Tim Olson
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description, tags
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| Generalizes pairs trading to N>2 historically correlated stocks within a cluster (e.g., an industry), buying underperformers and shorting overperformers relative to the cluster mean. |
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Mean-Reversion — Single Cluster (and Multiple Clusters)
Section: 3.9 / 3.9.1 | Asset Class: Stocks | Type: Mean-Reversion / Statistical Arbitrage
Overview
This strategy generalizes pairs trading to N > 2 stocks that are historically highly correlated — for example, stocks belonging to the same industry or sector. Each stock's return is demeaned relative to the cluster mean, and positions are taken proportional to negative demeaned returns: buy stocks that underperformed the cluster and short stocks that outperformed it.
Construction / Signal
Let R_i = ln(P_i(t_2) / P_i(t_1)) be the log return for stock i in the cluster of N stocks.
Cluster mean return and demeaned returns:
R_bar = (1/N) * sum_{i=1}^{N} R_i (293)
R_tilde_i = R_i - R_bar (294)
Short stocks with positive R_tilde_i (outperformers), buy stocks with negative R_tilde_i (underperformers).
Dollar-neutrality constraints:
sum_{i=1}^{N} P_i |Q_i| = I (295)
sum_{i=1}^{N} P_i Q_i = 0 (296)
A simple prescription for dollar positions D_i = P_i Q_i proportional to demeaned returns:
D_i = -gamma * R_tilde_i (297)
where gamma > 0 (short outperformers, buy underperformers). Eq. (296) is automatically satisfied; Eq. (295) fixes gamma:
gamma = I / sum_{i=1}^{N} |R_tilde_i| (298)
Entry / Exit Rules
- Entry: Compute demeaned returns over a short measurement window; enter positions D_i = -gamma * R_tilde_i.
- Exit: Close when demeaned returns converge back toward zero, or at a predefined time horizon.
Key Parameters
- Cluster definition: Industry group, sector, or any set of historically correlated stocks
- Measurement window: Short-term (days to weeks) for mean-reversion
- Position sizing: Dollar-neutral via gamma normalization (Eq. 298)
- Weights: Uniform modulus, or non-uniform (e.g., suppressed by volatility)
Variations
3.9.1 — Mean-Reversion: Multiple Clusters
Generalize to K > 1 clusters, where stocks within each cluster are historically highly correlated. Treat clusters independently and combine via linear regression (unified approach).
Let Lambda_{iA} be the N×K binary loadings matrix: Lambda_{iA} = 1 if stock i belongs to cluster A, else 0. Cluster sizes: N_A = sum_{i=1}^{N} Lambda_{iA} > 0, N = sum_{A=1}^{K} N_A.
Run a linear regression of stock returns R_i on Lambda_{iA} (no intercept, unit weights):
R_i = sum_{A=1}^{K} Lambda_{iA} f_A + epsilon_i (303)
Regression coefficients (cluster mean returns):
f = Q^{-1} Lambda^T R, Q = Lambda^T Lambda (304, 305)
Q_{AB} = N_A delta_{AB} (307)
R_bar_A = (1/N_A) sum_{j in J_A} R_j (308)
Demeaned return (residual) for stock i:
epsilon_i = R_i - R_bar_{G(i)} = R_tilde_i (309)
where G(i) is the cluster to which stock i belongs. These residuals are cluster-neutral:
sum_{i=1}^{N} R_tilde_i Lambda_{iA} = 0, A = 1,...,K (310)
Also automatically: sum_{i=1}^{N} R_tilde_i = 0 (dollar-neutral).
Investments can be allocated uniformly across the K independent cluster strategies.
Notes
- The single-cluster strategy (3.9) is the natural generalization of pairs trading to N stocks.
- The multiple-cluster (3.9.1) formulation uses linear regression to compute all cluster means simultaneously in a unified framework.
- The binary loadings matrix Lambda_{iA} ensures each stock belongs to exactly one cluster (no overlap assumed).
- The intercept is automatically included via the constraint that each stock belongs to one cluster (sum_A Lambda_{iA} = 1).
- Mean-reversion strategies work best when stocks are truly co-integrated or highly correlated; sector/industry groupings provide natural clusters.