MODELING THE OPTIMAL STRATEGY IN AN INCOMPLETE MARKET

We examine the optimal portfolio selection problem for a single agent who receives a unhedgeable endowment. The agent wishes to optimize his/her log-utility derived from his/her terminal wealth. We do not solve this problem analytically but rigorously prove that there exists a unique optimal portfolio strategy. We present a recursive computational algorithm which produces a sequence of portfolios converging to the optimal one. We present an ”intelligent” initial portfolio which requires, numerically, about 25% fewer corrective steps in the algorithm than a random initial portfolio, and outperforms the portfolio which ignores the unhedgeable risk of the endowment.

[1]  Mark D. Schroder,et al.  Optimal Lifetime Consumption-Portfolio Strategies Under Trading Constraints and Generalized Recursive Preferences , 2003 .

[2]  Stylianos Perrakis,et al.  Option pricing and replication with transaction costs and dividends , 2000 .

[3]  J. Kallsen,et al.  A Complete Explicit Solution to the Log-Optimal Portfolio Problem , 2003 .

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

[5]  Jakša Cvitanić,et al.  HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH12 , 1996 .

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

[7]  W. Schachermayer,et al.  The asymptotic elasticity of utility functions and optimal investment in incomplete markets , 1999 .

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

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

[10]  Andrew E. B. Lim.  Quadratic Hedging and Mean-Variance Portfolio Selection with Random Parameters in an Incomplete Market , 2004, Math. Oper. Res..

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

[12]  I. Karatzas,et al.  Optimal Consumption from Investment and Random Endowment in Incomplete Semimartingale Markets , 2001, 0706.0051.

[13]  D. Cuoco.  Optimal Consumption and Equilibrium Prices with Portfolio Constraints and Stochastic Income , 1997 .

[14]  Vicky Henderson,et al.  Explicit solutions to an optimal portfolio choice problem with stochastic income , 2005 .

[15]  J. Keppo.  Optimality with Hydropower System , 2002, IEEE Power Engineering Review.

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

[17]  W. Fleming,et al.  Hedging in incomplete markets with HARA utility , 1997 .

[18]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[19]  Monique Jeanblanc,et al.  Optimization of consumption with labor income , 1998, Finance Stochastics.

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

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

[22]  Gordan Zitkovic.  Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment , 2005, math/0503516.

[23]  David M. Kreps.  Arbitrage and equilibrium in economies with infinitely many commodities , 1981 .

[24]  Jussi Keppo.  Optimality with hydropower system , 2002 .

[25]  R. C. Merton,et al.  Continuous-Time Finance , 1990 .

[26]  S. Shreve,et al.  Martingale and duality methods for utility maximization in a incomplete market , 1991 .

[27]  Hui Wang,et al.  Utility maximization in incomplete markets with random endowment , 2001, Finance Stochastics.