Tim Olson
47471b7700
- Flink update_bars debouncing - update_bars subscription idempotency bugfix - Price decimal correction bugfix of previous commit - Add GLM-5.1 model tag alongside renamed GLM-5 - Use short Anthropic model IDs (sonnet/haiku/opus) instead of full version strings - Allow @tags anywhere in message content, not just at start - Return hasOtherContent flag instead of trimmed rest string - Only trigger greeting stream when tag has no other content - Update workspace knowledge base references to platform/workspace and platform/shapes - Hierarchical knowledge base catalog - 151 Trading Strategies knowledge base articles - Shapes knowledge base article - MutateShapes tool instead of workspace patch
80 lines
4.3 KiB
Markdown
80 lines
4.3 KiB
Markdown
---
|
||
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.
|