PORTFOLIO MANAGEMENT WITH TRANSACTION COSTS: AN ASYMPTOTIC ANALYSIS OF THE MORTON AND PLISKA MODEL

We examine the Morton and Pliska (1993) model for the optimal management of a portfolio when there are transaction costs proportional to a fixed fraction of the portfolio value. We analyze this model in the realistic case of small transaction costs by conducting a perturbation analysis about the no-transaction-cost solution. Although the full problem is a free-boundary diffusion problem in as many dimensions as there are assets in the portfolio, we find explicit solutions for the optimal trading policy in this limit. This makes the solution for a realistically large number of assets a practical possibility.     

BOUNDARY EVOLUTION EQUATIONS FOR AMERICAN OPTIONS

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.     

MULTIDIMENSIONAL PORTFOLIO OPTIMIZATION WITH PROPORTIONAL TRANSACTION COSTS

We provide a computational study of the problem of optimally allocating wealth among multiple stocks and a bank account, to maximize the infinite horizon discounted utility of consumption. We consider the situation where the transfer of wealth from one asset to another involves transaction costs that are proportional to the amount of wealth transferred. Our model allows for correlation between the price processes, which in turn gives rise to interesting hedging strategies. This results in a stochastic control problem with both drift-rate and singular controls, which can be recast as a free boundary problem in partial differential equations. Adapting the finite element method and using an iterative procedure that converts the free boundary problem into a sequence of fixed boundary problems, we provide an efficient numerical method for solving this problem. We present computational results that describe the impact of volatility, risk aversion of the investor, level of transaction costs, and correlation among the risky assets on the structure of the optimal policy. Finally we suggest and quantify some heuristic approximations.     

Optimal investment with correlated stochastic volatility factors

The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the situation with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. Our proposed approximation requires to solve numerically two linear equations in lower dimension instead of a fully nonlinear HJB equation. A rigorous accuracy result is derived by constructing sub- and super-solutions so that their difference is at the desired order of accuracy. We illustrate our result with a particular model for which we have explicit formulas for the approximation. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.     

OPTIMAL CONSUMPTION AND PORTFOLIO SELECTION WITH INCOMPLETE MARKETS AND UPPER AND LOWER BOUND CONSTRAINTS

We study an optimal consumption and portfolio selection problem for an investor by a martingale approach. We assume that time is a discrete and finite horizon, the sample space is finite and the number of securities is smaller than that of the possible securities price vector transitions. the investor is prohibited from investing stocks more (less, respectively) than given upper (lower) bounds at any time, and he maximizes an expected time additive utility function for the consumption process. First we give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. Also we state the existence of the unique primal optimal solutions. Next we formulate a dual control problem and establish the duality between primal and dual control problems. Also we show the existence of dual optimal solutions. Finally we consider the computational aspect of dual approach through a simple numerical example.     

OPTIMAL PORTFOLIO MANAGEMENT WITH FIXED TRANSACTION COSTS

We study optimal portfolio management policies for an investor who must pay a transaction cost equal to a fixed Traction of his portfolio value each time he trades. We focus on the infinite horizon objective function of maximizing the asymptotic growth rate, so me optimal policies we derive approximate those of an investor with logarithmic utility at a distant horizon. When investment opportunities are modeled as m correlated geometric Brownian motion stocks and a riskless bond, we show that the optimal policy reduces to solving a single stopping time problem. When there is a single risky stock, we give a system of equations whose solution determines the optima! rule. We use numerical methods to solve for the optima! policy when there are two risky stocks. We study several specific examples and observe the general qualitative result that, even with very low transaction cost levels, the optimal policy entails very infrequent trading.     

Optimal Stopping and the American Put

We show that the problem of pricing the American put is equivalent to solving an optimal stopping problem. the optimal stopping problem gives rise to a parabolic free-boundary problem. We show there is a unique solution to this problem which has a lower boundary. We identify an integral equation solved by the boundary and show that it is the unique solution to this equation satisfying certain natural additional conditions. the proofs also give a natural decomposition of the price of the American option as the sum of the price of the European option and an "American premium."     

A CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION

A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solving a (static) convex stochastic optimization problem in terms of the terminal wealth, with a budget constraint. In literature the latter is solved by assuming a priori that the problem is well-posed (i.e., the supremum value is finite) and a Lagrange multiplier exists (and as a consequence the optimal solution is attainable). In this paper it is first shown that, via various counter-examples, neither of these two assumptions needs to hold, and an optimal solution does not necessarily exist. These anomalies in turn have important interpretations in and impacts on the portfolio selection modeling and solutions. Relations among the non-existence of the Lagrange multiplier, the ill-posedness of the problem, and the non-attainability of an optimal solution are then investigated. Finally, explicit and easily verifiable conditions are derived which lead to finding the unique optimal solution.     

A NOTE ON THE QUANTILE FORMULATION

Many investment models in discrete or continuous‐time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change‐of‐variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank‐dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well‐posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law‐invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.     

OPTIMAL INVESTMENT IN CREDIT DERIVATIVES PORTFOLIO UNDER CONTAGION RISK

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.     

NEWS-GENERATED DEPENDENCE AND OPTIMAL PORTFOLIOS FOR n STOCKS IN A MARKET OF BARNDORFF-NIELSEN AND SHEPHARD TYPE

We consider Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein–Uhlenbeck type. The investor can trade in n stocks and a risk-free bond. We assume that the dependence between stocks lies in that they partly share the Ornstein–Uhlenbeck processes of the volatility. We refer to these as news processes, and interpret this as that dependence between stocks lies solely in their reactions to the same news. The model is primarily intended for assets that are dependent, but not too dependent, such as stocks from different branches of industry. We show that this dependence generates covariance, and give statistical methods for both the fitting and verification of the model to data. Using dynamic programming, we derive and verify explicit trading strategies and Feynman–Kac representations of the value function for power utility.     

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.     

OPTIONED PORTFOLIO SELECTION: MODELS AND ANALYSIS

The mean‐variance model of Markowitz and many of its extensions have been playing an instrumental role in guiding the practice of portfolio selection. In this paper we study a mean‐variance formulation for the portfolio selection problem involving options. In particular, the portfolio in question contains a stock index and some European style options on the index. A refined mean‐variance methodology is adopted in our approach to formulate this problem as multistage stochastic optimization. It turns out that there are two different solution techniques, both lead to explicit solutions of the problem: one is based on stochastic programming and optimality conditions, and the other one is based on stochastic control and dynamic programming. We introduce both techniques, because their strengths are very different so as to suit different possible extensions and refinements of the basic model. Attention is paid to the structure of the optimal payoff function, which is shown to possess rich properties. Further refinements of the model, such as the request that the payoff should be monotonic with respect to the index, are discussed. Throughout the paper, various numerical examples are used to illustrate the underlying concepts.     

ENHANCEMENT OF THE APPLICABILITY OF MARKOWITZ'S PORTFOLIO OPTIMIZATION BY UTILIZING RANDOM MATRIX THEORY

The traditional estimated return for the Markowitz mean-variance optimization has been demonstrated to seriously depart from its theoretic optimal return. We prove that this phenomenon is natural and the estimated optimal return is always       times larger than its theoretic counterpart, where       with  y  as the ratio of the dimension to sample size. Thereafter, we develop new bootstrap-corrected estimations for the optimal return and its asset allocation and prove that these bootstrap-corrected estimates are proportionally consistent with their theoretic counterparts. Our theoretical results are further confirmed by our simulations, which show that the essence of the portfolio analysis problem could be adequately captured by our proposed approach. This greatly enhances the practical uses of the Markowitz mean-variance optimization procedure.     

OPTIMAL LOT SOLUTION TO CARDINALITY CONSTRAINED MEAN–VARIANCE FORMULATION FOR PORTFOLIO SELECTION

The pioneering work of the mean–variance formulation proposed by Markowitz in the 1950s has provided a scientific foundation for modern portfolio selection. Although the trade practice often confines portfolio selection with certain discrete features, the existing theory and solution methodologies of portfolio selection have been primarily developed for the continuous solution of the portfolio policy that could be far away from the real integer optimum. We consider in this paper an exact solution algorithm in obtaining an optimal lot solution to cardinality constrained mean–variance formulation for portfolio selection under concave transaction costs. Specifically, a convergent Lagrangian and contour-domain cut method is proposed for solving this class of discrete-feature constrained portfolio selection problems by exploiting some prominent features of the mean–variance formulation and the portfolio model under consideration. Computational results are reported using data from the Hong Kong stock market.     

PORTFOLIO MANAGEMENT WITH CONSTRAINTS

The traditional portfolio selection problem concerns an agent whose objective is to maximize the expected utility of terminal wealth over some horizon. This basic problem can be modified by adding constraints. In this paper we investigate the portfolio selection problem for an investor who desires to outperform some benchmark index with a certain confidence level. The benchmark is chosen to reflect some particular investment objective and it can be either deterministic or stochastic. The optimal strategy for this class of problems can lead to nonconvex constraints raising issues of existence and uniqueness. We solve this optimal portfolio selection problem and investigate the procedure for both deterministic and stochastic benchmarks.     

LARGE PORTFOLIO ASYMPTOTICS FOR LOSS FROM DEFAULT

We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogeneous portfolios. The density of the limiting measure is shown to solve a nonlinear stochastic partial differential equation, and certain moments of the limiting measure are shown to satisfy an infinite system of stochastic differential equations. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustrate the accuracy of the approximation, and highlight its computational advantages over a direct Monte Carlo simulation of the original stochastic system.     

A symmetric LPM model for heuristic mean-semivariance analysis

While the semivariance (lower partial moment degree 2) has been variously described as being more in line with investors’ attitude towards risk, implementation in a forecasting portfolio management role has been hampered by computational problems. The original formulation by Markowitz (1959) requires a laborious iterative process because the cosemivariance matrix is endogenous and a closed form solution does not exist. There have been attempts at optimizing an exogenous asymmetric cosemivariance matrix. However, this approach does not always provide a positive semi-definite matrix for which a closed form solution exists. We provide a proof that converts the exogenous asymmetric matrix to a symmetric matrix for which a closed form solution does exist. This approach allows the mean-semivariance formulation to be solved using Markowitz's critical line algorithm. Empirical results compare the cosemivariance algorithm to the covariance algorithm which is currently the best optimization proxy for the cosemivariance. We also compare our formulation to Estrada's (2008) cosemivariance formulation. The results demonstrate that the cosemivariance algorithm is robust to a 45 security universe and is still effective at increasing portfolio skewness at a 150 security universe. There are four major benefits to a usable mean-semivariance formulation: (1) managers may engineer skewness into the portfolio without resorting to option strategies, (2) managers will be able to evaluate the skewness effect of option strategies within their portfolio, (3) a workable mean-semivariance algorithm leads to a workable n-degree lower partial moment (LPM) algorithms which provides managers access to a wider variety of investor utility functions including risk averse, risk neutral, and risk seeking utility functions, and (4) a workable LPM algorithm leads to a workable UPM/LPM (upper partial moment/lower partial moment) algorithm.     

HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH12

We derive a formula for the minimal initial wealth needed to hedge an arbitrary contingent claim in a continuous-time model with proportional transaction costs; the expression obtained can be interpreted as the supremum of expected discounted values of the claim, over all (pairs of) probability measures under which the “wealth process” is a supermartingale. Next, we prove the existence of an optimal solution to the portfolio optimization problem of maximizing utility from terminal wealth in the same model, we also characterize this solution via a transformation to a hedging problem: the optimal portfolio is the one that hedges the inverse of marginal utility evaluated at the shadow state-price density solving the corresponding dual problem, if such exists. We can then use the optimal shadow state-price density for pricing contingent claims in this market. the mathematical tools are those of continuous-time martingales, convex analysis, functional analysis, and duality theory.