Optimal Security Liquidation Algorithms

Abstract

This paper develops trading strategies for liquidation of a financial security, which maximize the expected return. The problem is formulated as a stochastic programming problem that utilizes the scenario representation of possible returns. Two cases are considered, a case with no constraint on risk and a case when the risk of losses associated with trading strategy is constrained by Conditional Value-at-Risk (CVaR) measure. In the first case, two algorithms are proposed; one is based on linear programming techniques, and the other uses dynamic programming to solve the formulated stochastic program. The third proposed algorithm is obtained by adding the risk constraints to the linear program. The algorithms provide path-dependent strategies, i.e., the fraction of security sold depends upon price sample-path of the security up to the current moment. The performance of the considered approaches is tested using a set of historical sample-paths of prices.

Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization

“…Step 1 Compute F k ≡ F α (x k ζ ) and ξ k g ≡ ((ξ k ) T , g k ) T via (4) and (8). If ξ k g < ε 0 , terminate the procedure with the current solution.…”

“…Due to this simple LP approach, and the aforementioned properties such as convexity and sub-additivity, CVaR has rapidly gained in popularity. Some recent applications include a medium-term integrated risk-management problem for a hydrothermal generation company [12], a short-term bidding strategy in power markets [11], and a trading strategy for the complete liquidation of a financial security under CVaR-based risk constraints [8].…”

A sample-path approach to optimal position liquidation

“…We compared their results for a 4-branch scenario tree, solution to which was presented among others, with the solution of problem (18) under a "tree" partition constructed by splitting each group into 4 groups of equal size at each time t. According to Proposition 1, "tree" partition reduces any decision rule (14) to the simple rule (12). We ran both algorithms on an instance of data containing 5120 paths, and the algorithm (18) slightly outperformed the dynamic programming approach, producing expected return of 1.0241 versus 1.02062 obtained by Butenko et al (2003). The similarity of solutions obtained by different techniques validates the correctness of our approach and that presented in Butenko et al (2003).…”

“…We ran both algorithms on an instance of data containing 5120 paths, and the algorithm (18) slightly outperformed the dynamic programming approach, producing expected return of 1.0241 versus 1.02062 obtained by Butenko et al (2003). The similarity of solutions obtained by different techniques validates the correctness of our approach and that presented in Butenko et al (2003).…”

“…The dynamic programming approach in application to the optimal closing problem in frictionless market was considered by Butenko et al (2003). The authors developed a technique for transforming a collection of sample paths into a scenario tree with a prescribed number of branches per node.…”