Statistical Arbitrage Week4

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