Portfolio Optimization for Binary Options Based on Relative 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>

Option Portfolio Selection with Generalized Entropic Portfolio Optimization

“…As previously presented in Mercurio et al (2020) [ 2 ], we use the extension of the Kelly criterion (Kelly, 1956) [ 21 ] to n wagers. For events with success probabilities , let w represent the total percentage of bankroll to be wagered.…”

“…As shown in our previous paper (2020) [ 2 ], relative entropy qualifies as a convex risk measure based on the relative entropy principle. We once again use this quantity as the discriminatory risk measure for option portfolio optimization.…”

“…Using the same combinatorial technique employed for REPO [ 1 ] and DEPO [ 2 ], the Shannon entropy of option portfolio returns can be estimated empirically via probability generating functions. For a collection of n discrete return assets over time period , let denote the cross-sectional n -dimensional vector of outcomes across one observational row of data, and let them be uniquely represented by the collection of ’s such that .…”

“…The return-entropy portfolio optimization (REPO) [ 1 ] allocates capital to an equity portfolio based on the expected return and Shannon entropy of portfolio returns. REPO was adapted to form the discrete entropic portfolio optimization (DEPO) [ 2 ] for application to portfolios comprised of assets with discrete distributed returns, like exotic instruments such as binary or digital options, or fixed-return options (FROs). In this paper we further extend DEPO to accommodate mixed returns with generalized entropic portfolio optimization (GEPO).…”

“…(2) Discrete entropic portfolio optimization (DEPO) [ 2 ]: for the selection of portfolios comprising strictly discrete return assets, such as binary options, digital options, fixed-return options (FRO), sports bets, or other betting wagers, and (3) Generalized entropic portfolio optimization (GEPO): for the selection of portfolios comprising assets with mixed returns (returns that can exhibit both continuous and discrete returns), like option strategies such as covered puts, married calls, credit spreads, straddles, long strangles, butterfly spreads, iron condors, and more. GEPO can handle portfolios that contain various types of options that are much more general than binary options.…”