Mean-Variance Policy for Discrete-time Cone Constrained Markets: The Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure

The discrete-time mean-variance portfolio selection formulation, a representative of general dynamic mean-risk portfolio selection problems, does not satisfy time consistency in efficiency (TCIE) in general, i.e., a truncated pre-committed efficient policy may become inefficient when considering the corresponding truncated problem, thus stimulating investors' irrational investment behavior. We investigate analytically effects of portfolio constraints on time consistency of efficiency for convex cone constrained markets. More specifically, we derive the semi-analytical expressions for the pre-committed efficient mean-variance policy and the minimum-variance signed supermartingale measure (VSSM) and reveal their close relationship. Our analysis shows that the pre-committed discrete-time efficient mean-variance policy satisfies TCIE if and only if the conditional expectation of VSSM's density (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our findings indicate that the property of time consistency in efficiency only depends on the basic market setting, including portfolio constraints, and this fact motivates us to establish a general solution framework in constructing TCIE dynamic portfolio selection problem formulations by introducing suitable portfolio constraints

Realtime market microstructure analysis: online Transaction Cost Analysis

Motivated by the practical challenge in monitoring the performance of a large number of algorithmic trading orders, this paper provides a methodology that leads to automatic discovery of the causes that lie behind a poor trading performance. It also gives theoretical foundations to a generic framework for real-time trading analysis. Academic literature provides different ways to formalize these algorithms and show how optimal they can be from a mean-variance, a stochastic control, an impulse control or a statistical learning viewpoint. This paper is agnostic about the way the algorithm has been built and provides a theoretical formalism to identify in real-time the market conditions that influenced its efficiency or inefficiency. For a given set of characteristics describing the market context, selected by a practitioner, we first show how a set of additional derived explanatory factors, called anomaly detectors, can be created for each market order. We then will present an online methodology to quantify how this extended set of factors, at any given time, predicts which of the orders are underperforming while calculating the predictive power of this explanatory factor set. Armed with this information, which we call influence analysis, we intend to empower the order monitoring user to take appropriate action on any affected orders by re-calibrating the trading algorithms working the order through new parameters, pausing their execution or taking over more direct trading control. Also we intend that use of this method in the post trade analysis of algorithms can be taken advantage of to automatically adjust their trading action.Comment: 33 pages, 12 figure