# Portfolio Optimization for Binary Options Based on Relative Entropy

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## Abstract

**:**

## 1. Introduction

#### Literature Review

## 2. Maximum Exponential Growth Rate

#### 2.1. The Kelly Criterion

#### 2.2. Extension of the Kelly Criterion to Multiple Wagers

## 3. Shannon Entropy of Discrete Returns

#### 3.1. Investments Versus Wagers

#### 3.2. Joint Entropy of a Portfolio of Discrete Return Assets

## 4. Minimum Relative Entropy

#### 4.1. Kullback–Leibler Divergence

#### 4.2. Relative Entropy as a Convex Risk Measure

- (i)
- Monotonicity. If $X\le Y$, then $\rho \left(X\right)\le \rho \left(Y\right)$,
- (ii)
- Translation invariance. If $c\in \mathbb{R}$, then $\rho (X+c)=\rho \left(X\right)+c$, and
- (iii)
- Convexity. $\rho (\lambda X+(1-\lambda \left)Y\right)\le \lambda \rho \left(X\right)+(1-\lambda )\rho \left(Y\right)$, for $0\le \lambda \le 1$.

**Proposition**

**1.**

**Proof.**

- (i)
- Monotonicity. For a risk measure $\rho (\xb7)$ to be monotonic it must satisfy: If $X,Y\in \mathbb{X}$ and $X\le Y$ almost surely then $\rho \left(X\right)\le \rho \left(Y\right)$ almost surely. Using $\rho \left(X\right)$ as stated, we have $\rho \left(X\right)=E\left(X\right)+k{D}_{KL}(P\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)$ and $\rho \left(Y\right)=E\left(Y\right)+k{D}_{KL}(Q\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)$ for discrete distributions P and Q. Because $X\le Y$ implies ${D}_{KL}(P\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)\le {D}_{KL}(Q\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)$ as a consequence of the data processing inequality (Cover, 1991) [30], it follows that$$\rho \left(X\right)=E\left(X\right)+k{D}_{KL}(P\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)\le E\left(Y\right)+k{D}_{KL}(Q\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)=\rho \left(Y\right),$$
- (ii)
- Translation invariance. For a risk measure $\rho (\xb7)$ to exhibit translation invariance it must satisfy: If $X\in \mathbb{X}$ then $\rho (X+c)=\rho \left(X\right)+c$. Recall the risk measure based on the relative entropy principle is of the form $\rho \left(X\right)=E\left(X\right)+k{D}_{KL}(P\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)$. Since $H(X+c)=H\left(X\right)$, for all c, it follows that ${D}_{KL}(P\left(X\right)\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)={D}_{KL}(P(X+c)\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)$ and, thus, we have (17),$$\rho (X+c)=E(X+c)+k{D}_{KL}(P(X+c)\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)=E\left(X\right)+c+k{D}_{KL}(P\left(X\right)\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}U)=\rho \left(X\right)+c.$$Therefore $\rho $ exhibits translation invariance.
- (iii)
- Convexity. A risk measure $\rho (\xb7)$ is convex if: For ${Z}_{1},{Z}_{2}\in \mathbb{X}$ and $\lambda \in [0,1]$ it follows that: $\rho (\lambda {Z}_{1}+(1-\lambda ){Z}_{2})\le \lambda \rho \left({Z}_{1}\right)+(1-\lambda )\rho \left({Z}_{2}\right)$. It is known that ${D}_{KL}(P\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}Q)$ is convex in the pair of probability mass functions $(P,Q)$. If $({P}_{1},{Q}_{1})$ and $({P}_{2},{Q}_{2})$ are two pairs of probability mass functions, then (18) follows,$${D}_{KL}(\lambda {P}_{1}+(1-\lambda ){P}_{2}\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}\lambda {Q}_{1}+(1-\lambda ){Q}_{2})\le \lambda {D}_{KL}({P}_{1}\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}{Q}_{1})+(1-\lambda ){D}_{KL}({P}_{2}\phantom{\rule{0.222222em}{0ex}}\parallel \phantom{\rule{0.222222em}{0ex}}{Q}_{2}).$$

#### 4.3. Minimum Risk Option Portfolios with Relative Entropy

#### 4.4. Discrete Entropic Portfolio Optimization (DEPO)

#### 4.5. Risk-Adjusted Performance

## 5. Portfolio Selection Examples with DEPO

#### 5.1. FOREX Binary Option Portfolio Example

#### 5.1.1. Data

#### 5.1.2. Efficient Frontier and Portfolio Selection

#### 5.1.3. Comparison to the Kelly Criterion over Time

#### 5.2. NFL Sportsbook Example

#### 5.2.1. Data

#### 5.2.2. Efficient Frontier and Portfolio Selection

#### 5.2.3. Comparison to the Kelly Criterion over Time

## 6. Conclusions

## 7. Materials and Methods

`R version 3.5.1`) used for the portfolio selection example demonstrated in this paper can be accessed from the following DropBox sharing links,

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AUD | Australian dollar |

BAL | Baltimore Ravens |

CAD | Canadian dollar |

CHF | Confoederatio Helvetica (Swiss) franc |

DEPO | Discrete entropic portfolio optimization |

EUR | Euro |

FOREX | Foreign Exchange |

FRO | Fixed-return option |

GBP | Great British pound (sterling) |

GROUND | Growth rate over uniform divergence |

IND | Indianapolis Colts |

JAC | Jacksonville Jaguars |

JPY | Japanese yen |

KC | Kansas City Chiefs |

KL | Kullback-Leibler |

LAC | Los Angeles Chargers |

LV | Las Vegas |

MIA | Miami Dolphins |

NADEX | North American Derivative Exchange |

NFL | National Football League |

REPO | Return-entropy portfolio optimization |

USD | US dollar |

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**Table 1.**Mean, in-the-money rate and estimated relative entropy (in bits) of FOREX binary options from January 2019 to January 2020.

Currency Pair | Short Name | Mean Outcome | % In-the-Money | Relative Entropy |
---|---|---|---|---|

Australian Dollar/Japanese Yen | AUD/JPY | −0.42591 | 0.287045 | 0.135127 |

Australian Dollar/US Dollar | AUD/USD | −0.868988 | 0.065506 | 0.651076 |

Euro/Pound Sterling | EUR/GBP | 0.0174 | 0.5087 | 0.000218 |

Euro/Japanese Yen | EUR/JPY | −0.243882 | 0.378059 | 0.04334 |

Euro/US Dollar | EUR/USD | −0.109232 | 0.445384 | 0.008624 |

Pound Sterling/Japanese Yen | GBP/JPY | −0.069511 | 0.465244 | 0.003488 |

Pound Sterling/US Dollar | GBP/USD | −0.500692 | 0.249654 | 0.18927 |

US Dollar/Canadian Dollar | USD/CAD | 0.468873 | 0.734437 | 0.164974 |

US Dollar/Swiss Franc | USD/CHF | −0.889943 | 0.055028 | 0.692615 |

US Dollar/Japanese Yen | USD/JPY | −0.370885 | 0.314557 | 0.101636 |

**Table 2.**Select FOREX binary options for 2 February 2020 11:00 p.m., with their respective contract strike price, market consensus edge, and estimated probability in-the-money.

Currency Pair | Contract Strike Price | Market Consensus Edge | P(In-the-Money) |
---|---|---|---|

Australian Dollar/Japanese Yen | AUD/JPY > 72.60 | 1.75% | 48.25% |

Australian Dollar/US Dollar | AUD/USD > 0.6700 | 2.25% | 52.25% |

Euro/Pound Sterling | EUR/GBP > 0.8420 | 0% | 50% |

Euro/Japanese Yen | EUR/JPY > 120.20 | 0% | 50% |

Euro/US Dollar | EUR/USD > 1.1080 | 1.5% | 51.5% |

Pound Sterling/Japanese Yen | GBP/JPY > 142.80 | 1% | 49% |

Pound Sterling/US Dollar | GBP/USD > 1.3180 | 1.5% | 51.5% |

US Dollar/Canadian Dollar | USD/CAD > 1.3240 | 2.625% | 47.375% |

US Dollar/Swiss Franc | USD/CHF > 0.9640 | 2.5% | 47.5% |

US Dollar/Japanese Yen | USD/JPY > 108.40 | 2.75% | 47.25% |

Option Contract | Buy or Sell | Probability of Success | Kelly Wager % |
---|---|---|---|

USD/JPY > 108.40 | Sell | 52.75% | 5.5% |

**Table 4.**Discrete entropic portfolio optimization (DEPO) portfolio of options for contracts expiring 2 February 2020 11:00 p.m.

Option Contract | Buy or Sell | Probability of Success | DEPO Wager % |
---|---|---|---|

USD/JPY > 108.40 | Sell | 52.75% | 0.7% |

USD/CAD > 1.3240 | Sell | 52.625% | 0.7% |

USD/CHF > 0.9640 | Sell | 52.5% | 0.7% |

AUD/JPY > 72.60 | Sell | 51.75% | 0.7% |

EUR/USD > 1.1080 | Buy | 51.5% | 0.7% |

GBP/USD > 1.3180 | Buy | 51.5% | 0.7% |

**Table 5.**Mean, cover rate and estimated relative entropy of NFL teams for 2011–12 to 2018–19 seasons.

Team Name | Short Name | Mean Outcome | Cover Rate | Relative Entropy (bits) |
---|---|---|---|---|

Arizona Cardinals | ARI | 0.056 | 0.528 | 0.002263 |

Atlanta Falcons | ATL | −0.079365 | 0.460317 | 0.004548 |

Baltimore Ravens | BAL | −0.081967 | 0.459016 | 0.004852 |

Buffalo Bills | BUF | −0.04 | 0.48 | 0.001154 |

Carolina Panthers | CAR | 0.080645 | 0.540323 | 0.004697 |

Chicago Bears | CHI | −0.031746 | 0.484127 | 0.000727 |

Cincinnati Bengals | CIN | 0.173554 | 0.586777 | 0.021838 |

Cleveland Browns | CLE | −0.121951 | 0.439024 | 0.010755 |

Dallas Cowboys | DAL | −0.02439 | 0.487805 | 0.00043 |

Denver Broncos | DEN | 0 | 0.5 | 0 |

Detroit Lions | DET | −0.064516 | 0.467742 | 0.003004 |

Green Bay Packers | GB | 0.072 | 0.536 | 0.003742 |

Houston Texans | HOU | −0.02439 | 0.487805 | 0.00043 |

Indianapolis Colts | IND | 0.088 | 0.544 | 0.005593 |

Jacksonville Jaguars | JAC | −0.114754 | 0.442623 | 0.00952 |

Kansas City Chiefs | KC | 0.095238 | 0.547619 | 0.006553 |

Los Angeles Chargers | LAC | −0.031746 | 0.484127 | 0.000727 |

Los Angeles Rams | LAR | −0.112903 | 0.443548 | 0.009215 |

Miami Dolphins | MIA | −0.04065 | 0.479675 | 0.001192 |

Minnesota Vikings | MIN | 0.193548 | 0.596774 | 0.027194 |

New England Patriots | NE | 0.2 | 0.6 | 0.02905 |

New Orleans Saints | NO | 0.129032 | 0.572581 | 0.015254 |

New York Giants | NYG | −0.00813 | 0.495935 | 0.000047 |

New York Jets | NYJ | −0.081967 | 0.459016 | 0.004852 |

Oakland Raiders | OAK | −0.064516 | 0.467742 | 0.003004 |

Philadelphia Eagles | PHI | −0.055118 | 0.472441 | 0.002192 |

Pittsburgh Steelers | PIT | 0.024 | 0.512 | 0.000415 |

Seattle Seahawks | SEA | 0.163934 | 0.581967 | 0.019473 |

San Francisco 49ers | SF | 0.008 | 0.504 | 0.000046 |

Tampa Bay Buccaneers | TB | −0.096774 | 0.451613 | 0.006767 |

Tennessee Titans | TEN | −0.196721 | 0.401639 | 0.028099 |

Washington Redskins | WAS | −0.015625 | 0.492188 | 0.000176 |

**Table 6.**Scheduled NFL games for Sunday 8 September 2019, with their respective Las Vegas point spreads and market consensus estimated probabilities of covering.

Away Consensus | Away Team | Home Team | Home Consensus | |
---|---|---|---|---|

70% | Kansas City Chiefs −3.5 | @ | Jacksonville Jaguars | 30% |

44% | Tennessee Titans | @ | Cleveland Browns −5.5 | 56% |

53% | Atlanta Falcons | @ | Minnesota Vikings −3.5 | 47% |

43% | Washington Redskins | @ | Philadelphia Eagles −10.5 | 57% |

67% | Baltimore Ravens −7.0 | @ | Miami Dolphins | 33% |

63% | Los Angeles Rams −1.5 | @ | Carolina Panthers | 37% |

46% | Buffalo Bills | @ | New York Jets −2.5 | 54% |

44% | Cincinnati Bengals | @ | Seattle Seahawks −9.5 | 56% |

34% | Indianapolis Colts | @ | Los Angeles Chargers −6.0 | 66% |

43% | San Francisco 49ers | @ | Tampa Bay Buccaneers −1.0 | 57% |

61% | Detroit Lions −2.5 | @ | Arizona Cardinals | 39% |

58% | New York Giants | @ | Dallas Cowboys −7.0 | 42% |

51% | Pittsburgh Steelers | @ | New England Patriots −5.5 | 49% |

**Table 7.**The Kelly criterion portfolio of wagers with percent allocation for NFL 2019-20 season week 1.

Bet to Cover | Bet to Not Cover | Probability of Success | Kelly Wager % |
---|---|---|---|

Kansas City Chiefs −3.5 | Jacksonville Jaguars | 60% | 20% |

Bet to Cover | Bet to Not Cover | Probability of Success | DEPO Wager % |
---|---|---|---|

Kansas City Chiefs −3.5 | Jacksonville Jaguars | 60% | 5.89% |

Baltimore Ravens −7.0 | Miami Dolphins | 58.5% | 5.89% |

Los Angeles Chargers −6.0 | Indianapolis Colts | 58% | 5.89% |

© 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/).

## Share and Cite

**MDPI and ACS Style**

Mercurio, P.J.; Wu, Y.; Xie, H. Portfolio Optimization for Binary Options Based on Relative Entropy. *Entropy* **2020**, *22*, 752.
https://doi.org/10.3390/e22070752

**AMA Style**

Mercurio PJ, Wu Y, Xie H. Portfolio Optimization for Binary Options Based on Relative Entropy. *Entropy*. 2020; 22(7):752.
https://doi.org/10.3390/e22070752

**Chicago/Turabian Style**

Mercurio, Peter Joseph, Yuehua Wu, and Hong Xie. 2020. "Portfolio Optimization for Binary Options Based on Relative Entropy" *Entropy* 22, no. 7: 752.
https://doi.org/10.3390/e22070752