Optimal Execution in a Market with Small Investors

Abstract

The author considers the dynamic trading strategies that minimize the expected cost of trading a large block of securities over a fixed finite number of periods. In this model, the market impact function that yields the execution prices for individual trades is endogeneously determined. This analysis is novel in that it introduces small investors, who do not affect the price flow, and a noise trader as market participants other than the institutional investors into a general equilibrium model. It is found that the institutional investor takes a rather complicated strategy to make use of its private information. As a result, the price impact not only changes over time but also depends on the trade history. Although there are several studies that deal with this topic in the recent empirical literature, it has remained unnoticed in the context of the theoretical optimal execution model.     

Optimal Execution and Block Trade Pricing: A General Framework

Abstract

In this article, we develop a general framework to study optimal execution and to price block trades. We prove existence of optimal liquidation strategies and provide regularity results for optimal strategies under very general hypotheses. We exhibit a Hamiltonian characterization for the optimal strategy that can be used for numerical approximation. We also focus on the important topic of block trade pricing and propose a methodology to give a price to financial (il)liquidity. In particular, we provide a closed-form formula for the price of a block trade when there is no time constraint to liquidate.     

Optimal Trade Execution Under Stochastic Volatility and Liquidity

Abstract

We study the problem of optimally liquidating a financial position in a discrete-time model with stochastic volatility and liquidity. We consider the three cases where the objective is to minimize the expectation, an expected exponential or a mean-variance criterion of the implementation cost. In the first case, the optimal solution can be fully characterized by a forward-backward system of stochastic equations depending on conditional expectations of future liquidity. In the other two cases, we derive Bellman equations from which the optimal solutions can be obtained numerically by discretizing the control space. In all three cases, we compute optimal strategies for different simulated realizations of prices, volatility and liquidity and compare the outcomes to the ones produced by the deterministic strategies of Bertsimas and Lo (1998; Optimal control of execution costs. Journal of Financial Markets, 1, 1–50) and Almgren and Chriss (2001; Optimal execution of portfolio transactions. Journal of Risk, 3, 5–33).     

Mean–Variance Optimal Adaptive Execution

Abstract

Electronic trading of equities and other securities makes heavy use of ‘arrival price’ algorithms that balance the market impact cost of rapid execution against the volatility risk of slow execution. In the standard formulation, mean–variance optimal trading strategies are static: they do not modify the execution speed in response to price motions observed during trading. We show that substantial improvement is possible by using dynamic trading strategies and that the improvement is larger for large initial positions.

We develop a technique for computing optimal dynamic strategies to any desired degree of precision. The asset price process is observed on a discrete tree with an arbitrary number of levels. We introduce a novel dynamic programming technique in which the control variables are not only the shares traded at each time step but also the maximum expected cost for the remainder of the program; the value function is the variance of the remaining program. The resulting adaptive strategies are ‘aggressive-in-the-money’: they accelerate the execution when the price moves in the trader's favor, spending parts of the trading gains to reduce risk.     

Mean–variance optimal trading problem subject to stochastic dominance constraints with second order autoregressive price dynamics

The efficient modeling of execution price path of an asset to be traded is an important aspect of the optimal trading problem. In this paper an execution price path based on the second order autoregressive process is proposed. The proposed price path is a generalization of the existing first order autoregressive price path in literature. Using dynamic programming method the analytical closed form solution of unconstrained optimal trading problem under the second order autoregressive process is derived. However in order to incorporate non-negativity constraints in the problem formulation, the optimal static trading problems under second order autoregressive price process are formulated. For a risk neutral investor, the optimal static trading problem of minimizing expected execution cost subject to non-negativity constraints is formulated as a quadratic programming problem. Whereas, for a risk averse investor the variance of execution cost is considered as a measure for the timing risk, and the mean–variance problem is formulated. Moreover, the optimal static trading problem subject to stochastic dominance constraints with mean–variance static trading strategy as the reference strategy is studied. Using Static approximation method the algorithm to solve proposed optimal static trading problems is presented. With numerical illustrations conducted on simulated data and the real market data, the significance of second order autoregressive price path, and the optimal static trading problems is presented.     

A Hamilton-Jacobi-Bellman approach to optimal trade execution

The optimal trade execution problem is formulated in terms of a mean-variance tradeoff, as seen at the initial time. The mean-variance problem can be embedded in a linear-quadratic (LQ) optimal stochastic control problem. A semi-Lagrangian scheme is used to solve the resulting nonlinear Hamilton-Jacobi-Bellman (HJB) PDE. This method is essentially independent of the form for the price impact functions. Provided a strong comparison property holds, we prove that the numerical scheme converges to the viscosity solution of the HJB PDE. Numerical examples are presented in terms of the efficient trading frontier and the trading strategy. The numerical results indicate that in some cases there are many different trading strategies which generate almost identical efficient frontiers.     

Optimal Basket Liquidation for CARA Investors is Deterministic

Abstract

We consider the problem faced by an investor who must liquidate a given basket of assets over a finite time horizon. The investor's goal is to maximize the expected utility of the sales revenues over a class of adaptive strategies. We assume that the investor's utility has constant absolute risk aversion (CARA) and that the asset prices are given by a very general continuous-time, multiasset price impact model. Our main result is that (perhaps surprisingly) the investor does no worse if he narrows his search to deterministic strategies. In the case where the asset prices are given by an extension of the nonlinear price impact model of Almgren [(2003) Applied Mathematical Finance, 10, pp. 1–18], we characterize the unique optimal strategy via the solution of a Hamilton equation and the value function via a nonlinear partial differential equation with singular initial condition.     

Efficient portfolios in financial markets with proportional transaction costs

In this article, we characterize efficient portfolios, i.e. portfolios which are optimal for at least one rational agent, in a very general multi-currency financial market model with proportional transaction costs. In our setting, transaction costs may be random, time-dependent, have jumps and the preferences of the agents are modeled by multivariate expected utility functions. We provide a complete characterization of efficient portfolios, generalizing earlier results of Dybvig (Rev Financ Stud 1:67–88, 1988) and Jouini and Kallal (J Econ Theory 66: 178–197, 1995). We basically show that a portfolio is efficient if and only if it is cyclically anticomonotonic with respect to at least one consistent price system that prices it. Finally, we introduce the notion of utility price of a given contingent claim as the minimal amount of a given initial portfolio allowing any agent to reach the claim by trading, and give a dual representation of it as the largest proportion of the market price necessary for all agents to reach the same expected utility level.     

Robust option pricing

In this paper, we combine robust optimization and the idea of ?$?$-arbitrage to propose a tractable approach to price a wide variety of options. Rather than assuming a probabilistic model for the stock price dynamics, we assume that the conclusions of probability theory, such as the central limit theorem, hold deterministically on the underlying returns. This gives rise to an uncertainty set that the underlying asset returns satisfy. We then formulate the option pricing problem as a robust optimization problem that identifies the portfolio which minimizes the worst case replication error for a given uncertainty set defined on the underlying asset returns. The most significant benefits of our approach are (a) computational tractability illustrated by our ability to price multi-asset, American and Asian options using linear optimization; and thus the computational complexity of our approach scales polynomially with the number of assets and with time to expiry and (b) modeling flexibility illustrated by our ability to model different kinds of options, various levels of risk aversion among investors, transaction costs, shorting constraints and replication via option portfolios.     

机构投资者的最优变现策略

在投资、变现等大宗交易过程中,资产交易价格与交易策略密切相关,因此,交易的完成过程需要很高的技巧.文章讨论了机构投资者的最优变现策略问题,假设证券价格服从几何布朗运动,以均值方差效用为目标函数,得到了最优变现策略所满足的二阶微分方程,并由差分法得到其数值解.最后,由参数的敏感性分析知:最优变现策略与瞬时冲击、市场波动率及风险厌恶系数等参数有关,但与永久冲击无关,且最优变现策略对市场波动率和瞬时冲击的变化较敏感.     

Regularized robust optimization: the optimal portfolio execution case

An uncertainty set is a crucial component in robust optimization. Unfortunately, it is often unclear how to specify it precisely. Thus it is important to study sensitivity of the robust solution to variations in the uncertainty set, and to develop a method which improves stability of the robust solution. In this paper, to address these issues, we focus on uncertainty in the price impact parameters in an optimal portfolio execution problem. We first illustrate that a small variation in the uncertainty set may result in a large change in the robust solution. We then propose a regularized robust optimization formulation which yields a solution with a better stability property than the classical robust solution. In this approach, the uncertainty set is regularized through a regularization constraint, defined by a linear matrix inequality using the Hessian of the objective function and a regularization parameter. The regularized robust solution is then more stable with respect to variation in the uncertainty set specification, in addition to being more robust to estimation errors in the price impact parameters. The regularized robust optimal execution strategy can be computed by an efficient method based on convex optimization. Improvement in the stability of the robust solution is analyzed. We also study implications of the regularization on the optimal execution strategy and its corresponding execution cost. Through the regularization parameter, one can adjust the level of conservatism of the robust solution.     

Optimal portfolio execution under time-varying liquidity constraints

In this article, we take an algorithmic approach to solve the problem of optimal execution under time-varying constraints on the depth of a limit order book (LOB). Our algorithms are within the resilience model proposed by Obizhaeva and Wang (2013) with a more realistic assumption on the order book depth; the amount of liquidity provided by an LOB market is finite at all times. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a one-dimensional root-finding problem which can be readily solved by standard numerical algorithms. When the depth of the order book is monotone in time, we apply the Karush-Kuhn-Tucker conditions to narrow down the set of candidate strategies. Then, we use a dichotomy-based search algorithm to pin down the optimal one. For the general case, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity.     

Martingale optimal transport and robust hedging in continuous time

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge–Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.     

Modified extragradient algorithms for solving equilibrium problems

ABSTRACT

In this paper, we introduce some new algorithms for solving the equilibrium problem in a Hilbert space which are constructed around the proximal-like mapping and inertial effect. Also, some convergence theorems of the algorithms are established under mild conditions. Finally, several experiments are performed to show the computational efficiency and the advantage of the proposed algorithm over other well-known algorithms.     

基于联系数的高职院校教师教学执行力评价数学模型

教学执行力是教师从事教学工作的能力,是教师素质的重要展示,科学、公平、公正地评价教师的教学执行力益于引导教师提高素质.采用"三选法"探讨教师教学执行力评价指标内容、用矩阵法确定指标权重、评价量化数值采用三参数区间数表示,并转化为联系数处理数据运算,方法更加方便、有效.     

我国保险企业执行力模糊评价模型与实证分析

执行力不足是现阶段我国保险企业普遍面临的困境.根据保险企业执行力的影响因素,可以建立保险企业执行力评价指标体系,并采用AHP与模糊评价相结合的方法对我国保险企业执行力进行模糊评价.各保险企业可以从评价结果中找出自身执行力的薄弱环节加以改进,以提升企业执行力。     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

Optimal Selling of an Asset with Jumps Under Incomplete Information

ABSTRACT

We study the optimal liquidation strategy of an asset with price process satisfying a jump diffusion model with unknown jump intensity. It is assumed that the intensity takes one of two given values, and we have an initial estimate for the probability of both of them. As time goes by, by observing the price fluctuations, we can thus update our beliefs about the probabilities for the intensity distribution. We formulate an optimal stopping problem describing the optimal liquidation problem. It is shown that the optimal strategy is to liquidate the first time the point process falls below (goes above) a certain time-dependent boundary.