Journal of Mathematics and Statistics Original Research Paper Application of Bi-Directional Grid Constrained Stochastic Processes to Algorithmic Trading Aldo Taranto and Shahjahan Khan School of Sciences, University of Southern Queensland, Australia Article history Abstract: Bi-directional Grid Constrained (BGC) Stochastic Processes Received: 23-11-2020 (BGCSP) become more constrained the further they drift away from the Revised: 01-03-2021 origin or time axis are examined here. As they drift further away from the Accepted: 05-03-2021 time axis, then the greater the likelihood of stopping, as if by two hidden reflective barriers. The theory of BGCSP is applied to a trading environment in Corresponding Author: Aldo Taranto which long and short trading is available. The stochastic differential equation of School of Sciences, University the Grid Trading Problem (GTP) is proposed, proved and its solution is of Southern Queensland, simulated to derive new findings that can lead to further research in this area Australia and the reduction of risk in portfolio management. Email: [email protected] Keyword: Grid Trading, Random Walks, Probability of Ruin, Stochastic Differential Equation, Bi-Directional Grids, Trending Grids, Mean Reversion Grids Introduction To visualize the impact of the BGC function Ψ(Xt,t) in (1.1), 1000 Itô diffusions were simulated both with Bi-directional Grid Constrained (BGC) Stochastic and without BGC, with zero drift coefficient µ and unit Processes (BGCSP) are described as Itô diffusions in diffusion coefficient σ, as shown in Fig. 1. which the further they drift from the origin or time axis, The zero drift in (a) is constrained in (b) the more it then the more they will be reflected back to the origin. deviates from the origin, causing the hidden reflective Definition 1.1. (SDE of BGC Stochastic Process) upper barrier and hidden reflective lower barrier to emerge, together with horizontal bands to form due to For a complete filtered probability space (Ω,,{}t0, the discretization effect of BGC. ℙ) and a BGC function Ψ(x): ℝ ℝ, x ∈ ℝ, then the BGCSP have applications by solving problems corresponding BGC Itô diffusion is defined as follows: involving the constraining of stochastic processes within two reflective barriers (often times hidden and not predefined) and that an event occurs when the barriers are dX fXtdtgXtdW t, t , t sgn Xt t , Xt t , , (1.1) hit. In the context of mathematical, quantitative (quant) and BGC computational finance and algorithmic trading, we define an application of BGCSP by the following definition. where, sgn[x] is defined in the usual sense as: Definition 1.2. [Bi-Directional] Grid Trading (BGT) BGT is the simultaneous placement of a Long and 1 ,x 0 Short trade at every grid width g level at, above and sgnxx 0 , 0, below the initial price rate R0 and the corresponding 1 ,x 0 taking profit of each trade at the nearest Take Profit level (also of width g) without any predetermined stop losses. This definition is illustrated in Fig. 2. and f(Xt,t), g(Xt,t) and Ψ(Xt,t) are convex functions. As the BGCSP evolves over time, it will collect (i.e., The drift function f(x): ℝ ℝ and the diffusion function close) many winning trades and also hold on to some losing g(x): ℝ ℝ, x ∈ ℝ, in the limit, they approach the typical trades, which should become profitable over time. It is constant expressions for the drift and diffusion coefficients: remarkable how such a simple trading strategy can generate profits most of the time due to frequent periods of low volatility (i.e., diffusion σ). However, when a strong trend limf x ,lim g x . (1.2) xx emerges, this strategy accumulates large losses and so the © 2021 Aldo Taranto and Shahjahan Khan. This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Aldo Taranto and Shahjahan Khan / Journal of Mathematics and Statistics 2021, Volume 17: 22.29 DOI: 10.3844/jmssp.2021.22.29 trades need to be closed down before they exceed the of magnitude, then the trading account can become ruined, account’s current capital or balance level. When the losing inducing a stopping time. We call this the Grid Trading trades grow too far in terms of either total count or in terms Problem (GTP). 80 60 20 40 10 20 0 X X 0 -20 -40 -10 -60 -20 -80 0 200 400 600 800 1000 0 200 400 600 800 1000 T T (a) (b) Fig. 1: Itô Diffusions with and without BGC R R T T 1 2 3 4 5 6 T = 6, W = 6, L = 21, P = -15 Fig. 2: Illustration of BGCSP in trading R = Rate, T = Time, W = Winning trades, L = Losing trades, P = Profit, E = Equity. (a) Horizontal blue lines represent when Long Trades occur and horizontal red lines represent when Short trades occur. The arrows represent the movement of the rate Rt over time t. (b) Dotted lines depict trades in profit and closed at their nearest Take Profit (TP). Solid lines depict trades that are held in loss until they reach their TP, closed down when loss becomes ‘too large’ or finally if an account is ruined 23 Aldo Taranto and Shahjahan Khan / Journal of Mathematics and Statistics 2021, Volume 17: 22.29 DOI: 10.3844/jmssp.2021.22.29 Literature Review n0, E00 E , n1, E E v v E , BGC stochastic processes are relatively new 1 0 0 n2, E E 2 v 3 v E v , (Taranto and Khan, 2020a-d). To the best of the authors’ 2 0 0 (3.1) knowledge, there is no additional formal academic nn 1 n t, E E vt v definition of BGC stochastic processes nor grid trading t 0 2 Wt available within all the references on the subject matter Lt (Mitchell, 2019; DuPloy, 2008; 2010; Harris, 1998; King, 2010; 2015; Markets, 2017; Work, 2018). These where, n is the grid level reached by the price Rt at secondary sources are not rigorous journal papers but time t. However, the markets do not trend indefinitely instead informal blog posts or software user manuals. and so Lt in (3.1) needs to be replaced with a Even if there were any academic worthey results found stochastic process. In a continuous time stochastic on grid trading, there is a general reluctance for traders framework t ∈ ℝ+, (3.1) becomes: to publish any trading innovation that will help other traders and potentially errode their own trading edge. dEt v Despite this, grid trading can be expressed vt dt n t n t 1, (3.2) Et 2 academically as a discrete form of the Dynamic Mean- Variance Hedging and Mean-Variance Potfolio where, E = E at t = 0 as an initial condition and adopting Optimization problem (Schweizer, 2010; Biagini et al., t 0 the simplest of 1-Dimensional Itô Diffusion processes: 2000; Thomson, 2005). There are many reasons why a firm would undertake a hedge, ranging from minimizing 1 the market risk of one of its client’s trades by trading in n t tt dt dW , (3.3) the opposite direction, through to minimizing the loss on g a wrong trade by correcting the new trade’s direction whilst keeping the old trade still open until a more noting that now we highlight that n is a function of t where: opportune time (Stulz, 2013). In the case of grid trading, it can be considered as a form of hedging of multiple µt is The drift (or direction) over time positions simultaneously over time, for the generation of σt is The diffusion (or volatility) over time, which trading profits whilst minimizing the total portfolio loss. are random and assumed independent of µt over time Wt is A Wiener Process (or Brownian motion) as dW dt with Methodology t t t (0,1) Derivation of Continuous Grid SDE We note that (3.2) is essentially a non-standard Geometric Brownian Motion (GBM). The reason why Theorem 3.1 we have not expressed it as a Arithmetic Brownian For a Bi-Directional grid trading constrained Itô Motion (ABM) is that we require the equity Itô diffusion process with a given grid width g, value ν per grid width, to be modelled as products of random factors and not drift (direction) µt and variance (risk) σt, then the change sums of random terms. GBM involves independently and in equity E over time t is: identically distributed ratios between successive factors. dW Furthermore, we require t 0, t as trading dE v v g E tt2.tg22 g dt dW t 22t t t Et 22 g g systems seek to exponentially compound E over time and an Et = 0 equates to ruin or bankruptcy. In fact, since Proof our drift and diffusion terms are non-constant over time, then our non-standard GBM is actually a form of a more In the discrete time framework t ∈ of Fig. 2, one generalised Itô Processes. Finally, we note that (3.2) does can see that the equity Et at any time t is comprised of the not appear at first glance to be a GBM as it does not exhibit initial equity E0, plus the sum of all the winning trades an explicit dWt term, even though it is implied due to (3.3). Wt, minus the sum of any losing trades Lt. We can Substituting (3.3) into (3.2) expands to: elaborate how the progression can evolve over time, in dE v 2 the worst case scenario of a strongly trending market, as t vt dt n t n t shown in Fig.