  In this article, we analyse optimal statistical arbitrage strategies from
stochastic control and optimisation problems for multiple co-integrated stocks
with eigenportfolios being factors. Optimal portfolio weights are found by
solving a Hamilton-Jacobi-Bellman (HJB) partial differential equation, which we
solve for both an unconstrained portfolio and a portfolio constrained to be
market neutral. Our analyses demonstrate sufficient conditions on the model
parameters to ensure long-term stability of the HJB solutions and stable growth
rates for the optimal portfolios. To gauge how these optimal portfolios behave
in practice, we perform backtests on historical stock prices of the S&P 500
constituents from year 2000 through year 2021. These backtests suggest three
key conclusions: that the proposed co-integrated model with eigenportfolios
being factors can generate a large number of co-integrated stocks over a long
time horizon, that the optimal portfolios are sensitive to parameter
estimation, and that the statistical arbitrage strategies are more profitable
in periods when overall market volatilities are high.
