Convex duality in optimal investment under illiquidity

We study the problem of optimal investment by embedding it in the general conjugate duality framework of convex analysis. This allows for various extensions to classical models of liquid markets. In particular, we obtain a dual representation for the optimum value function in the presence of portfolio constraints and nonlinear trading costs that are encountered e.g. in modern limit order markets. The optimization problem is parameterized by a sequence of financial claims. Such a parameterization is essential in markets without a numeraire asset when pricing swap contracts and other financial products with multiple payout dates. In the special case of perfectly liquid markets or markets with proportional transaction costs, we recover well-known dual expressions in terms of martingale measures.     

Arbitrage theory for non convex financial market models

We propose a unified approach where a security market is described by a liquidation value process. This allows to extend the frictionless models of the classical theory as well as the recent proportional transaction costs models to a larger class of financial markets with transaction costs including non proportional trading costs. The usual tools from convex analysis however become inadequate to characterize the absence of arbitrage opportunities in non-convex financial market models. The natural question is to which extent the results of the classical arbitrage theory are still valid. Our contribution is a first attempt to characterize the absence of arbitrage opportunities in non convex financial market models.     

A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for d assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman’s principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.     

Optimal and coherent economic-capital structures: evidence from long and short-sales trading positions under illiquid market perspectives

This paper broadens research literature associated with the assessment of modern portfolio risk management techniques by presenting a thorough modeling of nonlinear dynamic asset allocation and management under the supposition of illiquid and adverse market settings. Specifically, the paper proposes a re-engineered and robust approach to optimal economic capital allocation, in a Liquidity-Adjusted Value at Risk (L-VaR) framework, and particularly from the perspective of trading portfolios that have both long and short-sales trading positions. This paper expands previous approaches by explicitly modeling the liquidation of trading portfolios, over the holding period, with the aid of an appropriate scaling of the multiple-assets’ L-VaR matrix along with GARCH-M technique to forecast conditional volatility and expected return. Moreover, in this paper, the authors develop a dynamic nonlinear portfolio selection model and an optimization algorithm which allocates both economic capital and trading assets subject to some selected financial and operational rational constraints. The empirical results strongly confirm the importance of enforcing financially and operationally meaningful nonlinear and dynamic constraints, when they are available, on economic capital optimization procedure. The empirical results are interesting in terms of theory as well as practical applications and can aid in developing robust portfolio management algorithms that financial entities could consider in light of the aftermath of the latest financial crisis.     

A model for optimal execution of atomic orders

Atomic Orders are the basic elements of any algorithm for automated trading in electronic stock exchanges. The main concern in their execution is achieving the most efficient price. We propose two optimal strategies for the execution of atomic orders based on minimization of impact and volatility costs. The first considered strategy is based on a relatively simple nonlinear optimization model while the second allows re-optimization at some time point within a given execution time. In both cases a combination of market and limit orders is used. The key innovation in our approach is the introduction of a Fill Probability function which allows a combination of market and limit orders in the two optimization models we are discussing in this paper. Under certain conditions the objective functions of both considered problems are convex and therefore standard optimization tools can be applied. The efficiency of the resulting strategies is tested against two benchmarks representing common market practice on a representative sample of real trading data.     

Dual representation of superhedging costs in illiquid markets

This paper studies superhedging of contingent claims in illiquid markets where trading costs may depend nonlinearly on the traded amounts and portfolios may be subject to constraints. We give dual expressions for superhedging costs of financial contracts where claims and premiums are paid possibly at multiple points in time. Besides classical pricing problems, this setup covers various swap and insurance contracts where premiums are paid in sequences. Validity of the dual expressions is proved under new relaxed conditions related to the classical no-arbitrage condition. A new version of the fundamental theorem of asset pricing is given for unconstrained models with nonlinear trading costs.     

LMI Approach to Robust Model Predictive Control

This paper introduces a new approach to robust model predictive control (MPC) based on conservative approximations to semi-infinite optimization using linear matrix inequalities (LMIs). The method applies to problems with convex quadratic costs, linear and convex quadratic constraints, and linear predictive models with bounded uncertainty. If the MPC optimization problem is feasible at the initial control step (the first application of the MPC optimization), it is shown that the MPC optimization problems will be feasible at all future time steps and that the controlled system will be closed-loop stable. The method is illustrated with a solenoid control example. The authors thank the anonymous reviewers for suggestions that improved the presentation of this work. The work was supported in part by the EPRI/DoD Complex Interactive Networks/Systems Initiative under Contract EPRI-W08333-05 and by the US Army Research Office Contract DAAD19-01-1-0485.     

A Level Set Representation Method for N-Dimensional Convex Shape and Applications

In this work,we present a new method for convex shape representation,which is regardless of the dimension of the concerned objects,using level-set approaches.To the best of our knowledge,the proposed prior is the first one which can work for high dimensional objects.Convexity prior is very useful for object completion in computer vision.It is a very challenging task to represent high dimensional convex objects.In this paper,we first prove that the convexity of the considered object is equivalent to the convexity of the associated signed distance function.Then,the second order condition of convex functions is used to characterize the shape convexity equivalently.We apply this new method to two applications:object segmentation with convexity prior and convex hull problem(especially with outliers).For both applications,the involved problems can be written as a general optimization problem with three constraints.An algorithm based on the alternating direction method of multipliers is presented for the optimization problem.Numerical experiments are conducted to verify the effectiveness of the proposed representation method and algorithm.     

Portfolio optimization with linear and fixed transaction costs

We consider the problem of portfolio selection, with transaction costs and constraints on exposure to risk. Linear transaction costs, bounds on the variance of the return, and bounds on different shortfall probabilities are efficiently handled by convex optimization methods. For such problems, the globally optimal portfolio can be computed very rapidly. Portfolio optimization problems with transaction costs that include a fixed fee, or discount breakpoints, cannot be directly solved by convex optimization. We describe a relaxation method which yields an easily computable upper bound via convex optimization. We also describe a heuristic method for finding a suboptimal portfolio, which is based on solving a small number of convex optimization problems (and hence can be done efficiently). Thus, we produce a suboptimal solution, and also an upper bound on the optimal solution. Numerical experiments suggest that for practical problems the gap between the two is small, even for large problems involving hundreds of assets. The same approach can be used for related problems, such as that of tracking an index with a portfolio consisting of a small number of assets.     

Convex risk measures for portfolio optimization and concepts of flexibility

Due to their axiomatic foundation and their favorable computational properties convex risk measures are becoming a powerful tool in financial risk management. In this paper we will review the fundamental structural concepts of convex risk measures within the framework of convex analysis. Then we will exploit it for deriving strong duality relations in a generic portfolio optimization context. In particular, the duality relationship can be used for designing new, efficient approximation algorithms based on Nesterov's smoothing techniques for non-smooth convex optimization. Furthermore, the presented concepts enable us to formalize the notion of flexibility as the (marginal) risk absorption capacity of a technology or (available) resources. This paper is dedicated to R.T. Rockafellar for his stimulating and impressive work in convex optimization for decades. We thank you for the insights and inspirations we gained from your fundamental research.     

Proximal Point Algorithms for Convex Multi-criteria Optimization with Applications to Supply Chain Risk Management

We study a class of convex multi-criteria optimization problems with convex objective functions under linear constraints. We use a non-scalarization method—namely, two implementable proximal point algorithms—to obtain the Pareto optimum under multi-criteria optimization. We show that the algorithms are globally convergent. We apply the algorithms to a supply chain risk management problem under multi-criteria considerations.     

Convex analysis and financial equilibrium

Convexity has long had an important role in economic theory, but some recent developments have featured it all the more in problems of equilibrium. Here the tools of convex analysis are applied to a basic model of incomplete financial markets in which assets are traded and money can be lent or borrowed between the present and future. The existence of an equilibrium is established with techniques that include bounds derived from the duals to problems of utility maximization. Composite variational inequalities furnish the modeling platform. Models with and without short-selling are handled, moreover in the absence of any requirement that agents must initially have a positive amount of every asset, as is typical in equilibrium work in economics.     

Convexity in oriented matroids

We generalize to oriented matroids classical notions of Convexity Theory: faces of convex polytopes, convex hull, etc., and prove some basic properties. We relate the number of acyclic orientations of an orientable matroid to an evaluation of its Tutte polynomial.     

Stochastic optimization under constraints

We study a stochastic optimization problem under constraints in a general framework including financial models with constrained portfolios, labor income and large investor models and reinsurance models. We also impose American-type constraint on the state space process. General objective functions including deterministic or random utility functions and shortfall risk loss functions are considered. We first prove existence and uniqueness result to this optimization problem. In a second part, we develop a dual formulation under minimal assumptions on the objective functions, which are the analogue of the asymptotic elasticity condition of Kramkov and Schachermayer (1999).     

Safety first portfolio choice based on financial and sustainability returns

The aim of this paper is to expand the methodological spectrum of socially responsible investing by introducing stochastic sustainability returns into safety first models for portfolio choice. We provide a foundation of the notion of sustainability in portfolio theory and establish a general model for generalized safety first portfolio management with probabilistic constraints and three specifications of it. Moreover, we prove theorems about conditions for unique optimal solutions and for the constraints of one model being more restrictive than those of another. In an empirical part, we calculate the costs of investing according to our approach in terms of less financial return.     

Credit risk optimization with Conditional Value-at-Risk criterion

This paper examines a new approach for credit risk optimization. The model is based on the Conditional Value-at-Risk (CVaR) risk measure, the expected loss exceeding Value-at-Risk. CVaR is also known as Mean Excess, Mean Shortfall, or Tail VaR. This model can simultaneously adjust all positions in a portfolio of financial instruments in order to minimize CVaR subject to trading and return constraints. The credit risk distribution is generated by Monte Carlo simulations and the optimization problem is solved effectively by linear programming. The algorithm is very efficient; it can handle hundreds of instruments and thousands of scenarios in reasonable computer time. The approach is demonstrated with a portfolio of emerging market bonds. Received: November 1, 1999 / Accepted: October 1, 2000?Published online December 15, 2000     

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a 100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.

     

面对成比例交易成本的情况下外汇期权的定价

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor's expected utility. Option prices are determined from the investor's basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility.Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

FOREIGN CURRENCY OPTION PRICING WITH PROPORTIONAL TRANSACTION COSTS

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor＇s expected utility. Option prices are determined from the investor＇s basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility. Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

60 Years of portfolio optimization: Practical challenges and current trends

The concepts of portfolio optimization and diversification have been instrumental in the development and understanding of financial markets and financial decision making. In light of the 60 year anniversary of Harry Markowitz’s paper “Portfolio Selection,” we review some of the approaches developed to address the challenges encountered when using portfolio optimization in practice, including the inclusion of transaction costs, portfolio management constraints, and the sensitivity to the estimates of expected returns and covariances. In addition, we selectively highlight some of the new trends and developments in the area such as diversification methods, risk-parity portfolios, the mixing of different sources of alpha, and practical multi-period portfolio optimization.