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description, tags
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| Optimizes portfolio weights by maximizing the Sharpe ratio given expected stock returns and a covariance matrix, with an optional dollar-neutrality constraint enforced via a Lagrange multiplier. |
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Statistical Arbitrage — Optimization
Section: 3.18 / 3.18.1 | Asset Class: Stocks | Type: Statistical Arbitrage / Portfolio Optimization
Overview
This strategy determines optimal portfolio weights by maximizing the Sharpe ratio given expected returns E_i and a covariance matrix C_ij for N stocks. The unconstrained solution gives a closed-form inverse-covariance-weighted portfolio. A dollar-neutrality constraint can be imposed via a Lagrange multiplier, yielding a modified allocation that suppresses positions by stock volatilities.
Construction / Signal
Let C_ij be the N×N covariance matrix of stock returns, D_i the dollar holdings, I the total investment, and w_i = D_i / I the dimensionless weights.
Portfolio P&L, volatility, and Sharpe ratio:
P = sum_{i=1}^{N} E_i D_i (342)
V^2 = sum_{i,j=1}^{N} C_ij D_i D_j (343)
S = P / V (344)
Normalized: P = I * P_tilde, V = I * V_tilde, S = P_tilde / V_tilde where:
P_tilde = sum_{i=1}^{N} E_i w_i (347)
V_tilde^2 = sum_{i,j=1}^{N} C_ij w_i w_j (348)
with constraint sum_{i=1}^{N} |w_i| = 1.
Unconstrained Sharpe maximization (maximize S → max):
Solution:
w_i = gamma * sum_{j=1}^{N} C_ij^{-1} E_j (350)
where gamma is fixed by the normalization sum |w_i| = 1. These weights are not dollar-neutral in general.
Entry / Exit Rules
- Entry: At each rebalance, solve for optimal weights w_i using expected returns E_i and covariance matrix C_ij; establish long positions for w_i > 0 and short positions for w_i < 0.
- Exit: Rebalance periodically (daily or at the signal horizon); close positions that change sign or fall below a threshold.
Key Parameters
- Expected returns E_i: Can come from mean-reversion, momentum, ML signals, or other alpha sources
- Covariance matrix C_ij: Typically a multifactor risk model covariance (sample covariance is singular if T ≤ N+1)
- Total investment I: Scales all positions
- Rebalance frequency: Depends on signal horizon
Variations
3.18.1 — Dollar-Neutrality
To enforce dollar-neutrality (sum w_i = 0), use the Sharpe ratio's scale invariance to reformulate as a quadratic minimization with a Lagrange multiplier mu:
g(w, lambda) = (lambda/2) * sum_{i,j} C_ij w_i w_j - sum_i E_i w_i - mu * sum_i w_i (354)
g(w, mu, lambda) -> min (355)
Minimization w.r.t. w_i and mu gives:
lambda * sum_j C_ij w_j = E_i + mu (356)
sum_i w_i = 0 (357)
Dollar-neutral solution:
w_i = (1/lambda) * [sum_j C_ij^{-1} E_j - C_ij^{-1} * (sum_{k,l} C_kl^{-1} E_l) / (sum_{k,l} C_kl^{-1})] (358)
Lambda is fixed by the normalization sum |w_i| = 1. The weights w_i are approximately suppressed by stock volatilities sigma_i (since C_ii = sigma_i^2 and typically |E_i| ~ sigma_i), providing built-in risk management.
Notes
- The sample covariance matrix is singular if T ≤ N+1 (T = number of time observations); in practice a model covariance matrix (positive-definite, stable out-of-sample) is required.
- Eq. (350) is the unconstrained mean-variance (Markowitz, 1952) optimal portfolio.
- The dollar-neutrality solution (Eq. 358) removes market beta and is equivalent to imposing the constraint as a linear homogeneous condition on the quadratic objective.
- Expected returns E_i can be any alpha signal: mean-reversion residuals, momentum scores, ML predictions, etc.
- With a multifactor model C_ij, positions are approximately neutral to factor exposures; exact neutrality is achieved in the zero specific-risk limit (which reduces to weighted regression, see Section 3.10).
- In practice, trading costs, position/trading bounds, and nonlinear constraints are added; this generally breaks the equivalence between Sharpe maximization and quadratic minimization.