. Taking log-log slope between the first and last: $\log(5.5/2.0) / \log(80/10) = \log(2.75)/\log(8) \approx 1.012/2.079 \approx 0.487$. This is close to $H \approx 0.5$, suggesting near-ordinary Brownian behavior with only a slight anti-persistent tilt — not the pronounced trending ($H$ well above 0.5) that momentum traders might hope to find in a simple price series.

Worked example 2: rough volatility's much lower H

Empirical studies of realized volatility time series (rather than price) consistently estimate $H \approx 0.1$, far below 0.5. If typical volatility moves at $\Delta t = 1$ day have a scale of $0.05$, then moving to $\Delta t = 4$ days scales roughly as $4^{0.1} \approx 1.149$, giving a scale of about $0.05 \times 1.149 \approx 0.057$ — barely larger despite the fourfold time increase. Compare this to ordinary Brownian scaling, $4^{0.5}=2$, which would double the scale over the same horizon. This tiny $H$ is the mathematical signature behind "volatility is rough" (see Rough Volatility): vol moves in short, jagged bursts that don't smooth out the way ordinary diffusions do.

Drag the volatility control above and compare it to the jagged, low-H path in the figure — ordinary Brownian motion (H = 0.5) already looks choppy, but rough volatility's H around 0.1 is choppier still, reversing direction even more relentlessly than plain randomness.

What this means in practice

Realized and implied volatility surfaces empirically fit rough ($H \approx 0.1$) processes far better than the constant-volatility or ordinary-diffusion assumptions used in classic models — this observation launched the rough volatility literature and new option-pricing models built on fBM-driven variance. Outside volatility, some researchers have tested price series themselves for persistence ($H \neq 0.5$) as a signal of exploitable memory, though evidence for genuine, tradable persistence in liquid asset prices is weak and easily confused with statistical noise.