### Abstract

We discuss optimal trading strategies for general utility functions in portfolios of cash and stocks subject to small proportional transaction costs. We present a new interpretation of scalings found by Soner, Shreve, and others. To leading order in the small transaction cost parameter, the free boundary problem for the expected utility's value function is shown to be dual, in the sense of Lagrange multipliers for optimal design problems, to a free boundary problem minimizing a cost function. This cost function is the sum of a boundary integral corresponding to the rate of trading and an interior integral corresponding to opportunity loss that results from suboptimal portfolio allocation. Using the dual problem's formulation, we show that the quasi-steady state probability density of the optimal portfolio is uniform for a single stock but generally blows up even in the simple case of two uncorrelated stocks.

Original language | English (US) |
---|---|

Pages (from-to) | 1977-1998 |

Number of pages | 22 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 70 |

Issue number | 6 |

DOIs | |

State | Published - 2010 |

### Fingerprint

### Keywords

- Asymptotics
- Cost minimization
- General utility function
- Oblique boundary conditions
- Portfolio optimization
- Transaction costs

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*70*(6), 1977-1998. https://doi.org/10.1137/07070334X

**Balancing small transaction costs with loss of optimal allocation in dynamic stock trading strategies.** / Goodman, Jonathan; Ostrov, Daniel N.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 70, no. 6, pp. 1977-1998. https://doi.org/10.1137/07070334X

}

TY - JOUR

T1 - Balancing small transaction costs with loss of optimal allocation in dynamic stock trading strategies

AU - Goodman, Jonathan

AU - Ostrov, Daniel N.

PY - 2010

Y1 - 2010

N2 - We discuss optimal trading strategies for general utility functions in portfolios of cash and stocks subject to small proportional transaction costs. We present a new interpretation of scalings found by Soner, Shreve, and others. To leading order in the small transaction cost parameter, the free boundary problem for the expected utility's value function is shown to be dual, in the sense of Lagrange multipliers for optimal design problems, to a free boundary problem minimizing a cost function. This cost function is the sum of a boundary integral corresponding to the rate of trading and an interior integral corresponding to opportunity loss that results from suboptimal portfolio allocation. Using the dual problem's formulation, we show that the quasi-steady state probability density of the optimal portfolio is uniform for a single stock but generally blows up even in the simple case of two uncorrelated stocks.

AB - We discuss optimal trading strategies for general utility functions in portfolios of cash and stocks subject to small proportional transaction costs. We present a new interpretation of scalings found by Soner, Shreve, and others. To leading order in the small transaction cost parameter, the free boundary problem for the expected utility's value function is shown to be dual, in the sense of Lagrange multipliers for optimal design problems, to a free boundary problem minimizing a cost function. This cost function is the sum of a boundary integral corresponding to the rate of trading and an interior integral corresponding to opportunity loss that results from suboptimal portfolio allocation. Using the dual problem's formulation, we show that the quasi-steady state probability density of the optimal portfolio is uniform for a single stock but generally blows up even in the simple case of two uncorrelated stocks.

KW - Asymptotics

KW - Cost minimization

KW - General utility function

KW - Oblique boundary conditions

KW - Portfolio optimization

KW - Transaction costs

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UR - http://www.scopus.com/inward/citedby.url?scp=77951182026&partnerID=8YFLogxK

U2 - 10.1137/07070334X

DO - 10.1137/07070334X

M3 - Article

VL - 70

SP - 1977

EP - 1998

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 6

ER -