Stochastic Multi Criteria Decision Analytics and Artificial Intelligence in Continuous Automated

Trading for Wealth Maximization

By

Gordon H. Dash, Jr.1, Nina Kajiji2, John Forman3

1College of Business, University of Rhode Island2The NKD Group, Inc. and Computer Science and Statistics,

University of Rhode Island.3Thomson Reuters, Boston, MA

www.GHDash.net www.ninakajiji.netghdash@usa.net nina@nkd-group.com

XX1V Mini Euro ConferenceContinuous Optimization & Information-Based Technologies

in the Financial SectorIzmir, Turkey: June 23 – June 26, 2010

OutlineJustificationObjectiveContinuous-Time Stochastic Processes

Ito’s Stochastic Integral Buy-Hold Trading Strategy N-Dimensional Trading Strategy K4-RBF

Empirical Example – WINKSFirm FundamentalsConclusions

Justification

Complexities of trading across global markets

High frequency, or mathematical modeling, of stock price behavior to trade efficiently continues to gain in popularity

Stochastic methods are relevant as they rely upon random and pseudo-random methods to define an asset’s price

Objective

To assign stocks to wealth-building groups based on the historical trading performance of WINKS.To define the operational characteristics of the HF Trading system -- WINKS.To estimate the relative elasticity of firm-fundamental metrics to explain the production of WINKS trading profitability.

Automated Trading System (WINKS)

Mathematical Finance

Business Decision Theory

MCDA

Model for Equity Trading ProfitsLet Xt be a stock’s market price at time tWe assume that the price process X follows a geometric Brownian motion with a constant drift and volatility.Let θ represent a trading strategy.Let θt (ω) be the quantity of each security held in each state ω є Ω and at each time t. We assume that the trading strategy can only make use of available information at any time t. Thus preventing the possibility of unlimited gains trough high-frequency trading or flash-crash trading – i.e. θ is adapted.We thus can define the total financial gain as:

• between any time s and r – (Ito’s stochastic integral).

Buy and Hold Strategy (BH)

Position is initiated after some stopping time T and closed at a later stopping time U.

For a position size θt (ω), the trading strategy θ, is defined by θt = 1(T<t<U) θt (ω).

The gain from BH is:

The N-Dimensional Trading Strategy

• Suppose we have n different securities with price process: X1, … Xn and a trading strategy of θ = (θ1, … θn ) then the total gain is:

K4-RBFVarious ANN algorithmic topologies have been used in market prediction and forecast behavior.

The use of ANN to exploit complex financial optimization and mapping studies continues due to:

The RBF ANN can approximate any continuous function uniformly on compact sets by increasing the nodes in the hidden layer.

It is known for speed of modeling large datasets

We employ the K4-RBF ANN. The K4-RBF optimizes convergence by extending the traditional RBF to include a Bayesian closed form solution to the computation of the weight decay parameter(s).

K4-RBF Extensions

The Kajiji (2001) extension to the traditional RBF ANN specification introduced multiple objectives within a Bayesian RBF ANN framework. By adding a weight penalty term to the SSE optimization objective, the modified SSE is restated as the following cost function:

Under this specification the function to be minimized is stated as:

Trading Benefit the K4 ANN

Excellent mapping capabilities of the RBF topology allows for better forecasts for a system exhibiting Brownian motion – such as stock price movements.The algorithmic speed of the generalized RBF is greatly enhanced by the K4 enhancements making it suitable for HF valuation in N dimensional space.

WINKS Algorithm

A Sample of Trading ProfitabilityThe BH Strategy v/s HF Automated Trading:

June 01, 2009 to March 19, 2010

The database of 2,225 securities was established by random selection from the 8,000 tickers followed by Yahoo! Finance. These five were chosen for demonstration as they have the highest trading profit.Trading Efficiency = ((Profit or Loss) / Ini. Investment) *100. Note: All stocks have an ini. investment of $1000

Current status can be viewed at: http://www.nkd-group.com/index/index.htm

Do Firm Fundamental Factors Affect % Positive Trades?

Goal: To use the K4 mapping efficiency to estimate non-parametric quasi-elasticity metrics of individual firm fundamental variables in the production of K4 trading profitability.

Which firm fundamental variables to use? Fama & French (1993) found a three factor model efficiently

modeled the excess returns of individual firms. Subsequently, Carhart (1997) extended the FF model by

including a fourth factor. We approximate the Carhart four factor model:

Vasicek’s Beta (Bayesian corrected Beta) Standard CAPMMarket Capitalization FFM added factorBook to Last Trade Price FFM added factorPercent change from 50 day Moving Average Carhart extension

K4- Estimation of Production Scale Economies

Establish historical period: 01-Jun-2009 through 19-Mar-2010, inclusive.Create research sample (SAM):

Obtain 20-minute price observations for 2,225 securities. Eliminate all non-equity stocks, and stocks that do not have fundamental information

on Yahoo! -- sample size reduced to 1,765. For efficient cross-sectional modeling we sample from within the full content

population. The data sampling is guided by the use of the target variable of the study – percent positive trades (PPT).

We have 793 securities for the training set; and 972 securities for the validation set.

For SAM, obtain stock fundamentals (source: Yahoo Finance) Vasicek’s Beta - created from reported Yahoo beta. (P1) Market Capitalization (P2) Book to Last Trade Price (P3) Percent change from 50 day Moving Average (P4)

ResultsNumber of Positive Trades by Security for SAM of 1,765 Securities

Notice that the %positive trades form a band between 30% and 70%

The Production ModelUse K4 to estimate the double-log production theoretic model for positive trades.

• pi = f(P1, P2, P3, P4) Where:

pi = Ln(PPT)P1, P2, P3, P4 = Ln transformation of indicator variables as previously definedK4-RBF implemented with a softmax transfer function

K4 RBF weights are interpreted as quasi-production elasticity estimatesSum to capture system returns to scale for profitable trading

A Sample of Alternative Modeling ResultsK4-RBF Analysis Using Softmax Transfer Function

Dependent Variable: Ln(% Positive Trades)Indicator Variables: Ln(Fundamental Variables)

ResultsActual and Predicted

ResultsComparative K4 RBF Weights from the Alternative Models

Dependent Variable: Ln(% Positive Trades)

Model selected – Norm2Except for the variable Market Capitalization, all other variables show an inverse relationship to PPT. That is, Market Capitalization is the only variable that can improve PPT given any positive change in market size.Interestingly any increase in Book Value to Price tends to lessen the probability that WINKS will trade the stock profitably.The K4 production model exhibits decreasing returns to scale of 0.484 units. This implies a decrease in PPT given a simultaneous and proportionate change in all indicator variables (factors of production).

Conclusions

This research provides a synthesis of stochastic equity price behavior and the cognitive science of trading by implementation of the dual objective K4-RBF ANN as incorporated in WINKS.

WINKS is an MCDA trading algorithm that integrates the AI properties of two unique, but coordinated, high-frequency RBF ANNs.

Test results produced transaction cost adjusted trading profits that exceeded those generated by the simple buy-hold strategy.

WINKS performance was modeled using a four-factor firm pricing model to estimate the system-wide returns to scale of PPT.

The results suggest that the portfolio selection of stocks based on the estimated quasi-elasticity coefficients would greatly enhance trading profits. A test of this conclusion is left for future research.