Chapter 09

Medium

Mean-Reverting SDEs: The Ornstein–Uhlenbeck Process

00 · Symbol Glossary

$X_t$the OU process

Unlike GBM's $S_t$ , $X_t$ can be negative — it models a deviation from a level (a spread, a rate, a log-volatility), not a price with limited liability.

$\theta$mean-reversion speed

Governs how fast $X_t$ is pulled back toward its long-run level. Larger $\theta$ means faster reversion and a shorter half-life (Section 03).

$\kappa$long-run mean (level)

The value $X_t$ reverts toward. Some texts write $\mu$ for this; $\kappa$ is used here to avoid clashing with GBM's drift $\mu$ from Chapter 08, which plays a completely different role.

$\sigma$volatility (additive, not multiplicative)

Constant, not scaled by $X_t$ — additive noise in the classification of Chapter 07, which is exactly why the integrating factor solves this SDE directly with no transform needed first.

01 · The Model

Definition — Ornstein–Uhlenbeck (OU) Process

$X_t$ follows the Ornstein–Uhlenbeck process if it solves

dX_t = \theta(\kappa - X_t)\,dt + \sigma\,dW_t, \qquad \theta>0,\ \sigma>0

Plain language: the drift $\theta(\kappa-X_t)$ is positive when $X_t<\kappa$ and negative when $X_t>\kappa$ — a restoring force, proportional in strength to how far $X_t$ currently is from $\kappa$ , exactly like a spring pulling toward equilibrium. The noise term is additive and constant, unlike GBM's multiplicative noise, which is why $X_t$ can go negative: there is nothing in the equation preventing it, and nothing in the intended application (a spread, a rate, a deviation) requiring it to stay positive.

02 · Solving via Integrating Factor

This is the linear SDE that Chapter 07 flagged as the integrating-factor case, worked out here in full.

Step-by-step — Solving the OU equation

Rewrite the drift:

dX_t = (\theta\kappa - \theta X_t)\,dt + \sigma\,dW_t

, isolating the

-\theta X_t

term that the integrating factor must cancel.

Multiply by $e^{\theta t}$ : compute

d(e^{\theta t}X_t) = \theta e^{\theta t}X_t\,dt + e^{\theta t}\,dX_t

(ordinary product rule —

e^{\theta t}

is deterministic, so it contributes no second-order Itô correction). Substituting

dX_t

from Step 1, the

\theta e^{\theta t}X_t

terms cancel exactly, leaving

d(e^{\theta t}X_t) = \theta\kappa e^{\theta t}\,dt + \sigma e^{\theta t}\,dW_t

Integrate from $0$ to $t$ :

e^{\theta t}X_t - X_0 = \kappa(e^{\theta t}-1) + \sigma\int_0^t e^{\theta s}\,dW_s

Solve for $X_t$ : divide by

e^{\theta t}

X_t = \kappa + (X_0-\kappa)e^{-\theta t} + \sigma\int_0^t e^{-\theta(t-s)}\,dW_s

The first two terms are the deterministic path a noiseless version of the equation would take — decaying exponentially from $X_0$ toward $\kappa$ . The third term is a weighted sum of past noise, with weight $e^{-\theta(t-s)}$ decaying as the noise ages: recent shocks matter more than old ones, and old shocks fade geometrically rather than persisting.

Example — The distribution of $X_t$ for fixed $X_0$

Since $\int_0^t e^{-\theta(t-s)}\,dW_s$ is an Itô integral of a deterministic integrand, it is Gaussian with mean $0$ and, by the Itô isometry (Chapter 05), variance $\sigma^2\int_0^t e^{-2\theta(t-s)}\,ds = \dfrac{\sigma^2}{2\theta}\left(1-e^{-2\theta t}\right)$ . So conditional on $X_0$ ,

X_t \;\Big|\; X_0 \;\sim\; N\!\left(\kappa+(X_0-\kappa)e^{-\theta t},\ \frac{\sigma^2}{2\theta}\left(1-e^{-2\theta t}\right)\right)

03 · Stationary Distribution and Half-Life

Let $t\to\infty$ in the variance formula above: $e^{-2\theta t}\to 0$ , so the variance converges to a finite limit rather than growing without bound — a sharp contrast with Brownian motion's variance $t$ , which diverges.

Definition — Stationary Distribution and Half-Life

As $t\to\infty$ (for fixed $X_0$ ),

X_t \;\xrightarrow{d}\; N\!\left(\kappa,\ \frac{\sigma^2}{2\theta}\right) \qquad \textbf{(stationary distribution)}

The half-life — the time for the deterministic component of a deviation from $\kappa$ to decay by half — solves $e^{-\theta t_{1/2}}=\tfrac12$ :

t_{1/2} = \frac{\log 2}{\theta}

Plain language: no matter where $X_0$ starts, the process eventually forgets it and settles into one fixed distribution centered at $\kappa$ with a variance set by the balance of pull ( $\theta$ ) against noise ( $\sigma$ ) — stronger pull tightens the distribution, more noise widens it. The half-life converts the abstract rate $\theta$ into a number with direct intuition: a spread with half-life of two weeks has largely closed within a month, one with a half-life of two years has not.

Example — Reading $\theta$ off a half-life target

If a trading desk wants a mean-reverting spread model with a two-week half-life ( $t_{1/2}=14$ days), $\theta = \log 2 / 14 \approx 0.0495$ per day. This is the standard way calibration proceeds in practice: pick a half-life from data or intuition, back out $\theta$ , then fit $\kappa,\sigma$ from the residual behavior around the reverting level.

04 · OU vs. the Discrete AR(1)

Analogy to Time Series, not a requirement

The stationary AR(1) process $Y_t=\phi Y_{t-1}+\varepsilon_t$ (Time Series Chapter 02) is the discrete-time cousin of OU — both have a single parameter controlling how fast deviations decay ( $\phi$ here, $e^{-\theta\Delta t}$ if OU is sampled at spacing $\Delta t$ ), both converge to a stationary distribution, and both have a well-defined half-life. Sampling an OU process at a fixed frequency produces exactly an AR(1) with $\phi=e^{-\theta\Delta t}$ and Gaussian innovations — this is not a coincidence but an exact correspondence, and it is one clean bridge between the two subjects' otherwise separate toolkits. Nothing in this chapter requires that correspondence; it is offered as intuition, not as a solving technique.

05 · Quant Uses

OU is the standard building block wherever "reverts to a level, doesn't run away to infinity" is the right qualitative story: modeling interest rate deviations from a target (the short-rate models of Vasicek and CIR extend OU directly), a statistical-arbitrage spread between two cointegrated assets (Time Series Chapter 08 builds the discrete-time version of this same idea), or log-volatility itself, which tends to cluster around a long-run level rather than drift to $\pm\infty$ .

❌ Using OU for a price level instead of a deviation

Modeling a stock price $S_t$ directly with $dS_t=\theta(\kappa-S_t)\,dt+\sigma\,dW_t$ .

Why it breaks: OU's Gaussian increments allow $S_t<0$ with positive probability at every finite $t$ (the stationary distribution in Section 03 is Gaussian, hence supported on all of $\mathbb{R}$ ) — a structural contradiction for a price with limited liability, the same property GBM (Chapter 08) was built specifically to avoid.

Consequence: simulated paths eventually go negative, and any pricing formula relying on $\log S_t$ or $S_t^{-1}$ breaks down on those paths. OU is the right tool for a spread, rate deviation, or log-volatility, not for a raw asset price — that distinction is the main thing to get right before writing down the SDE.

06 · Exercises

EXERCISE 9.1

Differentiate $e^{\theta t}X_t$ using the product rule and substitute the OU SDE for $dX_t$ ; check which terms cancel.

$d(e^{\theta t}X_t) = \theta e^{\theta t}X_t\,dt + e^{\theta t}\big(\theta(\kappa-X_t)\,dt+\sigma\,dW_t\big) = \theta e^{\theta t}X_t\,dt + \theta\kappa e^{\theta t}\,dt - \theta e^{\theta t}X_t\,dt + \sigma e^{\theta t}\,dW_t = \theta\kappa e^{\theta t}\,dt + \sigma e^{\theta t}\,dW_t$ . The two $\theta e^{\theta t}X_t\,dt$ terms cancel — exactly Step 2 of Section 02, shown explicitly.

Verify Step 2 of Section 02 directly: show that $d(e^{\theta t}X_t) = \theta\kappa e^{\theta t}\,dt+\sigma e^{\theta t}\,dW_t$ , with the $X_t$ -dependent terms cancelling.

EXERCISE 9.2

Use $t_{1/2}=\log 2/\theta$ from Section 03, solved for $\theta$ .

$\theta = \log 2/t_{1/2} = \log 2/30 \approx 0.0231$ per day. Doubling the target half-life to $60$ days would halve $\theta$ to about $0.0116$ — half-life and $\theta$ are inversely proportional, so slower reversion always means a longer half-life, and the relationship is exact, not approximate.

A trader wants an OU spread model with a $30$ -day half-life. Find $\theta$ . If the target half-life is doubled to $60$ days, what happens to $\theta$ ?

EXERCISE 9.3

Compare the stationary variance formula from Section 03 to the sample variance of an AR(1) using the correspondence $\phi=e^{-\theta\Delta t}$ from Section 04.

The AR(1) stationary variance is $\gamma(0)=\sigma_\varepsilon^2/(1-\phi^2)$ (Time Series Chapter 02). Substituting $\phi=e^{-\theta\Delta t}$ and letting $\Delta t\to 0$ with $\sigma_\varepsilon^2$ scaled appropriately recovers $\sigma^2/(2\theta)$ — the continuous-time OU stationary variance from Section 03. The discrete formula is the exact statement; the continuous one is its $\Delta t\to 0$ limit, matching the general relationship between the two subjects described in Chapter 01's closing note.

Using the AR(1)–OU correspondence of Section 04, show how the discrete stationary variance $\sigma_\varepsilon^2/(1-\phi^2)$ relates to the continuous-time stationary variance $\sigma^2/(2\theta)$ of Section 03.

07 · Chapter Summary

Concept	Meaning
OU SDE	$dX_t=\theta(\kappa-X_t)\,dt+\sigma\,dW_t$
Solving method	integrating factor $e^{\theta t}$
Closed form	$X_t=\kappa+(X_0-\kappa)e^{-\theta t}+\sigma\int_0^t e^{-\theta(t-s)}\,dW_s$
Stationary distribution	$N(\kappa,\ \sigma^2/2\theta)$
Half-life	$t_{1/2}=\log 2/\theta$
Discrete cousin	sampled OU is exactly AR(1) with $\phi=e^{-\theta\Delta t}$
Use for	spreads, rate deviations, log-volatility — not raw prices

Next: Chapter 10 — Girsanov's Theorem and Change of Measure, which explains how a drift like OU's $\theta(\kappa-X_t)$ can be removed entirely by changing the probability measure, the key step toward risk-neutral pricing.