### Abstract

A classic dynamic asset allocation problem optimizes the expected final-time utility of wealth, for an individual who can invest in a risky stock and a risk-free bond, trading continuously in time. Recently, several authors considered the corresponding static asset allocation problem in which the individual cannot trade but can invest in options as well as the underlying. The optimal static strategy can never do better than the optimal dynamic one. Surprisingly, however, for some market models the two approaches are equivalent. When this happens the static strategy is clearly preferable, since it avoids any impact of market frictions. This paper examines the question: when, exactly, are the static and dynamic approaches equivalent? We give an easily tested necessary and sufficient condition, and many non-trivial examples. Our analysis assumes that the stock follows a scalar diffusion process, and uses the completeness of the resulting market model. A simple special case is when the drift and volatility depend only on time; then the two approaches are equivalent precisely if (μ(t) - r)/σ^{2}(t) is constant. This is not the Sharpe ratio or the market price of risk, but rather a nondimensional ratio of excess return to squared volatility that arises naturally in portfolio optimization problems.

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

Pages (from-to) | 173-183 |

Number of pages | 11 |

Journal | Quantitative Finance |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2006 |

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### Keywords

- Asset allocation
- Portfolio optimization

### ASJC Scopus subject areas

- Economics, Econometrics and Finance(all)

### Cite this

*Quantitative Finance*,

*6*(2), 173-183. https://doi.org/10.1080/14697680600580946

**On the equivalence of the static and dynamic asset allocation problems.** / Kohn, Robert; Papazoglu-Statescu, Oana M.

Research output: Contribution to journal › Article

*Quantitative Finance*, vol. 6, no. 2, pp. 173-183. https://doi.org/10.1080/14697680600580946

}

TY - JOUR

T1 - On the equivalence of the static and dynamic asset allocation problems

AU - Kohn, Robert

AU - Papazoglu-Statescu, Oana M.

PY - 2006/4/1

Y1 - 2006/4/1

N2 - A classic dynamic asset allocation problem optimizes the expected final-time utility of wealth, for an individual who can invest in a risky stock and a risk-free bond, trading continuously in time. Recently, several authors considered the corresponding static asset allocation problem in which the individual cannot trade but can invest in options as well as the underlying. The optimal static strategy can never do better than the optimal dynamic one. Surprisingly, however, for some market models the two approaches are equivalent. When this happens the static strategy is clearly preferable, since it avoids any impact of market frictions. This paper examines the question: when, exactly, are the static and dynamic approaches equivalent? We give an easily tested necessary and sufficient condition, and many non-trivial examples. Our analysis assumes that the stock follows a scalar diffusion process, and uses the completeness of the resulting market model. A simple special case is when the drift and volatility depend only on time; then the two approaches are equivalent precisely if (μ(t) - r)/σ2(t) is constant. This is not the Sharpe ratio or the market price of risk, but rather a nondimensional ratio of excess return to squared volatility that arises naturally in portfolio optimization problems.

AB - A classic dynamic asset allocation problem optimizes the expected final-time utility of wealth, for an individual who can invest in a risky stock and a risk-free bond, trading continuously in time. Recently, several authors considered the corresponding static asset allocation problem in which the individual cannot trade but can invest in options as well as the underlying. The optimal static strategy can never do better than the optimal dynamic one. Surprisingly, however, for some market models the two approaches are equivalent. When this happens the static strategy is clearly preferable, since it avoids any impact of market frictions. This paper examines the question: when, exactly, are the static and dynamic approaches equivalent? We give an easily tested necessary and sufficient condition, and many non-trivial examples. Our analysis assumes that the stock follows a scalar diffusion process, and uses the completeness of the resulting market model. A simple special case is when the drift and volatility depend only on time; then the two approaches are equivalent precisely if (μ(t) - r)/σ2(t) is constant. This is not the Sharpe ratio or the market price of risk, but rather a nondimensional ratio of excess return to squared volatility that arises naturally in portfolio optimization problems.

KW - Asset allocation

KW - Portfolio optimization

UR - http://www.scopus.com/inward/record.url?scp=33646580360&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646580360&partnerID=8YFLogxK

U2 - 10.1080/14697680600580946

DO - 10.1080/14697680600580946

M3 - Article

AN - SCOPUS:33646580360

VL - 6

SP - 173

EP - 183

JO - Quantitative Finance

JF - Quantitative Finance

SN - 1469-7688

IS - 2

ER -