Optimal execution of portfolio transaction with linear cost model

Abstract

In investment, wealth is allocated by forming portfolio which consists of risk free and risky assets (stocks). There are some factors that influence portfolio transaction, i.e. volatility risk and transaction costs. The aim of portfolio transaction is to produce a future cash flow in terms of dividend and capital gain. The cash flow is obtained by constructing an optimal portfolio subject to minimum risk of trading strategies. Optimal portfolio can be formed by minimizing cost function, which is a combination of volatility risk and transaction costs arising from permanent and temporary market impact. This optimization can also generally be done by minimizing implementation of shortfall costs using linear market impact cost model. The construction of efficient strategies for optimal portfolio is obtained by a constrained optimization problem that minimize the expected value of implementation shortfall for any levels of maximum implemented shortfall variance. The constrained optimization problem can be solved using Lagrange multiplier, so that the unconstrained optimization problem will minimize cost function. Cost function is assumed to be a quadrature, which is strictly convex for any positive risk levels. Therefore, by solving the optimization problem, one can obtain explicit trajectories of optimal strategies. By solving initial value problem of difference equation, one can obtain the specific optimal solution of a trading trajectory, and for each risk aversion there is a unique corresponding trading trajectory that minimize cost function. Key words: volatility risk, market impact, implementation shortfall, trading trajectory, trading strategy, cost function.