### Abstract

For small transaction costs, we determine the leading order optimal dynamic trading strategy of a portfolio of stock, cash, and options. Except for the transaction costs, our market assumptions are those of Black, Scholes, and Merton. Without transaction costs, the option is redundant in the portfolio. With transaction costs, however, we show that adding the option to the portfolio can significantly reduce overall trading costs compared to optimal strategies that use only stock and cash. The analysis is based on an asymptotic expansion with three scales: macroscopic, mesoscopic, and microscopic. The macroscopic analysis is Merton's optimal investment problem. Within a plane defined by the amount of stock and options held, the macroscopic analysis yields a Merton line of optimal portfolios. We show that there is a particular magic point on the Merton line that minimizes expensive stochastic movement away from the Merton line. The mesoscopic scale governs less expensive deviations of the portfolio away from the magic point but along the Merton line. The microscopic scale governs the more expensive deviations of the portfolio away from the magic point, transverse to the Merton line. The resulting strategy is related to commonly used Delta and Gamma hedging strategies, but our scale analysis implies that some rebalancings are much more effective than others. We do not give rigorous mathematical proofs, only arguments of formal applied mathematics.

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

Pages (from-to) | 512-537 |

Number of pages | 26 |

Journal | SIAM Journal on Financial Mathematics |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - 2011 |

### Fingerprint

### Keywords

- Asymptotics
- Merton strategy
- Optimal rebalancing
- Transaction costs

### ASJC Scopus subject areas

- Applied Mathematics
- Numerical Analysis
- Finance

### Cite this

*SIAM Journal on Financial Mathematics*,

*2*(1), 512-537. https://doi.org/10.1137/100798053

**An option to reduce transaction costs.** / Goodman, Jonathan; Ostrov, Daniel N.

Research output: Contribution to journal › Article

*SIAM Journal on Financial Mathematics*, vol. 2, no. 1, pp. 512-537. https://doi.org/10.1137/100798053

}

TY - JOUR

T1 - An option to reduce transaction costs

AU - Goodman, Jonathan

AU - Ostrov, Daniel N.

PY - 2011

Y1 - 2011

N2 - For small transaction costs, we determine the leading order optimal dynamic trading strategy of a portfolio of stock, cash, and options. Except for the transaction costs, our market assumptions are those of Black, Scholes, and Merton. Without transaction costs, the option is redundant in the portfolio. With transaction costs, however, we show that adding the option to the portfolio can significantly reduce overall trading costs compared to optimal strategies that use only stock and cash. The analysis is based on an asymptotic expansion with three scales: macroscopic, mesoscopic, and microscopic. The macroscopic analysis is Merton's optimal investment problem. Within a plane defined by the amount of stock and options held, the macroscopic analysis yields a Merton line of optimal portfolios. We show that there is a particular magic point on the Merton line that minimizes expensive stochastic movement away from the Merton line. The mesoscopic scale governs less expensive deviations of the portfolio away from the magic point but along the Merton line. The microscopic scale governs the more expensive deviations of the portfolio away from the magic point, transverse to the Merton line. The resulting strategy is related to commonly used Delta and Gamma hedging strategies, but our scale analysis implies that some rebalancings are much more effective than others. We do not give rigorous mathematical proofs, only arguments of formal applied mathematics.

AB - For small transaction costs, we determine the leading order optimal dynamic trading strategy of a portfolio of stock, cash, and options. Except for the transaction costs, our market assumptions are those of Black, Scholes, and Merton. Without transaction costs, the option is redundant in the portfolio. With transaction costs, however, we show that adding the option to the portfolio can significantly reduce overall trading costs compared to optimal strategies that use only stock and cash. The analysis is based on an asymptotic expansion with three scales: macroscopic, mesoscopic, and microscopic. The macroscopic analysis is Merton's optimal investment problem. Within a plane defined by the amount of stock and options held, the macroscopic analysis yields a Merton line of optimal portfolios. We show that there is a particular magic point on the Merton line that minimizes expensive stochastic movement away from the Merton line. The mesoscopic scale governs less expensive deviations of the portfolio away from the magic point but along the Merton line. The microscopic scale governs the more expensive deviations of the portfolio away from the magic point, transverse to the Merton line. The resulting strategy is related to commonly used Delta and Gamma hedging strategies, but our scale analysis implies that some rebalancings are much more effective than others. We do not give rigorous mathematical proofs, only arguments of formal applied mathematics.

KW - Asymptotics

KW - Merton strategy

KW - Optimal rebalancing

KW - Transaction costs

UR - http://www.scopus.com/inward/record.url?scp=84871060799&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871060799&partnerID=8YFLogxK

U2 - 10.1137/100798053

DO - 10.1137/100798053

M3 - Article

VL - 2

SP - 512

EP - 537

JO - SIAM Journal on Financial Mathematics

JF - SIAM Journal on Financial Mathematics

SN - 1945-497X

IS - 1

ER -