Portfolio Selection: A Review

This paper reviews portfolio selection models and provides perspective on some open issues. It starts with a review of the classic Markowitz mean-variance framework. It then presents the intertemporal portfolio choice approach developed by Merton and the fundamental notion of dynamic hedging. Martingale methods and resulting portfolio formulas are also reviewed. Their usefulness for economic insights and numerical implementations is illustrated. Areas of future research are outlined.

[1]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[2]  Jérôme Detemple,et al.  Intertemporal asset allocation: A comparison of methods ☆ , 2005 .

[3]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[4]  Massimo Marinacci,et al.  PORTFOLIO SELECTION WITH MONOTONE MEAN‐VARIANCE PREFERENCES , 2009 .

[5]  T. R. Hurd,et al.  Portfolio choice with jumps: A closed-form solution , 2009, 0906.2324.

[6]  Fernando Zapatero,et al.  Monte Carlo computation of optimal portfolios in complete markets , 2003 .

[7]  D. Heath,et al.  Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation , 1990, Journal of Financial and Quantitative Analysis.

[8]  David S. Bates Post-'87 crash fears in the S&P 500 futures option market , 2000 .

[9]  Jessica A. Wachter Risk Aversion and Allocation to Long-Term Bonds , 2001, J. Econ. Theory.

[10]  Carsten Sørensen,et al.  Dynamic Asset Allocation and Fixed Income Management , 1999, Journal of Financial and Quantitative Analysis.

[11]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[12]  Jakša Cvitanić,et al.  Convex Duality in Constrained Portfolio Optimization , 1992 .

[13]  Xiuli Chao,et al.  On martingale measures when asset returns have unpredictable jumps , 1996 .

[14]  Jérôme Detemple,et al.  Representation formulas for Malliavin derivatives of diffusion processes , 2005, Finance Stochastics.

[15]  Jérôme Detemple,et al.  Dynamic Asset Allocation: Portfolio Decomposition Formula and Applications , 2010 .

[16]  S. Ross,et al.  The valuation of options for alternative stochastic processes , 1976 .

[17]  M. Allais Le comportement de l'homme rationnel devant le risque : critique des postulats et axiomes de l'ecole americaine , 1953 .

[18]  H. Thompson,et al.  Jump‐Diffusion Processes and the Term Structure of Interest Rates , 1988 .

[19]  X. Zhou,et al.  MEAN–VARIANCE PORTFOLIO OPTIMIZATION WITH STATE‐DEPENDENT RISK AVERSION , 2014 .

[20]  Abraham Lioui,et al.  International asset allocation: A new perspective , 2002 .

[21]  M. Rothschild,et al.  Increasing risk: I. A definition , 1970 .

[22]  Jessica A. Wachter Portfolio and Consumption Decisions under Mean-Reverting Returns: An Exact Solution for Complete Markets , 2001, Journal of Financial and Quantitative Analysis.

[23]  Jean-Charles Rochet,et al.  CHANGES OF NUMERAIRE, CHANGES OF PROBABILITY MEASURE , 1995 .

[24]  Xing Jin,et al.  Decomposition of Optimal Portfolio Weight in a Jump-Diffusion Model and Its Applications , 2012 .

[25]  Larry G. Epstein,et al.  Stochastic differential utility , 1992 .

[26]  Stanley R. Pliska,et al.  A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios , 1986, Math. Oper. Res..

[27]  V. Bogachëv,et al.  Liens entre équations différentielles stochastiques et ordinaires en dimension infinie , 1995 .

[28]  Claus Munk,et al.  Optimal Consumption and Investment Strategies with Stochastic Interest Rates , 2001 .

[29]  J. Detemple,et al.  Asymptotic Properties of Monte Carlo Estimators of Diffusion Processes , 2006 .

[30]  J. Detemple,et al.  A Monte Carlo Method for Optimal Portfolios , 2000 .

[31]  J. Cox,et al.  Optimal consumption and portfolio policies when asset prices follow a diffusion process , 1989 .

[32]  Monique Pontier,et al.  Optimal portfolio for a small investor in a market model with discontinuous prices , 1990 .

[33]  Abraham Lioui,et al.  On optimal portfolio choice under stochastic interest rates , 2001 .

[34]  Abraham Lioui The asset allocation puzzle is still a puzzle , 2007 .

[35]  Jan Kallsen,et al.  Optimal portfolios for exponential Lévy processes , 2000, Math. Methods Oper. Res..

[36]  D. Nualart The Malliavin Calculus and Related Topics , 1995 .

[37]  S. Shreve,et al.  Optimal portfolio and consumption decisions for a “small investor” on a finite horizon , 1987 .

[38]  John P. Lehoczky,et al.  Martingale and duality methods for utuility [ i . e . , utility ] maximization in an incomplete market , 2015 .

[39]  David S. Bates Post-&Apos;87 Crash Fears in S&P 500 Futures Options , 1997 .

[40]  I. Karatzas,et al.  A generalized clark representation formula, with application to optimal portfolios , 1991 .

[41]  Xiuli Chao,et al.  Martingale Analysis for Assets with Discontinuous Returns , 1995, Math. Oper. Res..

[42]  Eduardo S. Schwartz,et al.  Strategic asset allocation , 1997 .

[43]  F. Jamshidian An Exact Bond Option Formula , 1989 .

[44]  Sanjiv Ranjan Das,et al.  Systemic Risk and International Portfolio Choice , 2002 .

[45]  R. C. Merton,et al.  Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case , 1969 .

[46]  Mark D. Schroder,et al.  Optimal Consumption and Portfolio Selection with Stochastic Differential Utility , 1999 .

[47]  Nicholas G. Polson,et al.  The Impact of Jumps in Volatility and Returns , 2000 .

[48]  D. Ellsberg Decision, probability, and utility: Risk, ambiguity, and the Savage axioms , 1961 .

[49]  Suleyman Basak,et al.  Dynamic Mean-Variance Asset Allocation , 2009 .

[50]  Larry G. Epstein,et al.  Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework , 1989 .

[51]  R. C. Merton,et al.  An Analytic Derivation of the Efficient Portfolio Frontier , 1972, Journal of Financial and Quantitative Analysis.

[52]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[53]  Jérôme Detemple,et al.  Chapter 21 Simulation Methods for Optimal Portfolios , 2007 .

[54]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[55]  Claudia Klüppelberg,et al.  Optimal portfolios when stock prices follow an exponential Lévy process , 2004, Finance Stochastics.

[56]  M. Rindisbacher,et al.  CLOSED‐FORM SOLUTIONS FOR OPTIMAL PORTFOLIO SELECTION WITH STOCHASTIC INTEREST RATE AND INVESTMENT CONSTRAINTS , 2005 .

[57]  R. C. Merton,et al.  Optimum Consumption and Portfolio Rules in a Continuous-Time Model* , 1975 .

[58]  Knut K. Aase,et al.  Optimum portfolio diversification in a general continuous-time model , 1984 .