---
description: "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."
tags: [stocks, statistical-arbitrage, optimization, portfolio-construction, sharpe-ratio]
---

# 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.