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    Heuristic Optimization Strategies in Finance - An Overview

    Marianna Lyra

    COMISEF Fellows’ Workshop: Numerical Methods and Optimization in Finance

    Financial support from the EU Commission through COMISEF is gratefully acknowledged

    June 18, 2009

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    Marianna Lyra Early Stage Researcher - COMISEF Research Network

    1 BA Business Administration - minor Finance, UCY

    2 MSc Finance, UCY 3 PhD student JLU Giessen, DE 4 Research Interests: Apply

    Optimization Heuristics in Finance (Credit Risk Management)

    5 Contact Info: http://comisef.wikidot.com/ mariannalyra

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    1 Introduction Financial World Complexity

    2 Heuristic Optimization Techniques Classical concept Differential Evolution

    3 Financial Applications Optimization heuristic strategies in finance

    4 Conclusion Have in mind

  • logo

    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    1 Introduction Financial World Complexity

    2 Heuristic Optimization Techniques Classical concept Differential Evolution

    3 Financial Applications Optimization heuristic strategies in finance

    4 Conclusion Have in mind

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Financial World

    Financial world

    Optimization

    Financial problems

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Complexity

    Least median of squares residuals as a function of α and β

    Multiple local minima (optima) Apply optimization heuristic techniques

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    1 Introduction Financial World Complexity

    2 Heuristic Optimization Techniques Classical concept Differential Evolution

    3 Financial Applications Optimization heuristic strategies in finance

    4 Conclusion Have in mind

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Classical concept

    Local search procedure

    1: Generate initial solution xc

    2: while stopping criteria not met do 3: Select xn ∈ N (xc) (neighbor to current solution) 4: if f (xn) < f (xc) then 5: xc = xn

    6: end if 7: end while

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Classical concept

    Heuristic Techniques

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    RRRR RRR

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    _ _ _ _ _ _Construction methods Local search methods

    Trajectory methods mmm

    mmm

    vvmmm mm Population search

    OOO OOO

    ''OO OO

    Threshold accepting

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    Differential evolution

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    Tabu search

    Genetic algorithms

    Hybrid Meta Heuristics

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Classical concept

    Heuristic Techniques

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    lll lll

    l

    ))RRR RRRR

    RRRR RRR

    _ _ _ _ _ _� �

    � �_ _ _ _ _ _Construction methods Local search methods

    Trajectory methods nnn

    nnn n

    wwnnnn nn Population search

    PPP PPP

    P

    ((PP P

    Threshold accepting

    � � �

    Differential evolution

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    Tabu search

    Genetic algorithms

    Hybrid Meta Heuristics

    99dd

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Population based heuristic with remarkable performance in continuous numerical problems.

    Example

    Capital Asset Pricing Model (CAPM): Risk-return equilibrium.

    ri,t − r st = α + β(rm,t − r st ) + εi,t (1)

    Least Median of Squares Estimators (LMS) (Rousseeuw and Leroy (1987)):

    min α,β

    (med(ε2i,t)) (2)

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    1 Generate random set values α and β (initial solutions)

    2 Evaluate initial solutions minimizing LMS 3 Generate new candidate solutions from the

    initial one 4 Evaluate new candidate solutions minimizing

    LMS 5 Repeat until a very good solution is found

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    1 Generate random set values α and β (initial solutions)

    2 Evaluate initial solutions minimizing LMS 3 Generate new candidate solutions from the

    initial one 4 Evaluate new candidate solutions minimizing

    LMS 5 Repeat until a very good solution is found

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    1 Generate random set values α and β (initial solutions)

    2 Evaluate initial solutions minimizing LMS 3 Generate new candidate solutions from the

    initial one 4 Evaluate new candidate solutions minimizing

    LMS 5 Repeat until a very good solution is found

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    1 Generate random set values α and β (initial solutions)

    2 Evaluate initial solutions minimizing LMS 3 Generate new candidate solutions from the

    initial one 4 Evaluate new candidate solutions minimizing

    LMS 5 Repeat until a very good solution is found

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    1 Generate random set values α and β (initial solutions)

    2 Evaluate initial solutions minimizing LMS 3 Generate new candidate solutions from the

    initial one 4 Evaluate new candidate solutions minimizing

    LMS 5 Repeat until a very good solution is found

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    Initialization Step 1 & 2: Generate and evaluate random set values α[−1, 2] and β[0, 1] (initial solutions)

    P(0)=

     d

    1 2

    np

     minα,β(med(ε2i,t))

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    Initialization Step 1 & 2: Generate and evaluate random set values α[−1, 2] and β[0, 1] (initial solutions)

    P(0)=

     d

    1 2

    np

     minα,β(med(ε2i,t))

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    Generate new candidate solutions from the initial one Step 3a: Differential mutation

    P(0)1,i =

     1 2 ... r3 ... r1 ... r2 ... np

     P(υ)1,i = P

    (0) 1,r1

    + F × (P(0)1,r2 - P (0) 1,r3

    )

    F scale factor ∈ [0, 1+] determines speed of shrinkage

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    ... ... u1 ud

    CR CR

    uniform crossover

    ... u1

    ud

    P(u)1,i . . . P (u) d ,i

    Random crossover Step 3b: Crossover elements from P(0)1,i and P

    (υ) 1,i

    1 Generate for each parameter a uniform random number, ud ∈ [0, 1]

    2 Determine the crossover probability, CR ∈ [0, 1]

    3 Only if u < CR, P(u)1,i = P (υ) 1,i

    4 else P(u)1,i = P (0) 1,i

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    Differential Evolution Optimal parameter estimation of CAPM using LMS estimators

    Generate new candidate solutions from the initial one Step 3a: Differential mutation

    P(0)1,i =

     1 2 ... r3 ... r1 ... r2 ... np  P(υ)2,i = P

    (0) 2,r1

    + F × (P(0)2,r2 - P (0) 2,r3

    )

    F scale factor ∈ [0, 1+] determines speed of shrinkage

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    Introduction Heuristic Optimization Techniques Financial Applications Conclusion

    Differential Evolution

    ... ... u1 ud

    CR CR

    uniform crossover