# Option Portfolio Selection with Generalized Entropic Portfolio Optimization

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## Abstract

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## 1. Introduction

#### 1.1. Literature Review

## 2. Maximum Exponential Growth Rate

#### 2.1. The Kelly Criterion for Multiple Wagers

#### 2.2. Extension of the Kelly Criterion to Option Strategies

#### 2.2.1. Covered Call

#### 2.2.2. Married Put

#### 2.2.3. Credit Spread

#### 2.2.4. Straddle

#### 2.2.5. Long Strangle

#### 2.2.6. Butterfly Spread

#### 2.2.7. Iron Condor

## 3. Minimum Relative Entropy

#### 3.1. Shannon Entropy

#### 3.2. Kullback–Leibler Divergence

## 4. Option Portfolio Selection Based on Growth Rate and Relative Entropy

#### 4.1. Generalized Entropic Portfolio Optimization (GEPO)

#### 4.2. Risk-Adjusted Performance

## 5. An Option Portfolio Selection Example with GEPO

#### 5.1. Data

#### 5.2. Efficient Frontier and Portfolio Selection

#### 5.3. Comparison to the Kelly Criterion Over Time

## 6. Conclusions

## 7. Materials and Methods

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ADP | Approximate dynamic programming |

AIG | American International Group |

DEPO | Discrete entropic portfolio optimization |

FB | Facebook Inc. |

GEPO | Generalized entropic portfolio optimization |

IBM | International Business Machines |

KL | Kullback–Leibler |

MCD | McDonald’s Corp |

MRK | Merck & Co. |

ORCL | Oracle Corp |

REPO | Return-entropy portfolio optimization |

WRDS | Wharton Research Data Services |

## References

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**Figure 1.**Sample asset return payoff function for class of assets used in return-entropy portfolio optimization (REPO).

**Figure 2.**Sample asset return payoff function for class of assets used in discrete entropic portfolio optimization (DEPO).

**Figure 3.**Sample asset return payoff function for class of assets used in generalized entropic portfolio optimization (GEPO).

**Table 1.**Mean outcome, average state probabilities and estimated relative entropy (in trits) of select equity credit spread options from July 2012 to January 2018.

Company Name | Symbol | Mean Outcome | p-prob | q-prob | $\mathit{\varrho}$-prob | ${\mathit{D}}_{\mathbf{KL}}$ |
---|---|---|---|---|---|---|

Apple Inc. | AAPL | 0.10124 | 52.3% | 42.9% | 4.9% | 0.226675 |

Accenture | ACN | 0.146462 | 54.3% | 38.1% | 7.7% | 0.183754 |

American Intl. Group | AIG | 0.121123 | 49.8% | 38.2% | 11.9% | 0.118573 |

Bank of America Corp | BAC | 0.139333 | 49.6% | 32.6% | 17.8% | 0.071421 |

Biogen | BIIB | −0.048327 | 51.4% | 46.1% | 2.4% | 0.281129 |

Caterpillar Inc. | CAT | 0.151175 | 53.7% | 38.5% | 7.8% | 0.180714 |

Capital One Financial Corp | COF | 0.013458 | 46.6% | 45.3% | 8.1% | 0.16507 |

Costco Wholesale Corp | COST | 0.177373 | 51.8% | 37.3% | 10.8% | 0.135782 |

Cisco Systems | CSCO | 0.340769 | 59.6% | 25% | 15.4% | 0.141729 |

Facebook Inc. | FB | 0.128127 | 53.5% | 42.3% | 4.2% | 0.242447 |

Intl. Business Machines | IBM | 0.148669 | 53.2% | 38.7% | 8.2% | 0.173439 |

Intel Corp | INTC | 0.143774 | 51.7% | 35.5% | 12.8% | 0.115093 |

Johnson & Johnson | JNJ | 0.290017 | 61.9% | 32.6% | 5.5% | 0.251597 |

JPMorgan Chase & Co. | JPM | 0.223986 | 58% | 36.4% | 5.6% | 0.230925 |

MasterCard Inc. | MA | 0.125993 | 53.3% | 40.8% | 5.9% | 0.209407 |

McDonald’s Corp | MCD | 0.160243 | 53% | 37.7% | 9.3% | 0.157863 |

3M Company | MMM | 0.18277 | 55.7% | 35.7% | 8.6% | 0.17661 |

Merck & Co. | MRK | 0.177165 | 54% | 37.9% | 8% | 0.17795 |

Microsoft | MSFT | 0.170696 | 55.3% | 38.5% | 6.2% | 0.209972 |

Oracle Corp | ORCL | 0.286192 | 60.1% | 30.6% | 9.3% | 0.191343 |

**Table 2.**Selected equity credit spreads on 12 January 2018, with their respective spread intervals, deltas and state projections.

Symbol | Spread Type | Spread Interval | Sell Delta | Buy Delta | p-proj | q-proj | $\mathit{\varrho}$-proj |
---|---|---|---|---|---|---|---|

AAPL | Put | [167.5, 170] | −0.496757 | −0.405426 | 50.3% | 40.5% | 9.1% |

ACN | Call | [149, 150] | 0.492771 | 0.447763 | 50.7% | 44.8% | 4.5% |

AIG | Call | [60, 61] | 0.489573 | 0.345599 | 51% | 34.6% | 14.4% |

BAC | Put | [28.5, 29] | −0.497381 | −0.401975 | 50.3% | 40.2% | 9.5% |

BIIB | Put | [317.5, 320] | −0.495617 | −0.448496 | 50.4% | 44.8% | 4.7% |

CAT | Put | [149, 150] | −0.497975 | −0.43738 | 50.2% | 43.7% | 6.1% |

COF | Call | [92.5, 93] | 0.498336 | 0.465034 | 50.2% | 46.5% | 3.3% |

COST | Put | [182.5, 185] | −0.497485 | −0.417452 | 50.3% | 41.7% | 8% |

CSCO | Put | [36.5, 37] | −0.496195 | −0.374409 | 50.4% | 37.4% | 12.2% |

FB | Put | [170, 172.5] | −0.487722 | −0.392952 | 51.2% | 39.3% | 9.5% |

IBM | Put | [150, 152.5] | −0.494561 | −0.311505 | 50.5% | 31.2% | 18.3% |

INTC | Put | [44, 44.5] | −0.496887 | −0.403415 | 50.3% | 40.3% | 9.3% |

JNJ | Call | [142, 143] | 0.499681 | 0.408484 | 50% | 40.8% | 9.1% |

JPM | Call | [105, 106] | 0.499166 | 0.435065 | 50.1% | 43.5% | 6.4% |

MA | Call | [146, 147] | 0.494829 | 0.433672 | 50.5% | 43.4% | 6.1% |

MCD | Put | [170, 172.5] | −0.499101 | −0.335054 | 50.1% | 33.5% | 16.4% |

MMM | Call | [242.5, 245] | 0.495376 | 0.398127 | 50.5% | 39.8% | 9.7% |

MRK | Call | [55, 55.5] | 0.48691 | 0.405147 | 51.3% | 40.5% | 8.2% |

MSFT | Call | [84.5, 85] | 0.498834 | 0.451614 | 50.1% | 45.2% | 4.7% |

ORCL | Call | [50, 51] | 0.490397 | 0.374001 | 51% | 37.4% | 11.6% |

**Table 3.**The Kelly criterion portfolio of options with percent allocation for expiration week 2, 12 January 2018.

Symbol | Spread Type | Spread Interval | p-proj | q-proj | $\mathit{\varrho}$-proj | Kelly Allocation % |
---|---|---|---|---|---|---|

IBM | Put | [150, 152.5] | 50.5% | 31.2% | 18.3% | 12% |

Symbol | Spread Type | Spread Interval | p-proj | q-proj | $\mathit{\varrho}$-proj | GEPO Allocation % |
---|---|---|---|---|---|---|

IBM | Put | [150, 152.5] | 50.5% | 31.2% | 18.3% | 1.5% |

AIG | Call | [60, 61] | 51% | 34.6% | 14.1% | 1.5% |

MCD | Put | [170, 172.5] | 50.1% | 33.5% | 16.4% | 1.5% |

ORCL | Call | [50, 51] | 51% | 37.4% | 11.6% | 1.5% |

FB | Put | [170, 172.5] | 51.2% | 39.3% | 9.5% | 1.5% |

MRK | Call | [55, 55.5] | 51.3% | 40.5% | 8.2% | 1.5% |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Mercurio, P.J.; Wu, Y.; Xie, H. Option Portfolio Selection with Generalized Entropic Portfolio Optimization. *Entropy* **2020**, *22*, 805.
https://doi.org/10.3390/e22080805

**AMA Style**

Mercurio PJ, Wu Y, Xie H. Option Portfolio Selection with Generalized Entropic Portfolio Optimization. *Entropy*. 2020; 22(8):805.
https://doi.org/10.3390/e22080805

**Chicago/Turabian Style**

Mercurio, Peter Joseph, Yuehua Wu, and Hong Xie. 2020. "Option Portfolio Selection with Generalized Entropic Portfolio Optimization" *Entropy* 22, no. 8: 805.
https://doi.org/10.3390/e22080805