{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:01:22Z","timestamp":1760238082382,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T00:00:00Z","timestamp":1594252800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["000000000000"],"award-info":[{"award-number":["000000000000"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The portfolio optimization problem generally refers to creating an investment portfolio or asset allocation that achieves an optimal balance of expected risk and return. These portfolio returns are traditionally assumed to be continuous random variables. In An Entropy-Based Approach to Portfolio Optimization, we introduced a novel non-parametric optimization method based on Shannon entropy, called return-entropy portfolio optimization (REPO), which offers a simple and fast optimization algorithm for assets with continuous returns. Here, in this paper, we would like to extend the REPO approach to the optimization problem for assets with discrete distributed returns, such as those from a Bernoulli distribution like binary options. Under a discrete probability distribution, portfolios of binary options can be viewed as repeated short-term investments with an optimal buy\/sell strategy or general betting strategy. Upon the outcome of each contract, the portfolio incurs a profit (success) or loss (failure). This is similar to a series of gambling wagers. Portfolio selection under this setting can be formulated as a new optimization problem called discrete entropic portfolio optimization (DEPO). DEPO creates optimal portfolios for discrete return assets based on expected growth rate and relative entropy. We show how a portfolio of binary options provides an ideal general setting for this kind of portfolio selection. As an example we apply DEPO to a portfolio of short-term foreign exchange currency pair binary options from the NADEX exchange platform and show how it outperforms leading Kelly criterion strategies. We also provide an additional example of a gambling application using a portfolio of sports bets over the course of an NFL season and present the advantages of DEPO over competing Kelly criterion strategies.<\/jats:p>","DOI":"10.3390\/e22070752","type":"journal-article","created":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T10:45:19Z","timestamp":1594291519000},"page":"752","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Portfolio Optimization for Binary Options Based on Relative Entropy"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0652-541X","authenticated-orcid":false,"given":"Peter Joseph","family":"Mercurio","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8858-1289","authenticated-orcid":false,"given":"Yuehua","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong","family":"Xie","sequence":"additional","affiliation":[{"name":"Manulife Financial Corp., Toronto, ON M4W 1E5, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mercurio, P., Wu, Y., and Xie, H. (2020). An Entropy-Based Approach to Portfolio Optimization. Entropy, 22.","DOI":"10.3390\/e22030332"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication: Part 1","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1002\/j.1538-7305.1948.tb00917.x","article-title":"A Mathematical Theory of Communication: Part 2","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_4","first-page":"77","article-title":"Portfolio Selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. Financ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1002\/j.1538-7305.1956.tb03809.x","article-title":"A New Interpretation of Information Rate","volume":"35","author":"Kelly","year":"1956","journal-title":"Bell Syst. Tech. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On Information and Sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_7","unstructured":"Kullback, S. (1959). Information Theory and Statistics, John Wiley and Sons."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Benello, A., van Biema, M., and Carlisle, T. (2016). Concentrated Investing: Strategies of the World\u2019s Greatest Concentrated Value Investors, John Wiley and Sons.","DOI":"10.1002\/9781119020516"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1080\/00029890.1992.11995955","article-title":"The Kelly Criterion and the Stock Market","volume":"99","author":"Rotundo","year":"1992","journal-title":"Am. Math. Mon."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.2307\/1428168","article-title":"Portfolio Choice and the Bayesian Kelly Criterion","volume":"28","author":"Browne","year":"1996","journal-title":"Adv. Appl. Probab."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"MacLean, L. (2010). The Kelly Capital Growth Investment Criterion: Theory and Practice, World Scientific.","DOI":"10.1142\/9789814293501"},{"key":"ref_12","unstructured":"Das, S. (2020, January 01). Data Science: Theories, Models, Algorithms, and Analytics. Available online: https:\/\/srdas.github.io\/Papers\/DSA_Book.pdf."},{"key":"ref_13","unstructured":"Lavinio, S. (2000). The Hedge Fund Handbook, McGraw-Hill, Irwin Library of Investment & Finance."},{"key":"ref_14","unstructured":"O\u2019Shaughnessy, D. (2012, January 11\u201313). Optimal Exchange Betting Strategy for Win-Draw-Loss Markets. Proceedings of the MATHSPORT 2012 11th Australasian Conference on Mathematics and Computers in Sport, Melbourne, Australia."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1287\/deca.2013.0271","article-title":"Optimal Betting Under Parameter Uncertainty: Improving the Kelly Criterion","volume":"10","author":"Baker","year":"2013","journal-title":"Decis. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Maclean, L., and Ziemba, W. (2013). The Handbook of the Fundamentals of Financial Decision Making, World Scientific.","DOI":"10.1142\/8557"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"65","DOI":"10.21314\/JOIS.2014.045","article-title":"Confidence Intervals for the Kelly Criterion","volume":"3","author":"Sinclair","year":"2014","journal-title":"J. Invest. Strateg."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1111\/itor.12174","article-title":"Multistage Portfolio Optimization with Stocks and Options","volume":"23","author":"Aminnayeri","year":"2016","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1017\/S0022109016000831","article-title":"Optimal Option Portfolio Strategies: Deepening the Puzzle of Index Option Mispricing","volume":"52","author":"Faias","year":"2017","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_20","first-page":"22","article-title":"Modified Kelly Criterion","volume":"14","author":"Chu","year":"2018","journal-title":"J. Quant. Anal. Sport."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1016\/j.ijforecast.2019.01.001","article-title":"Exploiting Sports-Betting Market Using Machine Learning","volume":"35","author":"Hubacek","year":"2019","journal-title":"Int. J. Forecast."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Vecer, J. (2020, July 03). Optimal Distributional Trading Gain: State Price Density Equilibrium and Bayesian Statistics. Working Paper. Available online: https:\/\/ssrn.com\/abstract=3616661.","DOI":"10.2139\/ssrn.3616661"},{"key":"ref_23","unstructured":"Khanna, N. (2020, March 01). The Kelly Criterion and the Stock Market. Available online: https:\/\/sites.math.washington.edu\/~morrow\/336_16\/2016papers\/nikhil.pdf."},{"key":"ref_24","unstructured":"MacKay, D. (2003). Information Theory, Inference and Learning Algorithms, Cambridge University Press."},{"key":"ref_25","unstructured":"Hardy, M. (2020, March 01). An Introduction to Risk Measures for Actuarial Applications. Available online: https:\/\/www.researchgate.net\/profile\/Mary_Hardy2\/publication\/242469445_An_Introduction_to_Risk_Measures_for_Actuarial_Applications\/links\/0a85e5342bd491d188000000\/An-Introduction-to-Risk-Measures-for-Actuarial-Applications.pdf."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Follmer, H., and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time, De Gruyter.","DOI":"10.1515\/9783110218053"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1007\/s10957-011-9968-2","article-title":"Entropic Value-at-Risk: A New Coherent Risk Measure","volume":"155","year":"2012","journal-title":"J. Optim. Theory Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/1467-9965.00068","article-title":"Coherent Measures of Risk","volume":"9","author":"Artzner","year":"1999","journal-title":"Math. Financ."},{"key":"ref_29","unstructured":"Kullback, S. (1996). Information Theory and Statistics, Dover Publications."},{"key":"ref_30","unstructured":"Cover, T., and Thomas, J. (1991). Elements of Information Theory, John Wiley and Sons."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1006\/obhd.1994.1075","article-title":"The Two Fallacies of Gamblers: Type I and Type II","volume":"60","author":"Keren","year":"1994","journal-title":"Organ. Behav. Hum. Decis. Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.jeconbus.2016.01.003","article-title":"Enhancing Mean\u2013Variance Portfolio Selection by Modeling Distributional Asymmetries","volume":"85","author":"Low","year":"2016","journal-title":"J. Econ. Bus."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1086\/294846","article-title":"Mutual Fund Performance","volume":"39","author":"Sharpe","year":"1966","journal-title":"J. Bus."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"54","DOI":"10.2469\/faj.v64.n3.6","article-title":"Does the Measure Matter in the Mutual Fund Industry?","volume":"64","author":"Eling","year":"2008","journal-title":"Financ. Anal. J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.1080\/14697688.2016.1164337","article-title":"The Profitability of Pairs Trading Strategies: Distance, Cointegration and Copula Methods","volume":"16","author":"Rad","year":"2016","journal-title":"Quant. Financ."},{"key":"ref_36","unstructured":"NADEX (2020, March 01). The Components of Binary Option Pricing. Available online: https:\/\/www.nadex.com\/learning-center\/courses\/binary-options\/components-binary-option-pricing."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/752\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:49:12Z","timestamp":1760176152000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/752"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,9]]},"references-count":36,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["e22070752"],"URL":"https:\/\/doi.org\/10.3390\/e22070752","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,7,9]]}}}