The purpose of this research is to investigate devising the optimal position liquidation strategies in financial markets. During the transaction of large block orders, investors need to maximize their profits against the losses caused by price slippage. We employed a sample-path based stochastic programming approach to obtain a dynamic lower-bound optimal trading strategy with Conditional Value-at-Risk constraint. We analyzed the optimization problems with different types of objective functions determined by distinct market impact functions. Nonanticipativity and risk constraints are discussed and properly imposed to avoid anticipating solutions and to control risk. A new formulation is proposed to obtain a lower-bound of the optimal position liquidation problem. A case study is implemented in the run-file environment of Portfolio Safeguard. Lower-bound optimal strategies are obtained from problems with and without risk constraints. The result also verifies the supposed improvement of computational efficiency in the new formulation.

This paper studies the risk-adjusted optimal timing to liquidate an option at the prevailing market price. In addition to maximizing the expected discounted return from option sale, we incorporate a path-dependent risk penalty based on shortfall or quadratic variation of the option price up to the liquidation time. We establish the conditions under which it is optimal to immediately liquidate or hold the option position through expiration. Furthermore, we study the variational inequality associated with the optimal stopping problem, and prove the existence and uniqueness of a strong solution. A series of analytical and numerical results are provided to illustrate the non-trivial optimal liquidation strategies under geometric Brownian motion (GBM) and exponential Ornstein-Uhlenbeck models. We examine the combined effects of price dynamics and risk penalty on the sell and delay regions for various options. In addition, we obtain an explicit closed-form solution for the liquidation of a stock with quadratic penalty under the GBM model.

A quasi-multi-period model for optimal position liquidation in the presence of market impact is proposed. Two features distinguish the approach from alternatives. First, a shrinking horizon framework is implemented to update intraday parameters by incorporating new information while maintaining standard non-anticipativity constraints. The method is data-driven, numerically tractable, and reactive to the market. Second, lower partial moments, a downside risk measure, is used which captures traders' increased risk aversion to losses better than symmetric risk measures. The performance of the proposed strategies is tested using historical, high-frequency New York Stock Exchange (NYSE) data. The proposed strategies outperform their benchmark on days with unfavorable market conditions, strongly supporting the use of lower partial moments as a risk measure. Additionally, results validate the use of a shrinking horizon framework as an adaptive, tractable alternative to dynamic programming for trading.

This monograph collects in one place the basic definitions, a careful description of the model, and discussion of how convex optimization can be used in multi-period trading, all in a common notation and framework.

Optimization problems involving stochastic models occur in almost all areas of science and engineering, such as telecommunications, medicine, and finance. Their existence compels a need for rigorous ways of formulating, analyzing, and solving such problems. This book focuses on optimization problems involving uncertain parameters and covers the theoretical foundations and recent advances in areas where stochastic models are available. Readers will find coverage of the basic concepts of modeling these problems, including recourse actions and the nonanticipativity principle. The book also includes the theory of two-stage and multistage stochastic programming problems; the current state of the theory on chance (probabilistic) constraints, including the structure of the problems, optimality theory, and duality; and statistical inference in and risk-averse approaches to stochastic programming.

Stochastic programming is the study of procedures for decision making under the presence of uncertainties and risks. Stochastic programming approaches have been successfully used in a number of areas such as energy and production planning, telecommunications, and transportation. Recently, the practical experience gained in stochastic programming has been expanded to a much larger spectrum of applications including financial modeling, risk management, and probabilistic risk analysis. Major topics in this volume include: (1) advances in theory and implementation of stochastic programming algorithms; (2) sensitivity analysis of stochastic systems; (3) stochastic programming applications and other related topics. Audience: Researchers and academies working in optimization, computer modeling, operations research and financial engineering. The book is appropriate as supplementary reading in courses on optimization and financial engineering.

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