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**1 - 5**of**5**### Unified Framework of Mean-Field Formulations for Optimal Multi-period Mean-Variance Portfolio Selection

, 2014

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### Working Paper Series Cone-Constrained Continuous-Time Markowitz Problems Cone-Constrained Continuous-Time Markowitz Problems

"... Abstract The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in ..."

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Abstract The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in a (possibly random and timedependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in L 2 . Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes L ± appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of L ± or equivalently into a coupled system of backward stochastic differential equations for L ± . We show how this can be used to both characterise and construct optimal strategies. Our results explain and generalise all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.

### Working Paper Series Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints out within the NCCR FINRISK project on "Mathematical Methods in Financial Risk Management" Convex duality in mean-variance hedging under convex trading constrain

"... Abstract We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictabl ..."

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Abstract We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L 2 of the space of all gains from trade (i.e., the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad hoc methods in specific frameworks. MSC 2010 Subject Classification: 60G48, 91G10, 93E20, 49N10, 60H05

### Impossible Frontiers ∗

, 2008

"... A key result of the Capital Asset Pricing Model (CAPM) is that the market portfolio— the portfolio of all assets in which each asset’s weight is proportional to its total market capitalization—lies on the mean-variance efficient frontier, the set of portfolios having meanvariance characteristics tha ..."

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A key result of the Capital Asset Pricing Model (CAPM) is that the market portfolio— the portfolio of all assets in which each asset’s weight is proportional to its total market capitalization—lies on the mean-variance efficient frontier, the set of portfolios having meanvariance characteristics that cannot be improved upon. Therefore, the CAPM cannot be consistent with efficient frontiers for which every frontier portfolio has at least one negative weight or short position. We call such efficient frontiers “impossible”, and derive conditions on asset-return means, variances, and covariances that yield impossible frontiers. With the exception of the two-asset case, we show that impossible frontiers are difficult to avoid. Moreover, as the number of assets n grows, we prove that the probability that a generically chosen frontier is impossible tends to one at a geometric rate. In fact, for one natural class of distributions, nearly one-eighth of all assets on a frontier is expected to have negative weights for every portfolio on the frontier. We also show that the expected minimum amount of shortselling across frontier portfolios grows linearly with n, and even when shortsales are constrained to some finite level, an impossible frontier remains impossible. Using daily and monthly U.S. stock returns, we document the impossibility of efficient frontiers in the data.

### © notice, is given to the source. Impossible Frontiers

, 2008

"... The views and opinions expressed in this article are those of the authors only, and do not necessarily represent the views and opinions of AlphaSimplex Group, MIT, Northwestern University, any of their affiliates and employees, or the National Bureau of Economic Research. The authors make no represe ..."

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The views and opinions expressed in this article are those of the authors only, and do not necessarily represent the views and opinions of AlphaSimplex Group, MIT, Northwestern University, any of their affiliates and employees, or the National Bureau of Economic Research. The authors make no representations or warranty, either expressed or implied, as to the accuracy or completeness of the information contained in this article, nor are they recommending that this article serve as the basis for any investment decision---this article is for information purposes only. Research support from AlphaSimplex Group and the MIT