Optimization in Computational Finance and Economics Edward Tsang Centre for Computational Finance and Economics University of Essex 15 September 2015All Rights Reserved, Edward Tsang

Synergy in computational finance and economics  Business expert provides behavioral models  Computing expert provides software 15 September 2015All Rights Reserved, Edward Tsang Amadeo Alentorn Old Mutual ex-CCFEA Andreas Krause Business School Bath Seminar at CCFEA: Herding behaviour (what happens when traders copy each other?) Produced software within a few hours (with graphical interface)

Herding Simulator by Alentorn 15 September 2015All Rights Reserved, Edward Tsang

Overview – Optimization in CFE 15/09/2015All Rights Reserved, Edward Tsang More complex modellingSimpler modelling Portfolio Optimization Basic algorithms Multi-objective Bargaining Complex reasoning Useful to approximate Economic Wind-Tunnel Complex model Complex strategies (Computational Finance and Economics)

Portfolio Optimization Edward Tsang EDDIE / GP Qingfu Zhang Optimisation Dietmar Maringer Portfolios Zhang, Q., Li, H., Maringer, D. & E.P.K. Tsang, E.P.K., MOEA/D with NBI-style Tchebycheff approach for Portfolio Management, Proceedings, Congress on Evolutionary Computation (WCCI 2010),WCCI 2010 Barcelona, Spain, July, Hui Li MOEA

Attention by a Computer Scientist  Surely you know what you want?  Tell me what you want to optimize  I promise to find you a solution  Some methods are better than others 15/09/2015All Rights Reserved, Edward Tsang Optimizer Computer Scientist’ attention Solution Objective

Attention by an Economist  How to model return?  How to model risk?  Once we know how to model them (Mathematically)  As a Rational Agent  Surely you can find the solution 15/09/2015All Rights Reserved, Edward Tsang Return Risk Objective Function Economists’ attention Solution In order to find solutions, they often have to make simplifying assumptions

Portfolio Optimization Overview  To succeed, one needs to see the full picture 15/09/2015All Rights Reserved, Edward Tsang Portfolio Return Risk Objective Function Economists’ attention Optimizer Computer Scientist’ attention

15 September 2015All Rights Reserved, Edward Tsang Portfolio Optimization  Typically: High risk  high return  Diversification reduces risk  Task: find a portfolio: m aximize return, minimize risk Risk Return

Modern portfolio theory  Expected return is a weighted sum of the individual returns  Expected risk depends on individual risks and correlations of the component assets  Diversification reduces risk 15/09/2015All Rights Reserved, Edward Tsang

Mean-Variance Efficiency Frontier 15/09/2015All Rights Reserved, Edward Tsang Fix risk Max return? Line from risk free rate?

Portfolio Optimization Problem Given:  W: budget  n: # of assets available  c i : unit price of asset i  r i : expected return of asset i  σ ij : covariance between assets i and j  K: maximum # of assets to buy Decision variables  x i : investment on asset i 15/09/2015All Rights Reserved, Edward Tsang

Further considerations  Assets come in units: There must be integers k i such that x i = g i (k i ) where g i is a function of investing in asset i, which may account for transaction costs  Short selling is not allowed: x i must be integers  Budget constraint: 15/09/2015All Rights Reserved, Edward Tsang

Two objectives  Maximize return R 15/09/2015All Rights Reserved, Edward Tsang  Minimize Risk V

Optimization in Economics  Dietmar Maringer and Manfred Gilli  Threshold Acceptance –Simplified Simulated Annealing  First to attack the realistic constraints –With integer variables, transaction cost, constraints 15/09/2015All Rights Reserved, Edward Tsang Single objective optimization: Fix risk, maximize return

Multi-objective Optimization  Given multiple objective functions f 1, f 2, …, f m  MOEA/D search in variable space x 1, x 2, …, x n  Decompose problem into single objective problems, each solved by a procedure  Neighbouring procedures exchange solutions –Assuming they have similar landscapes & solutions –Neighbouring defined by distance in weight vectors  Each procedure combines several solutions 15/09/2015All Rights Reserved, Edward Tsang

MOEA/D Performance  First to tackle portfolio optimization with two objectives  Compared favourably against NSGA-II 15/09/2015All Rights Reserved, Edward Tsang

MOPs Performance Criteria  Let A and B be Pareto Front approximations  Set Coverage: C(A,B) = |{u  B|  v  A: v dominates u}| / |B| 0 ≤ C(A,B) ≤ 1  Inverted Generational Distance (IGD-metric): –Measure both diversity and convergence 15/09/2015All Rights Reserved, Edward Tsang

Decomposition in MOEA/D  MOEA/D decomposes problems  Two commonly used methods –Weighted sum – good for convex MOPs –Weighted Tchebycheff – may handle convex MOPs  Both sensitive to scales of the problem  NBI (Normal Boundary Intersection) –Insensitive to scales, but not for MOEA/D  Combine Tchebycheff & NBI in MOEA/D 15/09/2015All Rights Reserved, Edward Tsang

Portfolio Optimization Conclusions  Economist focus on modelling –They assume that solutions can always be found –In reality, they rely on simplifying assumptions  Computer scientists focus on solving –They assume that we always know what we want –In reality, they are part of a loop to explore what is needed  There is synergy 15/09/2015All Rights Reserved, Edward Tsang

Automated Bargaining Edward Tsang CCFEA Constraints, Business models Nanlin Jin Computing Extending Rubinstein Model Evolving strategies Abhinay Muthoo Economics Game Theory Jin, N.Jin, N., Tsang, E. & Li, J., A constraint-guided method with evolutionary algorithms for economic problems, Applied Soft Computing, Vol.9, Iss.3, June 2009,

15 September 2015All Rights Reserved, Edward Tsang Bargaining in Game Theory  Rubinstein Model:  = Cake to share between A and B (= 1) A and B make alternate offers x A = A’s share (x B =  – x A ) r A = A’s discount rate t = # of rounds, at time Δ per round  A’s payoff x A drops as time goes by A’s Payoff = x A exp(– r A tΔ)  Important Assumptions: –Both players rational –Both players know everything  Equilibrium solution for A:  A = (1 –  B ) / (1 –  A  B ) where  i = exp(– r i Δ) Notice: No time t here 0  ? xAxA xBxB AB Optimal offer: x A =  A at t=0 In reality: Offer at time t = f (r A, r B, t) Is it necessary? Is it rational? (What is rational?)(What is rational?)

15 September 2015All Rights Reserved, Edward Tsang Evolutionary Rubinstein Bargaining, Overview  Game theorists solved Rubinstein bargaining problem –Subgame Perfect Equilibrium (SPE)  Slight alterations to problem lead to different solutions –Asymmetric / incomplete information –Outside option  Evolutionary computation –Succeeded in solving a wide range of problems –EC has found SPE in Rubinstein’s problem –Can EC find solutions close to unknown SPE?  Co-evolution is an alternative approximation method to find game theoretical solutions –Less time for approximate SPEs –Less modifications needed for new problems

15 September 2015All Rights Reserved, Edward Tsang Issues Addressed in EC for Bargaining  Representation Representation –Should t be in the language?  One or two population? One or two population  How to evaluate fitness –Fixed or relative fitness?  How to contain search space?  Discourage irrational strategies: Discourage irrational strategies –Ask for x A >1? –Ask for more over time? –Ask for more when  A is low? / AA BB  1 BB   1

15 September 2015All Rights Reserved, Edward Tsang Representation of Strategies  A tree represents a mathematical function g  Terminal set: {1,  A,  B }  Functional set: {+, , ×, ÷}  Given g, player with discount rate r plays at time t g × (1 – r) t  Language can be enriched: –Could have included e or time t to terminal set –Could have included power ^ to function set  Richer language  larger search space  harder search problem

15 September 2015All Rights Reserved, Edward Tsang Two populations – co-evolution  We want to deal with asymmetric games –E.g. two players may have different information  One population for training each player’s strategies  Co-evolution, using relative fitness –Alternative: use absolute fitness Evolve over time Player 1 … Player 2 ………

15 September 2015All Rights Reserved, Edward Tsang Incentive Method: Constrained Fitness Function  No magic in evolutionary computation –Larger search space  less chance to succeed  Constraints are heuristics to focus a search –Focus on space where promising solutions may lie  Incentives for certain properties in function returned: –The function returns a value in (0, 1) –Everything else being equal, lower  A  smaller share –Everything else being equal, lower  B  larger share Note: this is the key to our search effectiveness

15 September 2015All Rights Reserved, Edward Tsang Models with known equilibriums Complete Information  Rubinstein 82 model: –Alternative offering, both A and B know  A &  B  Evolved solutions approximates theoretical  Evolved solutions for problems with outside option Incomplete Information  Rubinstein 85 model: –B knows  A &  B –A knows  A and  B weak &  B strong with probability  weak  Evolved solutions approximates theoretical

15 September 2015All Rights Reserved, Edward Tsang Models with unknown equilibriums  Modified Rubinstein 85 models  Incomplete knowledge –B knows  B but not  A ; A knows  A but not  B  Asymmetric knowledge –B knows  A &  B ; A knows  A but not  B  Asymmetric, limited knowledge –B knows  A &  B –A knows  A and a normal distribution of  B  Also worked on limited knowledge, outside option  Future work: new bargaining procedures

15 September 2015All Rights Reserved, Edward Tsang Evolutionary Bargaining Conclusions  Demonstrated GP’s flexibility –Models with known and unknown solutionsknownunknown –Outside option –Incomplete, asymmetric and limited information  Co-evolution is an alternative approximation method to find game theoretical solutions –Relatively quick for approximate solutions –Relatively easy to modify for new models  Genetic Programming with incentive / constraints –Constraints used to focus the search in promising spaces

Evolving Agents Biliana Alexandrova-Kabadjova Cards Mexico Central Bank Andreas Krause Business Bath Alexandrova-Kabadjova, B., Artificial payment card market - an agent based approach, PhD Thesis, Centre for Computational Finance and Economic Agents (CCFEA), University of Essex, Edward Tsang EDDIE / GP

15 September 2015 Agent-based Payment Card Market ModelModel Costumer Merchant Interactions at the Point Of Sale Payment Card provider Costumer’s fees and benefits Merchant’s fees and benefits Government: public interest drives regulations Connected (topology) Possible Objectives: Maximize profit Maximize market share Learning optimal strategies Consistent patterns observed with static agents Decisions, decisions

Modelling is commonly used

Decisions dependency (from the bank’s point of view) 15/09/2015All Rights Reserved, Edward Tsang Customer Benefits Merchant Fixed Fee Customer’s decision to use the card Customer’s decision to hold a card Merchant’s decision to hold a card # of Merchants accepting the card Banks’ Profits Publicity Cost Customer Fixed Fee Merchant Benefits Banks’ Market Share # of Customers using the card # of Merchants using the card # of Customers having the card

Learning optimal strategies 15/09/2015All Rights Reserved, Edward Tsang  Each card makes the following decisions: Each card makes the following decisions –Publicity cost, fixed/variable fees to consumers/merchants  PBIL used to evolve strategies PBIL –Converged after 3,000 runs; observations being analysed Card 1 decisions Card 2 decisions Card n decisions … Probabilistic Model on the decisions Market simulation: Interaction between consumers and merchants

Population-based Incremental Learning (PBIL) 15/09/2015All Rights Reserved, Edward Tsang  Statistical approach  Related to ant-colonies, GA Model M: x = v1 (0.5) x = v2 (0.5) y = v3 (0.5) y = v4 (0.5) Sample from M solution X, eg Evaluation X Modify the probabilities

Economic Wind-tunnels Conclusions  Markets are complex systems  It is not easy to predict the consequences of actions  But modelling is better than wild-guessing  No model is correct  But some are useful  Useful for policy making as well as strategies development 15/09/2015All Rights Reserved, Edward Tsang

Conclusions

15 September 2015All Rights Reserved, Edward Tsang Optimization in Finance & Economics, Conclusions  Computer Scientists: –Surely you know what you want?  Economists: –Rational agents should find optimal solutions  Reality: –We don’t really know what we want –Perfect rationality doesn’t exist  Synergy in Computation + Finance/Economics –Optimization experts have key role to play

Game Theory Hall of Frame John Harsanyi John Nash Reinhard Selten Robert Aumann Thomas Schelling 1994 Nobel Prize 2005 Nobel Prize

15 September 2015All Rights Reserved, Edward Tsang 1978 Nobel Economic Prize Winner  Artificial intelligence  “For his pioneering research into the decision- making process within economic organizations"  “The social sciences, I thought, needed the same kind of rigor and the same mathematical underpinnings that had made the "hard" sciences so brilliantly successful. ”  Bounded Rationality –A Behavioral model of Rational Choice 1957 Sources: Herbert Simon (CMU) Artificial intelligence

15 September 2015All Rights Reserved, Edward Tsang Why Modelling?  Scientific Approach –Modelling allows scientific studies. –Human expert opinions are valuable, –But best supported by scientific evidences  Multiple Expertise –models can be built by multiple experts at the same time – The resulting model will have the expertise that no single expertise can have.  Models are investments –Models will never leave the institute as experts do. –Investments can be accumulated.

15 September 2015All Rights Reserved, Edward Tsang Why Agent Modelling  Agent modelling allows –Heterogeneity –Geographical distribution –Micro-behaviour to be modelled  Representative models don’t allow these  Micro-behaviour makes the market

15 September 2015All Rights Reserved, Edward Tsang Agent-based Payment Card Market Model Government: public interest drives regulations Possible Objectives: Maximize profit Maximize market share Costumer Merchant Interactions at the Point Of Sale Payment Card provider Costumer’s fees and benefits Merchant’s fees and benefits Connected (topology)topology Learning optimal strategies Consistent patterns observed with static agents Decisions, decisions

15 September 2015All Rights Reserved, Edward Tsang Research Profile, Edward Tsang ApplicationTechnology Finite Choices Decision Support, e.g. Assignment, Scheduling, Routing Constraint Satisfaction, Optimisation, Heuristic Search (Guided Local Search) Financial ForecastingGenetic Programming Automated BargainingGenetic Programming Wind Tunnel Testing for designing markets and finding winning strategies Mathematical Modelling, Machine Learning, Experimental Design Portfolio OptimisationMulti-objectives Optimisation Business Applications of Artificial Intelligence