2. Factor Model Statistical Arbitrage
A standard model for the dynamics of stock price is
This model can be enhanced by expanding the noise term
Where are risk factors associated with the market
In discrete time
Assume that , , , and that
F and are independent.

3. Covariance of Log Returns
If we have n observations and p factors:
Or in matrix form
Using

4. Principal Component Analysis
Spectra decomposition of matrix
where are the Eigen value, Eigen vector pair
Noise Reduction
We can approximate the model with a limited set of m Eigen vectors or Principal Components
Using the largest Eigen vectors will add the components that contribute most to the variance in the data

5. Stability of Principal Components
Comparison of the Stability/Evolution of the PCA
30 day initial data sample– Moved forward one day at a time.
10 largest Eigen cectors compared to the first sample using dot product
Two Subtle Problems
1. The Eigen vectors returned by PCA may be the inverse of the first set.
2. Since the Eigen vectors are given in descending order, a change in the relative magnitude of any components may swap their position. Therefore, comparisons must be made carefully.
Results
Eigen vectors are relatively stable over time.
After 10 Eigen vectors they become more unstable.

6. Stability of Principal Components

7. Stability of Principal Components

8. Statistical Distance vs Time of Day
Mahanalobis Distance
The distance a data point is from the center of the distribution
Procedure
The training set of 15 minute log return data was for 100 days.
The distance of the next 10 data points was calculated.
The training set was then shifted forward and the next 10 points measured.
The data was sorted by time of day to analyze the time of day that generated the most outliers.

9. Distance of new Test Data form the Training Data
Mahalanobis Distance
Conclusion – We can separate the market into two distinct time periods where the returns are generated by two different processes.

10. Generation of Residuals
Partial Least Squares
If X is the data set and Y is the component desired to regress from the data
then PCA analyzes
And PLS analyzes
PLS finds the matrix information associated with the first Eigen vector
Subtracts this information from the covariance matrix
Then finds the information for the second Eigen vector, etc.
Procedure
Test data : 100 day sample of 15 minute log returns on 500 stocks
Predict the next 10 points of data using PLS with largest 9 Eigen vectors
Test data moved forward
Results
Measure of fit

11. PLS First 45 Minutes of Market Removed

12. PLS First 45 Minutes of the Market

13. Calibrating OU Process: Problem Setup
Need to estimate κ, μ and σ in the OU-Process Equation:
The discrete form of the solution of the SDE can be written as:
κ: coefficient of mean reversion
∆: discretization time step
μ: long term mean of the residuals

14. Calibrating OU Process: OLS and MLE
Least Squares:
Basic idea: Fit parameters by minimizing sum of square of error terms.
Maximum Likelihood Estimation:
Basic idea: Find parameters by maximizing log-likelihood of the data.

15. Main Issue OLS and MLE tend to produce similar results.
However, MLE is known for overestimating the mean reversion speed κ:
example: Johnson, Thomas. “Approximating Optimal Trading Strategies Under Parameter Uncertainty: A Monte Carlo Approach”. Kellog Business School
Main idea: MLE typically overestimates the mean reversion speed and as a result, underestimates the noise σ.
Paper compares filtering trading strategy to MLE.
Filtering outperforms MLE every time.
Reason: Boguslavsky, Boguslavskaya. “Arbitrage Under Power”. February 2009.
MLE model suggests overly aggressive positions that can quickly lead the trader to bankruptcy.

16. Kalman Filtering
Idea: mathematical method to use noisy measurements to produced results that tend to be closer to the true value of the variable of interest.

17. Comparison of Estimation Methods
Parameter estimation by Kalman Filtering Produces produces more accurate estimates of the OU process parameters than either MLE or OLS.
Major disadvantage of EM Algorithm: Might take a long time to converge, computationally intensive for large window sizes.
Solution: Use MLE/OLS to produce initial guesses then use EM to refine estimation.

18. Optimal Trading of the Residuals-1
Implement the Boguslavsky/ Boguslavskyaya strategy described in: “Optimal Arbitrage Trading” (2003).
O-U process:
Conditional Distribution:
Utility Function
Normalization Process : Let α be the control variable and W the wealth at time t:
Value Function:

19. Optimal Trading of the Residuals-2
Solve for optimal control parameter using HJB equation:
Reduces to the PDE:
Solution: Let τ be the time left for trading,

20. Results on EvA residuals
∆ ~ 1 min, γ = -0.5, initial wealth = 100,000
Cumulative Wealth, Optimal Trading Position
Peak ~ 4,300,000
End ~ 3,700,000

21. Results on Our residuals using EvA’s data-XOM
∆ ~ 15 min, initialWealth = 100,000
Cumulative Wealth, γ = Cumulative Wealth,γ = -0.5
Peak ~ 530, Peak ~ 520,000
End ~ 490, End ~ 450,000

22. Incorporating TC-Separate Fund Allocation
All wealths curves will
lie between the red and
green curves.
Blue curve = no fixed cost
peak = 530,000, End = 490,000
Green curve
peak = 470,000, end = 420,000
Blue = no cost
Green = 10*fixed cost
Red = 1*fixed cost

23. Trading Residuals in Practice
Look at historical 15 minute data for ~500 stocks using a 100 days sliding window
For every stock i at time t
Generate partial least square representation using 10 components using the remaining 499 stocks last 100 days return sliding window
Generate a residual return by removing the PLS approximation from the stock return
Generate residue replicating portfolio weights
Pi = [-β1 –β2 …. -βi-1 1 -βi+1 …. -βn]

24. Available Data at Time t
Stock returns vector R(t)
Residuals returns Vector Rresidue(t)
Residuals means Vector μresidue(t)
Residuals standard deviations Vector σresidue(t)
Residuals replication matrix P(t)
Pij(t) is the weight of the jth stock in the portfolio replicating ith residue
If we have residuals positions vector V(t), the final investment portfolio will be V(t)P(t)

25. The Trading Strategy
Evaluate the market every 15 minutes to look for strong deviations of residuals from mean
Enter positions that exceed a entering threshold
Leave positions that cross the leaving threshold
Allocate money in a certain defined percentage equally between all opportunities invested in given a certain minimum cash position percentage
The dynamic rebalancing of portfolio is based on log optimal portfolio growth strategy of volatility pumping

26. The Secret Sauce: Trading Parameters
Long Enter threshold, Short Enter threshold
Long Exit Threshold, Short exit threshold
Minimum Cash percentage
Maximum single position percentage
Trading algorithm is robust with trading parameters (at least as far as I tested!)
Divided data sets into a training period and used matlab optimization toolbox to find parameters that maximizes sharpe ratio and applied the resulting parameters into a testing period
This strategy can be applied continuously to periodically recalibrate the trading parameters