Understand mean-reversion quant models

What Is Mean Reversion?

Three pillars:

1. Stationary process: The mean and variance don't drift wildly over time (unlike a trending stock that keeps climbing).
2. Reverting force: Economic forces (profit-taking, value buyers) pull prices back.
3. Quantifiable deviation: We measure "how far is too far" using standard deviations (z-scores) or Bollinger Bands.

The Mathematical Foundation: Ornstein-Uhlenbeck Process

Mean reversion is modeled by the Ornstein-Uhlenbeck (OU) process, the continuous-time version of an autoregressive process.

Derivation from First Principles

Step 1: Model the price change $dX_t$ as two competing forces:

• Drift toward mean: The term $\theta(\mu - X_t)$ says "if $X_t > \mu$, drift down; if $X_t < \mu$, drift up." The farther from $\mu$, the stronger the pull.
• Random shock: $\sigma dW_t$ is the unpredictable noise (news, liquidity shocks).

Why this step? We separate systematic reversion (predictable) from random walk (unpredictable).

Step 2: Solve for the expected value $E[X_t | X_0]$ (where will the price be on average?).

Ignoring the stochastic term for the expectation, we have: $\frac{dE[X_t]}{dt} = \theta(\mu - E[X_t])$

This is a first-order linear ODE. Rearrange: $\frac{dE[X_t]}{\mu - E[X_t]} = \theta \, dt$

Integrate both sides: $-\ln|\mu - E[X_t]| = \theta t + C$

At $t=0$, $E[X_0] = X_0$, so $C = -\ln|\mu - X_0|$.

Exponentiate: $\mu - E[X_t] =(\mu - X_0)e^{-\theta t}$

Why this step? We're solving how the expected price evolves. The exponential decay means deviations shrink over time.

Step 3: Rearrange to get the forecast:

Why this step? This formula is our trading signal. If current price deviates from $\mu$, we know the expected direction and can size our bet.

Practical Implementation: Z-Score Strategy

The OU process is continuous-time. In practice, we work with discrete returns.

Trading Rules:

1. Entry: If $z_t < -2$ (price is 2 SD below mean), BUY (expect reversion up).
2. Entry: If $z_t > +2$ (price is 2 SD above mean), SELL/SHORT (expect reversion down).
3. Exit: Close position when $z_t$ crosses back to0 (price returns to mean).
4. Stop-loss: If $|z_t| > 3$ (regime change? model breakdown?), exit to limit losses.

Why $\pm 2$ SD? Assuming normal distribution, ~95% of data falls within 2 SD. Breaching this is rare → high probability of reversion. But we must confirm stationarity.

Example1: Single-Stock Mean Reversion

Setup: Stock XYZ has been trading around $\mu = 100$ for months. Today, bad earnings → price drops to $P_t = 85$. Historical volatility $\sigma = 5$.

Step 1: Compute z-score. $z_t = \frac{85 - 100}{5} = \frac{-15}{5} = -3$

Why this step? We quantify how extreme the deviation is. $z = -3$ is 3 SD below mean, highly unusual.

Step 2: Decision $z < -2$ triggers a BUY signal. We expect reversion to $\mu = 100$.

Why this step? The model says extreme deviations are temporary. We're betting the panic is overdone.

Step 3: Estimate time to revert. If we estimated $\theta = 0.3$ per day (from historical fitting), half-life: $t_{1/2} = \frac{\ln 2}{0.3} \approx 2.3 \text{ days}$

In ~2-3 days, we expect half the deviation to close. Target exit: $z \approx 0$, price around $97-103$.

Why this step? Knowing the reversion speed helps set holding period and profit expectations.

Outcome: Three days later, stock rebounds to $98$ as earnings panic fades. $z \approx -0.4$. Exit for $\approx 15\%$ gain.

Example 2: Pairs Trading (Relative Mean Reversion)

Setup: Stocks A and B are in the same sector (e.g., Coca-Cola vs. Pepsi). Historically, their ==price spread $S_t = P_A - P_B$== hovers around $\mu_S = 5$. Today, $P_A = 50$, $P_B = 38$, so $S_t = 12$. $\sigma_S = 2$.

Step 1: Compute z-score of the spread. $z_S = \frac{12 - 5}{2} = 3.5$

Why this step? The spread is 3.5 SD above normal. A has outperformed B excessively.

Step 2: Trade the pair to bet on convergence.

• Short A (overpriced relative to B).
• Long B (underpriced relative to B).
• This is market-neutral: if the whole sector drops, we lose on B but gain on the short A. We only care about the spread narrowing.

Why this step? By trading the ratio, we isolate mean-reversion and hedge out market risk.

Step 3: Exit when $z_S \approx 0$ (spread returns to $\mu_S = 5$). Say $P_A = 48$, $P_B = 43$. Spread = 5.

Profit calculation:

• Short A: Sold at 50, bought back at 48 → gain $+2$ per share.
• Long B: Bought at 38, sold at 43 → gain $+5$ per share.
• Net (assuming equal position sizes): $+7$ per unit pair.

Why this step? Pairs trading profits from relative mispricing, not absolute direction.

Testing for Mean Reversion: The ADF Test

Not all price series revert. A trending stock is non-stationary. We use the Augmented Dickey-Fuller (ADF) test.

Why this step? Applying mean-reversion to a trending stock loses money (you keep shorting an uptrend). ADF prevents this.

Common Mistakes

Active Recall Checkpoints

Connections

• Statistical Arbitrage Strategies – mean reversion is the engine behind stat-arb
• Bollinger Bands – practical visualization of mean ± 2 SD thresholds
• Cointegration Testing – pairs trading requires cointegration, not just correlation
• Kalman Filter – advanced: dynamically estimate time-varying $\mu$ and $\theta$
• Risk Management in Quant Trading – position sizing using z-score magnitude
• Stationarity Tests (ADF, KPSS) – prerequisites before trusting mean-reversion signals

#flashcards/stock-market

What is the core assumption of mean-reversion models? :: Prices that deviate significantly from their historical mean are experiencing temporary overeactions and will revert back toward the mean over time.

What is the key risk of mean-reversion models? :: Regime shifts—when the fundamental mean permanently changes due to company deterioration, market structure change, or crisis. The price won't revert to the old mean.

Concept Map

Hinglish (regional understanding)