UC Irvine

We study the problem of constructing sparse and fast mean reverting portfolios based on a set of financial data arising in convergence trading. The problem is formulated as a generalized eigenvalue problem with a cardinality constraint \cite{ismp}. We develope a new proxy of mean reversion coefficient, the direct OU estimator, which can be used for both stationary and non-stationary data. In addition, we introduce two different methods to enforce the sparsity of the solutions instead of predetermining the cardinality. One method uses the ratio of $l_1$ and $l_2$ norms and the other one uses $l_1$ norm and prior knowledge. We analyze various formulations of the resulting non-convex optimization problems and develop efficient algorithms to solve them on portfolio sizes as large as hundreds. By adopting a simple convergence trading strategy, we test the performance of our sparse mean reverting portfolios on both generated and historical real market data. In particular, the $l_1$ norm regularization method gives robust results on large out-of-sample data set. We formulated a new type of problems for recovering fastest mean reverting process. It is a generalization of recovering sparse element in a subspace. From the numerical tests, we successfully recovered the hidden fastest OU process.