On Stochastic Multi Criteria Decision Analytics and Artificial Intelligence for Efficient Stock Trading By Gordon H. Dash, Jr. 1, Nina Kajiji 2, John Forman 3 1 College of Business, University of Rhode Island 2 Center for School Improvement and Social Policy, University of Rhode Island 3 Thomson-Reuters, Boston, MA Preliminary X111 International Conference Applied Stochastic Models and Data Analysis June 30 – July 3, 2009

Justification Increasing complexities of global markets New mathematical modeling of stock price behavior gaining popularity Traditional Brownian Motion Model assume stock price follow a random walk Traditional Brownian Motion Model assume stock price follow a random walk Geometric Brownian Motions assumes stock returns follow a random walk Geometric Brownian Motions assumes stock returns follow a random walk Stochastic methods are gaining popularity since they rely upon random and pseudorandom methods to define an asset’s price

Objective To join stochastic multi-criteria decision analytics with neural network based modeling to assign expected stocks to classification groups based on their trading profitability. To examine the time-series efficiency of the DK4-AT via a double log (restricted Cobb-Douglas (CD)) production model

A Trading System Factors that define a trading system are: An identification of the markets to trade An identification of the markets to trade Position quantities to buy/sell Position quantities to buy/sell Entry and exit decision that indicate when to buy/sell Entry and exit decision that indicate when to buy/sell When to exit a winning (losing) position When to exit a winning (losing) position DK4-AT incorporates any number of advanced trading rules that conform to these factor decisions

The Stock Trading Model Shreve (2004) provides the framework for use of the stochastic integral to characterize uncertain stock trading. Specifically: Define the random variable X t of a stock’s market price, at time t. The probability space (Ω, Ѵ,Р), a measure space with P(Ω) = 1, as well as filtration. Define the random variable X t of a stock’s market price, at time t. The probability space (Ω, Ѵ,Р), a measure space with P(Ω) = 1, as well as filtration.

The Model (cont) That is, Г i is loosely viewed as the set of events whose outcomes are certain to be revealed to investors as true or false by, or at, time t. For any event, A, the probability assigned to A by investors is P(A). The price process X is said to be adapted if for all t, X t is V t measurable

The Trading Strategy We assumes a market that is not characterized by the no-risk unlimited profit arbitrage effects of trading on advanced knowledge. We define a trading strategy θ that determines the quantity θ t (ω) of each security held in each state ω Є Ω and at each time t.

The Relation Hence, given a price process X and a trading strategy θ that satisfies the no arbitrage conditions, the total financial gain between any times s and t ≥ s is defined as a stochastic integral

Buy-Hold Strategy A short-horizon element of the DK4-AT trading strategy captured by θ where an investor initiates a position immediately after some stopping time T and closes it at some later stopping time U. Thus for a position size that is V t measurable, the trading strategy θ is defined by θ = 1 (T< t ≤ U) and the gain is:.

The n-dimensional Trading Strategy Therefore, for n different securities, with price process X 1,…, X n the investor can choose an associated n-dimensional trading strategy θ = {θ 1,…, θ n } or some allowable set Ѳ, for which the total gain- from-trade process is:

Why ANN? Prediction capabilities of ANNs for high frequency stock market (Refenes, 1996) Neural networks do not require a parametric system model They are relatively insensitive to chaotic data patterns

The RBF ANN Topology

AT Algorithm

Production System for a Profitable Stock Pick a starting date – Case Study List Creation Date: 24-Jan-2009 Establish historical period: 01-Jan-2008 through 1-Jan-2009, inclusive. Create research sample (SAM): Number of trades ≥ 25 throughout the historical period. Number of trades ≥ 25 throughout the historical period. Identify tickers where 50% or more of the trades generated a dollar profit. Identify tickers where 50% or more of the trades generated a dollar profit. Identify the research sample → 915 securities. Identify the research sample → 915 securities. For SAM, obtain stock fundamentals (source: Yahoo) EPS – estimate current year EPS – estimate current year Market Capitalization Market Capitalization 52Wk Range – real time 52Wk Range – real time Percent change from 50 day Moving Average Percent change from 50 day Moving Average Average Daily Volume Average Daily Volume EPS estimate next year EPS estimate next year EPS estimate next quarter EPS estimate next quarter Day’s Range Day’s Range

Production System for a Profitable Stock Execute K-SOM Target variable: Number of Positive Trades for the ith security Target variable: Number of Positive Trades for the ith security Predictor variables: fundamentals Predictor variables: fundamentals 1x1 classification structure – primarily to obtain distance measure 1x1 classification structure – primarily to obtain distance measure Create weighted probability of profitable trade – that is, % profitable x distance Create weighted probability of profitable trade – that is, % profitable x distance Use K4 to estimate the CD production of the weighted probability of positive trades Use K4 with softmax transfer function Use K4 with softmax transfer function Identify production elasticity for each fundamental variable Identify production elasticity for each fundamental variable Interpret the returns to scale for profitable trading Interpret the returns to scale for profitable trading

Results Number of Positive Trades by Security

Results KSOM Centroid Distance – First 819 Securities

Results K4 Analysis Using Softmax Transfer Function Dependent Variable: Weighted % Positive Trades Independent Variables: Ln(Fundamental Variable)

Results Plot of Actual and Predicted of Weighted % Positive Trades using K4

Results Zoom in View – Actual and Predicted

Results Weights from Comparative K4 Models Dependent Variable: Weighted % Positive Trades Model Chosen – Norm2 An increase in the 52 Wk Range or the Day’s Range increases the Weighted % Positive Trades. That is, higher the price differential higher the profit potential Mkt. Cap also exhibits a positive relationship. That is, higher the mkt. cap the higher the stock’s propensity to trade. The other five variables all have a negative relationship to Weighted % Positive Trades.

Pseudo Elasticity Estimates PTCP: % Positive Trades weighted by K-SOM Centroid Proximity

Conclusions The production system exhibits decreasing returns to scale (0.338); hence, a simultaneous 1% change in all fundamentals will result in a.34% increase in the % of weighted profitable trades (volatility is good). The DK4-AT proved to be an efficient “engine” for predicting high- frequency stock trades. A K-SOM 20-Minute Cluster produce Centroid proximity scores the weighted the % profitable trade in a meaningful manner for prediction estimation. A double-log (restricted CD) production function estimated by the K4 RBF with Norm:2 data transformation on fundamental variables produced meaningful production elasticity estimates