Multi-Portfolio Optimization: A Potential Game Approach

Abstract

In modern asset management, portfolio managers address the multi-account investment decision problem by optimizing each account's portfolio separately based on the trading requirements and portfolio constraints of the individual clients. However, trades associated with the individual accounts are usually pooled together for execution, therefore amplifying the level of the so-called market impact on all accounts. If this aggregate market impact is not considered when each account is individually optimized, the actual market impact can be severely underestimated. Multi-portfolio optimization aims at finding the optimal rebalancing of the multiple accounts by considering their joint effects while adhering to account-specific constraints. In this paper, we first model this phenomenon as a Nash Equilibrium problem (NEP) and thereafter consider a generalized NEP (GNEP) for the case where there are global constraints imposed on all accounts, adopting as a desirable outcome the concept of Nash Equilibrium (NE). For both game problems, we give a complete characterization of the NE, including its existence and uniqueness, and devise various distributed algorithms with provable convergence. Interestingly, the proposed methodology heavily hinges on a number of well-known and important signal processing techniques.

A Unified Successive Pseudoconvex Approximation Framework

“…In the context of the MIMO IC, (1) is a nonconvex problem and NPhard [5]. As another example, consider portfolio optimization in which f (x) represents the expected return of the portfolio (to be maximized) and the set X characterizes the trading constraints [10]. Furthermore, in sparse (l 1 -regularized) linear regression, f (x) denotes the least square function and g(x) is the sparsity regularization function [11,12].…”

“…. , x K ) is convex, the parallel algorithms converge if the stepsize is inversely proportional to the number of block variables K. This choice of stepsize, however, tends to be overly conservative in systems with a large number of block variables and inevitably slows down the convergence [2,10,18].…”

“…2) The stepsizes can be determined based on the problem structure, typically resulting in faster convergence than in cases where constant stepsizes [2,10,18] and decreasing stepsizes [8,19] are used. For example, a constant stepsize can be used when f (x) is given as the difference of two convex functions as in DC programming [21].…”

On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

“…In a practical market incorporating multiple investors, trades of diverse investors are usually pooled and executed together [35,54]. The transaction cost for a single investor may depend on the overall trading levels in the market and not just its own trading.…”

“…The problem(54) has a solution if and only if ∆(A, b) is nonempty and the following condition is satisfied:If x, v ∈ R n are such that x ∈ ∆(A, b), v ≥ 0, Av = 0, and v T Dv = 0, then (Dx + c) T v ≥ 0.Before providing the result, we state a simple result about the quadratic programming which may be proved directly or by invoking[20, Cor. 2.3.7].…”