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Gambler's Ruin Problem and Bi-Directional Grid Constrained Trading and Investment Strategies

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Gambler's Ruin Problem and Bi-Directional Grid Constrained Trading and Investment Strategies “Gambler’s ruin problem and bi-directional grid constrained trading and investment strategies” Aldo Taranto https://orcid.org/0000-0001-6763-4997 AUTHORS Shahjahan Khan https://orcid.org/0000-0002-0446-086X Aldo Taranto and Shahjahan Khan (2020). Gambler’s ruin problem and bi- ARTICLE INFO directional grid constrained trading and investment strategies. Investment Management and Financial Innovations, 17(3), 54-66. doi:10.21511/imfi.17(3).2020.05 DOI http://dx.doi.org/10.21511/imfi.17(3).2020.05 RELEASED ON Friday, 14 August 2020 RECEIVED ON Wednesday, 24 June 2020 ACCEPTED ON Monday, 27 July 2020 LICENSE This work is licensed under a Creative Commons Attribution 4.0 International License JOURNAL "Investment Management and Financial Innovations" ISSN PRINT 1810-4967 ISSN ONLINE 1812-9358 PUBLISHER LLC “Consulting Publishing Company “Business Perspectives” FOUNDER LLC “Consulting Publishing Company “Business Perspectives” NUMBER OF REFERENCES NUMBER OF FIGURES NUMBER OF TABLES 35 9 0 © The author(s) 2020. This publication is an open access article. businessperspectives.org Investment Management and Financial Innovations, Volume 17, Issue 3, 2020 Aldo Taranto (Australia), Shahjahan Khan (Australia) Gambler’s ruin problem BUSINESS PERSPECTIVES and bi-directional grid LLC “СPС “Business Perspectives” Hryhorii Skovoroda lane, 10, constrained trading Sumy, 40022, Ukraine www.businessperspectives.org and investment strategies Abstract Bi-Directional Grid Constrained (BGC) trading strategies have never been studied academically until now, are relatively new in the world of financial markets and have the ability to out-perform many other trading algorithms in the short term but will almost surely ruin an investment account in the long term. Whilst the Gambler’s Ruin Problem (GRP) is based on martingales and the established probability theory proves that the GRP is a doomed strategy, this research details how the semimartingale frame- work is required to solve the grid trading problem (GTP), i.e. a form of BGC financial markets strategies, and how it can deliver greater return on investment (ROI) for the same level of risk. A novel theorem of GTP is derived, proving that grid trading, whilst still subject to the risk of ruin, has the ability to generate significantly more profitable returns in the short term. This is also supported by extensive simulation and distribu- tional analysis. These results not only can be studied within mathematics and statistics in their own right, but also have applications into finance such as multivariate dynamic hedging, investment funds, trading, portfolio risk optimization and algorithmic loss recovery. In today’s uncertain and volatile times, investment returns are between 2%- 5% per annum, barely keeping up with inflation, putting people’s retirement at risk. BGC and GTP are thus a rich source of innovation potential for improved trading and Received on: 24th of June, 2020 investing. Accepted on: 27th of July, 2020 Published on: 14th of August, 2020 Keywords probability, ruin, gambler, semimartingales, martingales, stopping, bi-directional © Aldo Taranto, Shahjahan Khan, 2020 JEL Classification G11, G14, G17 INTRODUCTION Aldo Taranto, Researcher in Mathematics and Statistics, School of Sciences, Faculty of Health, There are numerous tales and claims from gamblers (or traders) with Engineering and Sciences, University of Southern Queensland, Toowoomba, an “infallible” strategy that will beat the house (or market) and guar- Queensland, Australia. (Corresponding antee riches. However, one rarely sees long-term independently au- author) dited evidence of these people making positive net profits from the Shahjahan Khan, Ph.D., Professor of Statistics, School of Sciences, Faculty strategy itself but instead from examples such as selling copies of their of Health, Engineering and Sciences, books or courses. One trading strategy that is entirely different to University of Southern Queensland, Toowoomba, Queensland, Australia. these and many other strategies is the so-called (Bi-Directional) grid trading strategy (or grid trading for short) that can start to generate large profits quickly, but can also lead to sudden ruin, a problem which is referred to as GTP. However, to dismiss grid trading as a doomed GRP strategy with no benefit would be a gross over-simplification of the resulting stochastic system. A novel theorem and resulting meth- odology are developed to estimate and quantify this risk of ruin for grid trading systems, which are related to hedging and portfolio op- This is an Open Access article, timization. This paper shows that GTP is more complex and yet more distributed under the terms of the profitable than GRP. Creative Commons Attribution 4.0 International license, which permits unrestricted re-use, distribution, and The GRP, defined formally in the Literature Review section, involves reproduction in any medium, provided the original work is properly cited. one of the most popular betting strategies, the so-called Martingale Conflict of interest statement: Strategy (not to be confused with Martingales of Probability Theory). Author(s) reported no conflict of interest In its simplest form, this involves the gambler winning $1 from the ca- 54 http://dx.doi.org/10.21511/imfi.17(3).2020.05 Investment Management and Financial Innovations, Volume 17, Issue 3, 2020 sino if a coin is facing up (U) and loses $1 if the coin is facing down (D). When the gambler is faced with one or more consecutive losing moves, they double their bet, 2n as shown in Figure 1. It is clear that such a strategy would eventually work if a fair coin was involved as sooner or later, one’s coin side would come up, depending on the size of one’s bankroll and the casino’s betting limits, other- wise ruin is inevitable. The GTP, on the other hand, in its simplest form, involves the simultaneous placement of long and short orders at equally spaced horizontal grid levels. Trades are closed at the next nearest grid level when they are in profit. This means that no matter where the market price moves to, it will trigger two new trades that form a hedge, close down a previous trade now in profit, and carry its losing trades(s) as shown in Figures 2 and 9. Given that financial markets are range bound (horizontally) most of the time, around 70%-80% of the time (Pukthuanthong-Le, Levich, & Thomas, 2007 and the references within), then grid trading will work most of the time. Periods of an initial trend that reverts back to the mean (i.e. exhibiting mean reversion) also plays to the strengths of grid trading, as it can also be considered as a form of ranging (albeit on a diagonal range). Grid trading is referred to as GTP because its weaknesses are revealed when markets have explosive strong trends, in which the losing trades grow faster than the winning trades and can lead to ruin if some of the trades are not closed down in time. Likewise, momentum strategies take advantage of how a trend tends to ‘run out of steam’ towards the end of its run. Whilst the start and end of a trend cannot be predicted consistently over time, GTP can benefit from momentum indicators as an early warning sign as to when a trend is about to end and therefore serves as a trigger to commence grid trading. 1. LITERATURE REVIEW the opposite direction, through to minimizing the loss on a wrong trade by correcting the new To the best of the authors’ knowledge, there are trade’s direction whilst keeping the old trade still only informal definitions of grid trading availa- open until a more opportune time. This research ble within all the references on the subject matter extends the traditional 2-trade hedge to any num- (DuPloy, 2008, 2010; Harris, 1998; King, 2010, 2015; ber of trades. Admiral Markets, 2017; ForexStrategiesWork.com, 2018). These are not rigorous academic journal pa- Another academic framework for grid trading pers but instead informal blog posts or software is the consideration of the series of open losing user manuals. Even if there were any academic trades in a grid system as a portfolio of stocks. worthy results found on grid trading, there is a The Merton problem – a question about optimal general reluctance for traders to publish any trad- portfolio selection and consumption in continu- ing innovation that will help other traders and po- ous time – is indeed ubiquitous throughout the tentially erode their own trading edge. mathematical finance, economics and economet- ric literature. Since Merton’s seminal paper in Despite this, grid trading can be expressed aca- 1971 (Merton, 1971), many variants of the origi- demically as a discrete form of the Dynamic Mean- nal problem have been put forward and extensive- Variance Hedging and Mean-Variance Portfolio ly studied to address various issues arising from Optimization problem (Schweizer, 2010; Biagini, economics. For example, Fleming and Hernández Guasoni, & Pratelli, 2000; Thomson, 2005). Hernández (2003) considered the case of optimal investment in the presence of stochastic volatili- There are many reasons why a firm would under- ty. This paper examines the concept that certain take a hedge, ranging from minimizing the mar- losing trades should be closed at certain points ket risk of one of its client’s trades by trading in in time to optionally grow the portfolio (Davis, http://dx.doi.org/10.21511/imfi.17(3).2020.05 55 Investment Management and Financial Innovations, Volume 17, Issue 3, 2020 Norman, 1990; Dumas & Luciano, 1991), and the ruin probability and duration of the game, more recently Guasoni and Muhle-Karbe (2012) whereas most papers studied only duration of and Muhle-Karbe and Liu (2012) addressed op- the game. Some generalizations to a higher di- timal portfolio selection under transaction costs. mension were studied in Rocha and Stern (2004), Rogers and Stapleton (2002) considered opti- Kmet and Petkovssek (2002).
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