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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

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

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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

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

uulll lll

lll lll

l

))RRR RRRR

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

uulll lll

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