Statistical Arbitrage Week4

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course notes of statistical arbitrage class

Portfolio Optimization

Structure of Optimizer

  • Risk Model
    • Shrinkage estimator of the covariance matrix of stock returns
  • Transaction Cost Model
    • \(1 \text{bp} + \frac12 \text{bid-ask spread}\)
  • Alpha
    • Weighted blend of various standardized, winsorized alphas

Input

  • position as of close of business on day \(t-1\)
  • alphas using data observed up to day \(t-1\)
  • transaction costs using data observed up to day \(t-1\)
  • risk model using data observed up to day \(t-1\)
  • constraints using data observed up to day \(t-1\)

Objectives & Constraints

  • Minimize risk: \(x’ \Sigma x\)
  • Maximize exposure to alpha: \(\alpha’ x\)
  • Neutralize exposure to beta: \(\beta’ x = 0\)
  • Minimize transaction costs: \(\tau’ \lvert x-w \rvert\)
  • Other constraints:
    • maximum trade size
    • maximum position size
    • maximum industry and country exposure
  • Notations
Variable Dimension Definition
\(\mathbf{x}\) \(n \times 1\) desired portfolio weights
\(\mathbf{w}\) \(n \times 1\) initial portfolio weights
\(\boldsymbol{\Sigma}\) \(n \times n\) covariance matrix of stock returns
\(\boldsymbol{\alpha}\) \(n \times 1\) aggregate alphas
\(\boldsymbol{\beta}\) \(n \times 1\) historical betas
\(\boldsymbol{\tau}\) \(n \times 1\) transaction costs

The parametric problems formed as:

\[max_{\mathbf{x}} \boldsymbol{\alpha}' \mathbf{x} - \lambda \boldsymbol{\tau}' \lvert \mathbf{x}-\mathbf{w} \rvert - \mu \mathbf{x}' \boldsymbol{\Sigma} \mathbf{x}\]

subject to:

  • beta neutrality: \(\boldsymbol{\beta}’ \mathbf{x} = 0\)
  • max trade and position: \(\boldsymbol{\gamma} \leq \mathbf{x} \leq \boldsymbol{\delta}\)
    • Max trade size for \(i^{th}\) stock: \(\theta_i\)
      • \(\Rightarrow w_i - \theta_i \leq x_i \leq w_i + \theta_i\)
    • Max position size for \(i^{th}\) stock: \pi_i
      • \(\Rightarrow -\pi_i \leq x_i \leq \pi_i\)
  • industry constraint: \(-r^* \cdot \mathbf{1} \leq \mathbf{R}’ \mathbf{x} \leq r^* \cdot \mathbf{1}\)
    • Sectors are a factor of risk
    • Difficult to time sector performance
    • Constrain industry exposure
    • But not to zero (will incur transaction cost)
    • \(r^* = $ 300,000 \text{ limit for }$ 50 \times 50M \text{ book size}\)
    • Industry Dummy
      • \(\rho\) industries
      • Boolean matrix \(\mathbf{R}\) of dimension \(n \times \rho\)
      • \(R(i, j) = 1\) if \(i^{th}\) stock belongs to \(j^{th}\) industry else 0
      • Every row of matrix \(\mathbf{R}\) has exactly one entry equal to 1; all other entries are equal to 0
  • country constraint: \(-f^* \cdot \mathbf{1} \leq \mathbf{F}’ \mathbf{x} \leq f^* \cdot \mathbf{1}\)
    • Countries are a factor of risk
    • Difficult to time country performance
    • Constrain country exposure
    • But not to zero (will incur too much transaction cost)
    • \(f^* = $ 100,000 \text{ limit for } $ 50 \times 50M \text{ book size}\)
    • Industry Dummy
      • \(\varphi \) industries
      • Boolean matrix \(\mathbf{F}\) of dimension \(n \times \varphi\)
      • \(F(i, j) = 1\) if \(i^{th}\) stock belongs to \(j^{th}\) industry else 0
      • Every row of matrix \(\mathbf{F}\) has exactly one entry equal to 1; all other entries are equal to 0

Output

trade to be executed on day \(t\)

\[\text{Final position}(t+1) = \text{Final position}(t) + \text{trade}(t)\]

Backtest Process

  1. Load all necessary data into memory
  2. Create the alphas
  3. Start from portfolio with zero dollar invested
  4. Loop over all days in backtest period
    • Every day: call optimizer to find optimal rebalancing trade given initial position
    • End-of-day position becomes initial position of next day
  5. Compute P&L