An optimal Strategy for Hedging with Short-Term Futures Contracts

The search for an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-dated futures contracts leads to a class of intrinsic optimization problems. We give an explicit analytic solution for this optimization problem if the market price of the commodity is based on a simple Gaussian model, thereby replacing previously used incomplete approximations to the optimal strategy.     

LIQUIDATION OF AN INDIVISIBLE ASSET WITH INDEPENDENT INVESTMENT

We provide an extension of the explicit solution of a mixed optimal stopping–optimal stochastic control problem introduced by Henderson and Hobson. The problem examines whether the optimal investment problem on a local martingale financial market is affected by the optimal liquidation of an independent indivisible asset. The indivisible asset process is defined by a homogeneous scalar stochastic differential equation, and the investor's preferences are defined by a general expected utility function. The value function is obtained in explicit form, and we prove the existence of an optimal stopping–investment strategy characterized as the limit of an explicit maximizing strategy. Our approach is based on the standard dynamic programming approach.     

PORTFOLIO OPTIMIZATION AND STOCHASTIC VOLATILITY ASYMPTOTICS

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean‐reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast‐moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single‐factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.     

DYNAMIC PORTFOLIO OPTIMIZATION WITH A DEFAULTABLE SECURITY AND REGIME‐SWITCHING

We consider a portfolio optimization problem in a defaultable market with finitely‐many economical regimes, where the investor can dynamically allocate her wealth among a defaultable bond, a stock, and a money market account. The market coefficients are assumed to depend on the market regime in place, which is modeled by a finite state continuous time Markov process. By separating the utility maximization problem into a predefault and postdefault component, we deduce two coupled Hamilton–Jacobi–Bellman equations for the post‐ and predefault optimal value functions, and show a novel verification theorem for their solutions. We obtain explicit constructions of value functions and investment strategies for investors with logarithmic and Constant Relative Risk Aversion utilities, and provide a precise characterization of the directionality of the bond investment strategies in terms of corporate returns, forward rates, and expected recovery at default. We illustrate the dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a detailed economic and numerical analysis.     

OPTIMAL EXECUTION OF A VWAP ORDER: A STOCHASTIC CONTROL APPROACH

We consider the optimal liquidation of a position of stock (long or short) where trading has a temporary market impact on the price. The aim is to minimize both the mean and variance of the order slippage with respect to a benchmark given by the market volume‐weighted average price (VWAP). In this setting, we introduce a new model for the relative volume curve which allows simultaneously for accurate data fit, economic justification, and mathematical tractability. Tackling the resulting optimization problem using a stochastic control approach, we derive and solve the corresponding Hamilton–Jacobi–Bellman equation to give an explicit characterization of the optimal trading rate and liquidation trajectory.     

Should Stochastic Volatility Matter to the Cost-Constrained Investor?

Significant strides have been made in the development of continuous-time portfolio optimization models since Merton (1969) . Two independent advances have been the incorporation of transaction costs and time-varying volatility into the investor's optimization problem. Transaction costs generally inhibit investors from trading too often. Time-varying volatility, on the other hand, encourages trading activity, as it can result in an evolving optimal allocation of resources. We examine the two-asset portfolio optimization problem when both elements are present. We show that a transaction cost framework can be extended to include a stochastic volatility process. We then specify a transaction cost model with stochastic volatility and show that when the risk premium is linear in variance, the optimal strategy for the investor is independent of the level of volatility in the risky asset. We call this the Variance Invariance Principle.     

关于人口城市化劳动力转移的思考——兼谈户籍制度改革的障碍和建议

在中国城市化进程中,面对骤然激增的农民工流向大中城市的现实,政府虽采取了以小城镇为主的户籍制度改革来缓解矛盾,但农村劳动力转移到城市后所遇到的主要障碍还在于至今尚未得到根本性改革的户籍制度.三农问题的症结是城市化滞后和制度创新的迟滞.要解决三农问题,就要消除不利于城市化发展的体制和政治障碍,把户籍改革扩大到城市;建立完善法律法规体系;建立全国统一的劳动力流动制度.     

Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein-Uhlenbeck type

We study Merton's classical portfolio optimization problem for an investor who can trade in a risk-free bond and a stock. The goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of non-Gaussian Ornstein-Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff-Nielsen and Shephard (2001) . Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman-Kac formulas are derived and verified for power utilities. Some numerical examples are also presented.     

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.     

OPTIMAL LIQUIDATION IN A LIMIT ORDER BOOK FOR A RISK‐AVERSE INVESTOR

In a limit order book model with exponential resilience, general shape function, and an unaffected stock price following the Bachelier model, we consider the problem of optimal liquidation for an investor with constant absolute risk aversion. We show that the problem can be reduced to a two‐dimensional deterministic problem which involves no buy orders. We derive an explicit expression for the value function and the optimal liquidation strategy. The analysis is complicated by the fact that the intervention boundary, which determines the optimal liquidation strategy, is discontinuous if there are levels in the limit order book with relatively little market depth. Despite this complication, the equation for the intervention boundary is fairly simple. We show that the optimal liquidation strategy possesses the natural properties one would expect, and provide an explicit example for the case where the limit order book has a constant shape function.     

The Secretary Problem Revisited - The Group Decision-Making Perspective

We consider a decision situation where there is an initial set of alternatives that may be augmented, a variant of the problem known in the literature as the "secretary problem". We focus the discussion on the special case of group decision-making, where a group or committee is charged with the responsibility of negotiating the decision. We investigate situations of explicitly stated (multiple) criteria and the simpler situation of no such stated criteria. The former case includes the congenial, the mixed, and the uncongenial cases, where the individuals agree on the direction of all, some, or none of the criteria, respectively. We offer a framework within which a group of individuals can be supported in such a decision process. In the case of explicit criteria, we provide the decision-makers with probability information of the likelihood of finding more preferred alternatives provided the initial set of alternatives is expanded. The framework is tested using a simulated real-world choice situation.     

Optimal Financing of a Corporation Subject To Random Returns

We consider the problem of finding an optimal financing mix of retained earnings and external equity for maximizing the value of a corporation in a stochastic environment. We formulate the problem as a singular stochastic control for a diffusion process. We show that the value function satisfies a free-boundary problem. We characterize the value function and show that the optimal policy can be characterized in terms of two threshold parameters. With asset level below the lower threshold, optimal policy is to finance the firm's growth by retaining all earnings and raising the required external equity financing. With asset level above the higher threshold, optimal policy is to pay all retained earnings as dividends and to bring in no new equity. Between the two thresholds, the optimal policy is to retain all earnings but not raise any external equity. We obtain an explicit solution for the value function when there is no brokerage commission in floating external equity. We provide economic interpretations of the results obtained in the paper.     

Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes

In a market driven by a Lévy martingale, we consider a claim ξ. We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for ξ: one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark-Haussmann-Ocone theorem.     

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 Nelson–Siegel Family

We study a problem posed in Bj"ork and Christensen (1999): Does there exist any nontrivial interest rate model that is consistent with the Nelson–Siegel family? They show that within the Heath–Jarrow–Morton framework with deterministic volatility structure the answer is no. In this paper we give a generalized version of this result including stochastic volatility structure. For that purpose we introduce the class of consistent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson–Siegel family. We characterize the consistent state space Itô processes in terms of their drift and diffusion coefficients. By solving an inverse problem we find their explicit form. It turns out that there exists no nontrivial interest rate model driven by a consistent state space Itô process.     

Predictable forward performance processes: Infrequent evaluation and applications to human-machine interactions

We study discrete-time predictable forward processes when trading times do not coincide with performance evaluation times in a binomial tree model for the financial market. The key step in the construction of these processes is to solve a linear functional equation of higher order associated with the inverse problem driving the evolution of the predictable forward process. We provide sufficient conditions for the existence and uniqueness and an explicit construction of the predictable forward process under these conditions. Furthermore, we find that these processes are inherently myopic in the sense that optimal strategies do not make use of future model parameters even if these are known. Finally, we argue that predictable forward preferences are a viable framework to model human-machine interactions occurring in automated trading or robo-advising. For both applications, we determine an optimal interaction schedule of a human agent interacting infrequently with a machine that is in charge of trading.     

ROBUST UTILITY MAXIMIZATION WITH LÉVY PROCESSES

We study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets, that is, possible instantaneous drift, volatility, and jump characteristics of the price process. We show that an optimal investment strategy exists and compute it in semi‐closed form. Moreover, we provide a saddle point analysis describing a worst‐case model.     

WIENER CHAOS: A NEW APPROACH TO OPTION HEDGING

This paper addresses the problem of estimating and analyzing the residual risk that is not hedged by a discrete hedging strategy. the use of die chaotic representation allows an elegant decomposition of the residual risk to be hedged by adequate assets. Alternative strategies to the classical delta hedging and optimization under the risk-neutral and historical probabilities are discussed.     

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.