Alternative trading strategies for software agents in an artificial stock market Sönke Wendt Erasmus Computing Session (e.g.

Transcription

1 Alternative trading strategies for software agents in an artificial stock market Sönke Wendt Erasmus Computing Session (e.g., 2005/2006) The candidate confirms that the work submitted is their own and the appropriate credit has been given where reference has been made to the work of others. I understand that failure to attribute material which is obtained from another source may be considered as plagiarism. (Signature of student)

2 Summary This report shows the requirements for self-trading software agents in an artificial stock market. As an example, it was tried to implement a numerically based self-trading software agent based on the downhill simplex into an artificial stock market. Therefore the author chose the often-used Santa Fe market. The failed implementation shows the problems of this approach. In the second part of this report the behaviour of fix-strategy agents in artificial stock markets are analysed. A direct relation between the strategy of the agents and their profit is shown. Key Words: downhill simplex, artificial stock market, numerical based optimisation methods

3 Table of Contents 1 Introduction Artificial Stock market Aim of the Projects Minimum Requirements Objectives of the Project Problems during the implementation phase Software tools used Project Management Project procedures and management Project schedule Project intensity Which problems occurred and how they were solved Software technique problems Software evaluation The artificial stock market model Agents Different types of agents What is the best strategy? Risk aversion Assets Trading procedure Find a market for the project The implementation of the Santa Fe market and the choice The optimisation algorithm Requirements for the algorithm Analysing the genetic algorithm The optimisation problem and communication with the market maker Choose of a optimisation algorithm Numerical based algorithms Downhill Simplex How does the simplex works... 20

4 5.3.2 Main idea of the downhill simplex and its power as a forecasting algorithm Adapting the problem to the downhill simplex Initialisation phase Swing in time Step reduction Step enlargement How to define the forecasting quality of this point Adapt the shape of the triangle to the solution Tuning ideas Problem with the Simplex The price adjustment function Prohibit the simplex to run out of range What is left of the main idea of the downhill simplex? General concerns of usage of numerical optimisation strategies Result and resume of the implementation Evaluation of single strategy agents Aim of the analysis Fixed strategy agents Configuration of the market Evaluation Model Evaluation and trials Behaviour of the GA-Agents Fixed strategy agents without variance and fixed risk aversion Optimistic set of agents Fixed strategy agents with the use of the variance Different risk aversions Special selection of agents Results Conclusion... 47

5 1 Introduction 1.1 Artificial Stock market Artificial stock markets are simulation models which allow to model and analyse stock markets and the behaviour of the market participants. The market participants are designed as software agents. The price of the assets on the market is generated by the demand of its participants. One goal for the designer is to create a market behaviour which is similar to normal stock markets. Also the trading behaviour of the agents should follow any reasonable strategy. 1.2 Aim of the Projects Self-trading non-ga-based software agents in an artificial stock market. 1.3 Minimum Requirements Understand the actual implementation of the stock market and the uses of the genetic algorithm. Develop and implement a new algorithm Define test scenes and evaluation rules for comparing my algorithm with existing ones. Based on the results of the test scenes, optimize and adjust my optimization algorithm. 1.4 Objectives of the Project I will develop and implement a new trading agent into the artificial stock market beside the genetic algorithm one. The only way to test and analyse the agent is to implement this agent into the stock market. Then I am able to run trials on this market and can see how well the agent works. 1 of 47

6 1.5 Problems during the implementation phase This project report focus on two main topics. The first topic in chapter five talks about the implementation of a numerically based optimisation strategy into an artificial stock market. The second focus of this work is on the evaluation of the trading behaviour of fixed-strategy agents. At the end of the research work into the downhill simplex I realized that the chosen optimisation strategy cannot be used to work for this aim in this environment. Accidentally, I ran some trials on this stock market with a fixed-forecasting strategy and stock price expectation for each agent. In the second part I focus on the results of these fixed-forecasting strategy agents. 1.6 Software tools used The simulation program of the artificial stock market is written in Java I decided to use Eclipse as the development environment. To visualise the work of the simplex GNUPlot [GNU-Plot team] was used. The visualisation of the data was done with MS Excel Project Management In the mid term report I decided to describe my work procedure with the expedition procedure model [see: for instance Mack,J]. This model describes a project as a expedition. During such a expedition a lot of problems can occur and setbacks need to be coped with. In this chapter I describe my expedition of a stock market project, show my project schedule and talk about the technical and organisational problems during the project. At the end, I will focus on my personal experience and the impact which the problem that occurred had on the project. 2.1 Project procedures and management At the beginning of my expeditions I marked my project route with several milestones to accomplish the minimum requirements. The main milestones are: 1. Background reading about artificial stock markets. 2. Select a source code of an artificial stock market. 3. Understand the actual implementation of the stock market and the source code. 4. Understand the uses of the genetic algorithm. 2 of 47

7 5. Find an idea for a new algorithm. 6. Implement this strategy into the market. 7. Define test scenes and evaluation rules for comparing my algorithm with existing ones. 8. Based on the results of the test scenes, optimize and adjust my optimization algorithm. These milestones should give orientation and motivation during my work. For the exact planning of reaching each milestone, I set several tasks with corresponding goals for each week. This myopic, exact planning as part of the agile methods is part of the expedition view. But the final goal of the project and the milestones should be never forgotten. Especially at the beginning of the project I wanted to make too much progress. So I thought that I had reached a milestone but later realised that I had not understood it as correctly and well as I needed to discover the next milestone correctly. So I often was at the point of stepping back and covering the milestones correctly. This fact was counter-productive for my motivation and confidence during the process. As I reached to milestone 5, I realised that the preparation work had not been enough to ensure a smooth implementation of the downhill simplex. So I realised in the middle of the project that I had quality problems with my work. The project management for agile methods recommends, among other measures, tests, cooperative work, prototyping and documentation to ensure the quality of the work. These methods are designed to ensure the quality of software products and especially its production process. I decided to focus more on the quality of the work and added extra times for analysis and correction of my work to my schedule. Instead of cooperative work I decided to roughly write down the results of the working sessions and start the next session with a short analysis of the documents to detect failures of previous working sessions. Writing and reading acts as a double check of the work and prevents silly mistakes. However, for the results of the first week I realised that it is difficult to find all failures of the previous work to improve the quality without redoing everything, so that at the end of the project I had to struggle with these starting mistakes. To reach the last milestones, I tried a lot of adjustments and tuning ideas to design good forecasting values for this trading agent. But at the end I came to the conclusion that all necessary adjustments to make this optimisation strategy stable would almost completely replace the main strategy of this forecasting algorithm. 3 of 47

8 During the project time I often evaluated the behaviour of the simplex. Accidentally, I ran plenty of trials which fixed strategy agents. I did not realise this and started to evaluate the behaviour of agents with this strategy. As I started to analyse the behaviour of the agents and noticed the statical behaviour I realised the mistake. But the result of the market which are generated by fixed strategy agents look quite interesting. Most ideas I developed to evaluate market effects of the downhill simplex can be used for fixed strategy agents as well, so that during my expedition I found new interesting areas in the stock market and decided to add a new topic to this project. Due to the effects of implementing the downhill simplex I learned a lot about the optimisation problem and the general problems of implementing numerical optimisation problems on artificial stock markets. The results of the market with fixed-strategy agents compared with the GA ones are interesting as well. So I think that my expedition did not discover the planned map, but the new ways I accidentally discovered were also interesting. 2.2 Project schedule Shows only the schedule The project started on the 11th of October with the first meeting with my supervisors. We decided to meet every Tuesday morning. A chronological overview of my working schedule for this project: 11 October to 18 October brief orientating readings understanding the research field start to think about the aim of the project. 18 October to 25 October decided to work on an artificial stock market find abstract ideas about the aim of the report. present the ideas on 25 October start reading about artificial stock markets. 25 October to 1 November 4 of 47

9 looking for an appropriate source code reading define exact project aim and present it on 1 November 1 November to 8 November reading about genetic algorithms analysing the different source codes decide to use the source code of Norman Ehrentreich start to understand the source code 8 November to 15 November continue analysing the source code start reading about on-line problems and optimizations find the idea of the downhill simplex 15 November to 22 November implement the first version of the downhill simplex run first trials 22 November to 29 November analyse the problems of the simplex find ideas to reduce the step length implement some new ideas think about next steps of the project 29 November to 6 December write mid term report 6 December to 14 December find first problems with the downhill simplex implement export file for GNU Plot 14 December to 21 December 5 of 47

10 implement plugs for data evaluation 22 December to 3 January semester break 4 January to 20 January preparing for the exam and exam period planing resizing of the shape 21 January to 31 January implement resizing of the shape 31 January to 7 February analyse and correct resizing of the shape 7 February to 14 February found further bugs and corrected them 14 February to 21 February focus mainly on essay and coursework 21 February to 28 February think about statistical evaluation and implement data implement random number observer 28 February to 6 March evaluate results 6 March to 14 March realise the bug which creates good values tried to find more adjustments strategies for the downhill simplex realised that implementation will not work 6 of 47

11 14 March to 21 March 16 March: Progress Meeting 21 March to End write project report evaluate work of fixed strategy agents 2.3 Project intensity Due to the previously discussed quality problems, the project was not straight-forward like the waterfall model. The project work happened more in loops and a lot of backtracking was needed. Naturally the work intensity grew before the mid-term report, progress meeting and due date of the report. Due to working continuously nearly each week, I could avoid hard working sessions at night and only had particularly stressful times before the crucial dates. As a non-native speaker I had to plan more time for reading and report writing. 2.4 Which problems occurred and how they were solved In this chapter I focus on the problems which occurred in this project and on how they influenced the progress and the results of the project Software technique problems Simulation software tools are often designed to be used by experts who are familiar with programming and the area of research where the simulation model is designed for. So the model provides a high flexibility for experts but has less facilities which help less experienced users to become familiar with it. The market I chose is an extreme example of such design. It was conceived as a Java version of the common Santa Fe market and was only used to gain data for two papers [Ehrentreich, Norman(2002)and (2005)]. As a result, I spent more time than I expected on understanding the source code. The main problems were: Due to the use of older methods of Java, the program only runs on Java Especially the Repast-Framework created a lot of error messages with later Java versions. Java version sometimes produces the Java.Lang.outOfMemory error. Later 7 of 47

12 versions of Java do not have this problem. This memory problems occurs especially in long simulation runs. The reading and understanding of the source code was very difficult because a lot of values of variables were constantly changed and used, but in the end they had no effect on the model. A lot of variables are badly named. Due the fact that there was no documentation from a second source, misunderstandings often were realised very late Software evaluation Firstly, the downhill simplex only returns the two values PdCoeff and Offset for each round. Out of this values the demand of stocks will be computed. [see: 5.1.2]But to understand the generation of these values a lot of special observation techniques are needed. I decided to use GNU Plot to visualise all important points in the coordinate system. I needed to observe a lot of iterations. Due to the adoption [ ] of the shape of the downhill simplex a lot of different ranges and sizes of the coordinate systems are generated by the simulation model. GNU Plot is not able to scroll between different plots or manually zoom in to and out of the picture. So all diagrams that show the behaviour of the downhill simplex must be slightly adapted manually, which makes evaluation of the programming very time-consuming. Secondly, I always have to ensure that the behaviour of the downhill simplex was not accidentally good, but rather driven by the real behaviour of the optimisation algorithm. Especially when the program ran and gave proper values I had to step back and analyse the latest changes to ensure real functionality. 3 The artificial stock market model Artificial stock markets are agent-based simulation models. The first artificial stock market was created by Cohen, Maier, Schwartz and Whitcomb in 1983 [LeBaron, Blake (2002)]. These simulation models allow a bottom-up view on trading behaviour and co-operative acting of agents. Instead of a field visit you can collect a lot more specific data from the whole market and from each market participant and the user can exactly trade on the market he needs. Hence the artificial stock markets can create a linking point between the macro level and the micro level in this field of research. There actually are plenty of different types of 8 of 47

13 stock markets available. Most of them were designed for different, often very specialised tasks and no market is a completely abstract model of a stock market. [Tesfation, Leigh (2002]. The stock markets are designed as a round- and event-based simulation model. Figure one shows the process of one round of the simulation model. [Fig1] In practice it is often difficult to adjust the agents and trading procedure correctly. If the set of agents is too homogeneous the market can easily settle to an equilibrium and if the set of agents is trading too aggressively the market will give very strange price curves. These price curve can have unusually high discrepancies and prices can be very high or near zero. It is also difficult to distinguish between real market effects and market effects that are only effect of these kind of implementation [Tesfation, Leigh (2002]. 3.1 Agents The software agents are the market participants. It is their aim to trade with assets and generate their demand of assets. To compute their demand of stocks for the next round the agents need models which allow them to predict future asset prices. Generally fixed models are used to generate the demand of stocks for the next round based on the forecast asset price. One example is given in This use of software agents is a classical examples for the use of software agent. General definition of software agents is given in Ciancaring, P Wooldridge, MJ Ch The designer can choose between a lot of different trading strategies and 9 of 47

14 programming ideas to drive the behaviour of the agent. An overview is given in chapter The agents often have some attributes which drive their decisions and should create more variety in the set of agents. This is discussed in chapter Different types of agents There is a large variety in the set of agents. At one end of the scale are noise traders with a random forecast value, while at the other end are complex self-learning agents based on generic algorithms or neuronal nets. The agents can be characterized in a two dimensional system. The first dimension describes which optimisation algorithm is used; the second dimension explain the main trading strategy and the aim of the agents. Very often the more clever agents are driven by a genetic algorithm or different kinds of evolution strategies. The genetic algorithm approach is also used in the Santa Fe market and will be explained in chapter There are some new approaches of using neuronal nets. [see: LeBaron, B.(2001b):] Agents with these strategies sometimes are called learning agents. Beside the learning agents are a set of more simple agents. These agents trade completely randomly or according to fixed strategies. These agents can be used for bringing noise into the market or as a competitor against the learning agents. The aim and trading strategy of the agent can be ordered in this list: 1.) Meta strategies or normal trading strategies. Meta strategy agents decide between different trading strategies. These trading strategies are forecasting the asset prices for the next round. Normal strategy agents have only got one trading strategy (these could be more complex). 2.) Noise traders. The trading strategies of the noise traders have not the goal to maximize profit. The strategy of using this agent is to create noise in the market by unexpected behaviour of the agents. 3.) Use of a random function. Some agents use a random function to find new trading strategies or asset prices. The GA based agents use a random function for their mutation of the strategies. A normal 10 of 47

15 simplex based agent does not use random functions. 4.) Technical Traders. Technical traders use the history of the market for forecasting future prices or finding a new strategy. Non-technical traders only analyse the actual situation on the market. 5.) Forecasting the behaviour of other agents. These agents try to forecast the next price by trying to forecast the forecasting value of the other agents. 6.) Different scope of the forecasting Nearly all agents are myopic and forecast only the price for the next round. But it could be useful to forecast the asset price for more than one round What is the best strategy? The previous chapter shows a lot of different strategies of agents. There is no one best strategy for software agents. It depends on the behaviour of all other agents and the exact market parametrisation which strategy will trade best. So the aim of strategy designer is on one hand to generate a competitive trading agents but on the other hand to design a special strategy. This strategy should be comprehensible and be designed for a special aim. The competitiveness of an agent can be evaluated if the agent runs against other normally used agents in the market. Another often-used competitor is an agent which generates completely random numbers. This test needs statistical evaluation to exclude all noises out of the market. Chapter 6 goes into further details of the evaluation problem Risk aversion To prevent the market from easily coming to an equilibrium and to give each agent a more specific behaviour, risk aversions are used. In business the risk aversion is a very common vocabulary to describe how risky a trader is. The trader with a high risk aversion prefers the option with more secure output instead of an insecure higher output (by the same expectancy value). For example a person with a high risk aversion prefers secure output of 50 instead of a gamble with a 50% chance of winning 100 (for low risk aversion or risk friendly vice versa). The value of risk aversion describes the price the person will pay to play the game 11 of 47

16 with the more secure output. 3.2 Assets Very often, including the Santa Fe market, the simple approach with two types of assets is used. One asset is a fixed income bond and the other asset is a risky stock. The price of the stock is often set by the demand of all market participants. In some models each asset pays a dividend to its holder. The dividend is often set randomly. A lot models allow the purchase of parts of assets. So the market comes closer to the idea of an ideal market with dividable goods and services, while agents respond to slightly different demands of stocks with different trading behaviour. 3.3 Trading procedure The trading procedure is the heart of the artificial market and it is not trivial to find a good trading procedure because in nearly all markets the agents have to finish all their decisions before it comes to a negotiation of price. In the Santa Fe market the pricing of the stock works in the cycle shown in fig. 2: [Fig2] The process terminates if the amount of buy and sell orders are the same or if the limit of passes has been reached. The actual limit is set up to ten iterations. The pricing is very simple. If there are more buy orders than sell orders the price increases, or vice versa. Each agent has to make its decision on how many stocks it will buy for each price and has to return it to the trading procedure. If the pricing finds an equilibrium between the amount of stocks to buy and sell the stocks will be traded for that price and superfluous stock will be bought by the trading procedure and lost out of the market. On the other side the trading procedure can sell some extra stocks to satisfy the demand that has not been met. Generally this tool is only used to 12 of 47

17 reduce price negotiation time and the gap between bought and sold stocks is so small that it has no effect on the market. 4 Find a market for the project While some stock market models were widely used for a lot of papers, for example the Santa Fe market, some authors designed there own stock market for their particular question, for example in [Ladley and Bullock (2004)]. Before I started my search for an appropriate stock market I defined some requirements for it. programmed in an imperative language the source code should be available for free run on a Windows or Linux system the source code should have some comments in German or English first focus on the often used markets at least one agent should be already implemented These requirements helped immediately to find a small set of markets, on which I took a closer look to evaluate the quality of the source code. I also wanted to use a market which was more often used, because I expected a higher level of software quality. These simulation models are generally used by a small user group with a high knowledge of the whole system. Hence, the systems will be tested only for their particular purpose. So I expected a lower level of documentation and a source code that looks unfinished. 4.1 The implementation of the Santa Fe market and the choice First I found the Santa Fe market. It fits to all of my requirements. There are plenty of papers which describe this market model. The market is available in four implementations [SFIASMTeam]. 1.) The original Santa Fe artificial Stock Market (is written in Objective C) 2.) The Java-Version of these Objective C-Code by Norman Ehrentreich 3.) The Java-Version written by Steve Phelps 4.) The Java-Version written by Jose Manuel Galan Luis R. Izquierdo (documented only in Spanish) I could easily skip 4, because I was unable to understand the documentation. Number 3 13 of 47

18 looked very massive and the classes seemed disorganized. I have never used Objective C, so I took a stronger look at number 2. The 10 Java classes looked very similar to the original Objective C code and I found documentation in the source code. For all simulation functionalities the repast framework is used[repast-team] which has the advantage, that they are programmed on a higher level and have a better documentation. 5 The optimisation algorithm The aim of the project is to find an agent which is able to trade in this artificial stock market. In the next part of this chapter I will take a look at the actual implemented genetic agent and analyse the technical requirements for a new optimisation algorithm in this market. Then I will take a brief look at possible optimisation techniques and describe the adaptation and improvement of the chosen downhill simplex. In the last sections I will discuss the work of the downhill simplex. 5.1 Requirements for the algorithm The aim of an optimisation algorithm in an artificial stock market is to make a decision on how many assets to buy in the next round. No human being or computer can really look into the future. All these forecasting processes are based on special models or are guessed. In general for a good forecasting algorithm there are two general requirements. 1.) The forecasting strategy should be competitive against other traders in the market. 2.) The forecasting strategy should based on a describable model. The word model is used here as a very wide word. Even a random selection of numbers is a forecasting strategy as well, but these strategies should be comprehensible and describable. These requirements should allow us to evaluate several forecasting strategies Analysing the genetic algorithm The genetic algorithm is the most used strategy to drive the behaviour of agents in artificial stock markets.[lebaron,blake (2002)] This algorithm is also implemented in the Santa Fe market. In the discussion about genetic algorithms the authors often speak about learning. [for instance: LeBaron,Blake (1999)]. Learning means that the genes of each agent represent the knowledge of that agent. This knowledge will be improved by learning with each iteration of 14 of 47

19 the genetic algorithm. The design of the gene is the first crucial design question in designing a genetic algorithm. This decision is driven by the different trading procedures and desired behaviour of the agent. In the Santa Fe market each agent contains a set of 100 genes. Trading rules are part of the genes of the genetic algorithms. For the genetic algorithms each agents acts individual. Each gene contains a condition part, a forecast part, its fitness value, and its forecast accuracy. The forecasting accuracy is the weighted variance of the performance during the last rounds of the gene. The fitness value stores the actual fitness value of these genes. The forecasting part contains the values PdCoeff and Offset. The condition part describes in which market situation the trading rule will be used [Ehrentreich, Norman(2002)]. Very often more then one trading rule is in the set of the allowed rules. The set of genes are optimised in a normal genetic algorithm [See for example: Mitchell, M., (1996)]. [Fig3] This process is the GA process, which is used in the artificial stock market. The circle starts with a random set of genes, which are benchmarked by the fitness function. The GA process has a lot of parameters or options which can be used to drive the optimisation process. I will give a brief introduction about the main ones for this problem and show how they are set in the Santa Fe market. The parameterisation of the GA is one field of research in artificial stock markets. So there is a big variety of possible parameterisations. [LeBaron,Blake (1999)]. 15 of 47

20 Fitness-function: The fitness-function judges each gene by the forecasting value of the last round and by the fitness of the condition part. The fitness function uses the forecasting quality of these genes for the actual round and the use of technical functions to judge the gene. Replace the genes: the number of replaced genes is one driver for the learning speed of the agents. The more genes are replaced, the higher the learning speed will become. In this implementation, 20 out of 100 genes will be replaced. Mutation: The mutation could be interpreted as learning by experiment or by unintended mistakes. [Ehrentreich, Norman(2002)] The mutation will have some some random changes of its strategy, and any changes of the main forecast values PdCoeff and Offset will be taken into account. There are several strategies to compute the main forecasting values out of the set of Genes. The best gene can be chosen, or an average of a number of genes, which are selected by a specific criterion.[see: Ehrentreich, Norman(2002) p. 5] For my work I take the average value of all genes. That avoids extreme forecasting values and engenders a more moderate trading strategy The optimisation problem and communication with the market maker The expected (E) price for the stock and the dividend at the time point t+1 is defined as a linear function of the price (p) and the dividend (d) of the actual round (time point t). (1) The aim of an optimisation algorithm is to find values for the variables PdCoeff and Offset. The numbers of stocks to own (x) is computed by this formula. The amount of stocks to buy or to sell is the difference between x and the actual owned stock of the agent. (2) [Ehrentreich, Norman(2002)] 16 of 47

21 Where: p: Price of Stock; d: dividend of stock; λ: Risk aversion of agent;σ2 variance of the the forecasting quality of this agent of the previous rounds. The part of the formula (1+rf)p describes the alternative investment into the fixed income bond. Here rf is the rate of interest. In the actual model the interest is 10%. In the actual implementation, the variance of forecasting from the previous rounds is computed with the formula. (3) A = 1-b and b = 1/75. With the setting of the value b the designer can define the scope of influence of the forecasting quality of the last round on the variance. Actually the value for b and the influence is small. If an agent has got a small variance, then it relies more on its forecasting quality and will buy more stocks than if it has a great variance. To find the right stock price, the trading procedure gives a trial stock price to formula 2. Each agent returns their demand of stocks for that trial stock price to the market maker. The demand is computed by formula 2. Then the trading procedure decides to increase or decrease the price of the stocks for the next trial round. Agents cannot change the PdCoeff and Offset values during the price finding period. The price setting function terminates after a fixed number of rounds (currently set to 20), or if the buy and sell orders are equal. Hence there are two requirements for the optimisation algorithm to find good values for the variables PdCoeff and Offset. 1) Forecast the asset price for the next round 2) The forecast value should be able to interact with the market maker. A lot of combinations of PdCoeff and Offset can satisfy the first requirements, but the second requirement makes the allowed range for both values smaller. 17 of 47

22 offset pdcoeff [Figure 4] The graph in figure 5 shows all possible combinations of PdCoeff and Offset which satisfy the first constraint. But only a small range of Offset and PdCoeff satisfy the second constraint and give a good market effect. 5.2 Choose of a optimisation algorithm As written in the minimum requirements, the chosen optimisation techniques should be different to any mutation selection processes, genetic algorithms or other evolutionary based approaches. The idea to work with a set of genes is essentially part of all these strategies. Only the generating, judging and erasing processes differ between all these strategies. The selected algorithm should be implementable in a appropriate amount of time. In no paper I read that the behaviour of more intelligent agents is driven numerically. Numerical based optimisation algorithms try to find an extreme point of a function. The geometrical approach allows an intuitive feeling of problem solving and provides a lot of adjustment and tuning possibilities. The main task in using a numerical based optimisation techniques is to find a model which satisfies the requirements of the artificial stock market and generates competitive forecasting values, and to find out which of the available numerically based optimisation techniques is a technique which can be adapted to the problem. All these numerical based optimisation techniques are generally designed to solve offline optimisation problems. That means that an algorithm gets a fixed function and should find the best function value in a certain amount of time. This forecasting problem is different to that 18 of 47

23 method. Here the design of the function changes very frequently, so the chosen optimisation algorithm should be adaptable so that it can solve this forecasting problem. After adaptation of the algorithm, the main optimisation strategy should be found and be used as a main part of the forecasting strategy. If the main strategy is adapted out of the optimisation algorithm, then it is a design of a completely new optimisation algorithm and not aim of the project Numerical based algorithms Numerically based optimisation algorithms are used to find extreme points of functions without knowing the derivation of them. This technique is mainly used for functions with plenty of parameters which makes derivation impossible. Nearly all these optimisation techniques can be modelled in an n-dimensional coordinate plane. Each point of the solution space describes one available parameterisation for the function. So often, these optimisation techniques are understood as a search algorithm in the solution space, so the process in all of these optimisation algorithms are very similar. The algorithms choose a point of the solution space, look for its function value and find the next point. The strategy for finding the next point is different in each optimisation algorithm. Beside a real describable function other things for judging each point of the solution space are possible. One example is a computer based simulation model with plenty of parameters.1 The results of each parametrised simulation run is then used to judge each point of the solution space. So the optimisation function can be used to find the best parameterisation for this simulation model. A look into the book Numerical recipes [Press, William H.(2002)] offers a big bunch of possible optimisation algorithms. Based on the optimisation problem there are some optimisation methods which cannot be well adapted to be used for my aim. First the optimisation strategy should be able to optimise a function with at least two parameters (PdCoeff and Offset). In several optimisation algorithms the derivative, for example the Gradient Method[Press, William H.(2002) p.317] is used. The derivative gives a hint into which direction or how far away the optimisation algorithm should try to find the next possible value. For that strategy there are no equivalent in the artificial stock market forecasting problem and none of these 1 One example for such a simulation model is a agent based simulation to evaluate boarding times for air planes. [Czogalla,R] 19 of 47

24 strategies can be used. Also strategies which mainly perform a random selection of the next point cannot be used as well (For example: Simulated Annealing[Press, William H.(2002)S.343]). These methods are not a good starting point to design a stock forecasting algorithm. The power of these methods is often the judgement function and strategies that show their power when plenty of iterations will be done, but if the problem changes every iteration this strategy cannot show its advantages. Out of this requirement and concern I decided to use and adapt a downhill simplex. 5.3 Downhill Simplex The downhill simplex is a numerical based optimisation technique. It can be used to find the minimum or a maximum of a function.[press, William H.(2002) p. 306] Because on its graphical representation and its very robust ground algorithm it is very flexible to use. The ground algorithm can be retained by the adaptation. In the following part of this chapter I will describe the basic work of the simplex, adapting it to this problem and ideas to improve its work How does the simplex works The downhill simplex is due to Nelder and Mead. It is a geometrical approach for finding the minimum or maximum of a function with n parameters. Therefore it can be used in Ndimensional solution space and equilateral N+1-dimensional polyhedron. [Fig5] In figure 4, the points A;B;C define a triangle which runs in a two-dimensional space. The 20 of 47

25 optimisation process starts with a random set of the starting points of the triangle. Each starting point returns a value of the function. In figure four it is pictured in the third dimension as level curves. When all starting points are set, the algorithm chooses the value with the worst function value. To find a new location this point will be point reflected by the middle point of the polyhedron (or line by a triangle), which is created by all non-chosen points. Then the function value of this location will be computed. In the next iteration, the method selects again the point with the worst function value and so on. To give this method more flexibility a lot of step reduction and enlargement techniques are developed. I take a look on it in and Main idea of the downhill simplex and its power as a forecasting algorithm A really good price forecasting strategy which is in every case better then all other strategies can only be found if humankind can find a way to predict the future, so the focus of creating the trading agent is just to create a good trading agent. But it is also interesting to understand their trading strategies and behaviour. [see and 3.1.2] The downhill simplex is an algorithm which started with a big polyhedron and this polyhedron will be reduced during the optimisation time to find a (local) optimum of a function. For the forecasting strategy I will use the idea of storing a small amount of knowledge about solution space in the shape of the triangle. The knowledge is stored in the function value of each point. The downhill simplex tries to improve its position by resetting the point with worst function value. For finding this new position it uses the location of the two other points, so the two points which gave the best results will not be visited again, and the new point lies a edge length of the polyhedron away from the best points. So I predict that the downhill simplex will be a very uneven trader on market effects. Its forecasting strategy will be spread widely and it can set a market into a stronger oscillation. Also the geometrical approach allows easy implementation of adaptation and tuning ideas. The representation of only a small amount of knowledge (here the location of two good forecasting values) has never been tried before. The step length of the simplex is the distance between the old and the new location of one point of the simplex triangle. The length of a step is one of the main drivers of the forecasting quality. A wide step represents a greater changes of the values PdCoeff and Offset and vice 21 of 47

26 versa. The exact change depends on the direction of the step Adapting the problem to the downhill simplex The aim of each optimisation algorithm in this artificial stock market is to find values for PdCoeff and Offset. Hence the downhill simplex needs to run as a triangle in a two dimensional coordinate system. For further explanation the x-axis of the coordinate system is set for the value PdCoeff and the Y-axis is set for the value Offset. Each point of the triangle represents one combination of values for the variable PdCoeff and Offset. The three vertices of the triangle are named in the following as A,B,C. The return value of the function is the forecasting quality of this point. The downhill simplex was designed to solve mainly off-line optimisation problems. An offline optimisation problem is that the function and the optimal solution never changed during the optimisation period. In the problem of an artificial stock market the best combination of PdCoeff and Offset changes in each round. So the simplex needs other, more problem specific, techniques to find a good next step. Specifically the step-reduction techniques [for example Press, William H.(2002) p. 307] needed to be changed and adapted. But by all this adaptation and tuning the triangle as main idea of the optimisation process should be retained. The simplex runs as a round based optimisation algorithm. For the adaptation to the asset price forecasting problem each iteration (round) of the simplex algorithm returns a new forecasting value. It could be possible to run more then one iteration to generate one new forecasting value. But then a function is needed to choose a result out of the iterations. The system knowledge is only knowledge about past simulation runs and the real prediction about the future should be taken out of the simplex triangle. This function needed to be design up from scratch and does not has anything in common with the basic idea of the downhill simplex. I think that the downhill simplex should only be adapted if it is useful. In the sub parts of this chapter I will discuss all changes that are needed to use this optimisation procedure in the Santa Fe market Initialisation phase Simplex can be initialised with three randomly chosen points as well as the normal simplex. But in this area the shape of the triangle is very important and drives the behaviour of the agent immediately. So it is appropriate to set the third point of the triangle in a non-random 22 of 47

27 fashion. It is the best way to set the third point of the triangle not randomly to greater a special shape of the triangle. So the designer has a parameter to drive the behaviour of the triangle. In an equilateral triangle each step in each direction using the normal step techniques has the same length. Later on this technique can be used to create other shapes of triangle. By creating these third point scale translation [see ] must be concerned. The triangle must be in the shape if the triangle is in the mute to move Swing in time Eespecially the first optimisation values of the downhill simplex after the initialisation are very inaccurate. Often the algorithm predicts negative or unbelievably high prices and the simplex needs a lot of integration to juggle to good forecasting values. This problem is based on the fact that the triangle starts often with a big shape. It could be possible to start the simplex algorithm with a very small shape and to inflate it later with the step enlargement technique, but it is challenging to find a good step enlargement technique to avoid that the simplex enlarges the step length too fast and generates a big triangle again or the simplex is set for a long time on one small area of the solution space. In both strategies only systems knowledge which gives the ability to predict a good triangle shape at the beginning, which is similar to the triangle that will be generated after the swing in time can reduce the swing in time. However in this simulation model every simulation run is different and influenced by the behaviour of all agents. So only rough guesses about the shape of the triangle are possible and both starting strategies need, with smaller or greater triangle, an experiment to see which strategy could be more practicable for the problem Step reduction The step length of the simplex controls the quality of the changes for the values PdCoeff and Offset. A long step changes the values more then a short step. The step reduction and enlargement should not change the shape of the triangle. If a change of the shape of the triangle is wished, then it should be done independently from the decision of changing the step length and vice versa. In the normal use of the downhill simplex the step reduction will be done if one point in the coordinate system is used twice [Press, William H.(2002) p. 307]. Step enlargement is only be used if the simplex has found a local optimum and the simplex should move to next optimum. However in the price forecasting problem a step enlargement is essential because 23 of 47

28 that optimisation process never ends and aim of the algorithm is not to find exactly the local optimum. So a completely new strategy is needed to detect whether a step enlargement or reduction is needed. For the price forecasting model a new technique is needed to identify situations where a step reduction or enlargement is appropriated. For a better localisation a straight line is drawn through the coordinate system. This straight line shows all possible combination of PdCoeff and Offset which forecast the right stock price for the actual round. This straight line can be define by the function: (5) Pricet stands for stock price of the actual round and Pricet-1 represents the price for the previous round. By the design of this function is the fact ignored that PdCoeff and Offset forecast the combination of Price and dividend [see: Ch ]. This relaxes the calculation of the function. The inexactness is not so dramatic because that only the graphical relation offset between the simplex triangle and the straight line is used. B C A pdcoeff [Figure6] Now we can see how far each vertex of the triangle is away from the optimal combination and Offset. When the line cuts the triangle every move of each vertex would bring the triangle further away from the optimal position of PdCoeff and Offset (see fig.6). So a reduced step towards the optimal line would bring the next vertices nearer towards the straight line. If the 24 of 47

29 straight line does not cut the the triangle a normal step towards the line will be automatically done by the triangle and a step reduction is only needed if the triangle is cut by the line. Move the vertex the reduce step The first step is to select the vertex with the worst forecasting value. To retain the shape of the triangle a second vertex is needed. If only one point of a triangle is moved all angles in the triangle will change. This vertex will be chosen randomly. With more system knowledge and experience it could be possible to find a more intelligent way to choose the second vertex. offset Both vertices now move on the edges towards the third vertex.[see Figure 6] C B' B A' A pdcoeff [Figure 7] The points A' and B' show the moved positions of the chosen points A and B. To retain the shape both points must move relatively to their actual distance to fix point of the triangle, closer to it. A random number decides about the new size of the triangle. The range for the random number is between 0.1 and 0.9. As distribution for the random number is actually a normal distribution implemented. This random number is multiplied with the actual distance between the fix points and the moving points to compute the new distance between them. The new location for the vertex with the previously worst forecasting value will be returned as the new forecasting value Step enlargement Step enlargement is appropriate if the simplex triangle is far away from the the straight line, which represented the optimal combination of PdCoeff and Offset. There are two models in 25 of 47

30 the optimisation problem: 1. If the distance between downhill simplex and straight line is greater then a specific level, then a step enlargement is appropriate. 2. If the downhill simplex has not been cut by the straight line for number of iteration a step enlargement is appropriate. In both cases the fixed number of iterations or the distance needed to be found. It is also possible to replace this fixed number by a random number. For instance this random number can be generated by a normal distribution. That would bring more randomness into the optimisation procedure. I decided to implement the second idea. Due to the adoption in this idea is easier to implement and a definition for the distance between the point and the line is not needed. Later on the other idea could be implemented as well for further evaluations if the algorithm is stable. The step enlargement works in a very similar way to the step reduction. The aim is here to retain the shape of the triangle. First I have to find out which point has got the best forecasting value (B in fig. 7). Then a randomly chosen point for the second other vertices of the triangle which is needed to be moved to retain the shape (A in fig. 7) is needed. Analogous to the step reduction the step enlargement procedure moves both points away from the fixed one. In offset figure 7 the two new points are named as A' and B'. B' B C A A' pdcoeff [Figure 8] The step length of the enlargement will be set randomly by a normal distribution analogous to 26 of 47