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Heuristic Optimization Strategies inFinance - An Overview

Marianna Lyra

COMISEF Fellows’ Workshop: Numerical Methods and Optimization in Finance

Financial support from the EU Commission through COMISEF is gratefullyacknowledged

June 18, 2009

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Marianna LyraEarly Stage Researcher - COMISEFResearch Network

1 BA Business Administration -minor Finance, UCY

2 MSc Finance, UCY3 PhD student JLU Giessen, DE4 Research Interests: Apply

Optimization Heuristics inFinance (Credit RiskManagement)

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

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

1 IntroductionFinancial WorldComplexity

2 Heuristic Optimization TechniquesClassical conceptDifferential Evolution

3 Financial ApplicationsOptimization heuristic strategies in finance

4 ConclusionHave in mind

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

1 IntroductionFinancial WorldComplexity

2 Heuristic Optimization TechniquesClassical conceptDifferential Evolution

3 Financial ApplicationsOptimization heuristic strategies in finance

4 ConclusionHave 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 IntroductionFinancial WorldComplexity

2 Heuristic Optimization TechniquesClassical conceptDifferential Evolution

3 Financial ApplicationsOptimization heuristic strategies in finance

4 ConclusionHave 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 do3: Select xn ∈ N (xc) (neighbor to current solution)4: if f (xn) < f (xc) then5: xc = xn

6: end if7: end while

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Classical concept

Heuristic Techniques

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_ _ _ _ _ _Construction methods Local search methods

Trajectory methodsmmmmmm

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Thresholdaccepting

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Differentialevolution

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Tabusearch

Geneticalgorithms

Hybrid Meta Heuristics

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Classical concept

Heuristic Techniques

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

Trajectory methodsnnnnnnn

wwnnnnnn Population search

PPPPPPP

((PPP

Thresholdaccepting

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Differentialevolution

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Tabusearch

Geneticalgorithms

Hybrid Meta Heuristics

99dd

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionPopulation based heuristic with remarkable performance incontinuous numerical problems.

Example

Capital Asset Pricing Model (CAPM): Risk-returnequilibrium.

ri,t − r st = α + β(rm,t − r s

t ) + εi,t (1)

Least Median of Squares Estimators (LMS) (Rousseeuwand Leroy (1987)):

minα,β

(med(ε2i,t)) (2)

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM usingLMS estimators

1 Generate random set values α and β (initialsolutions)

2 Evaluate initial solutions minimizing LMS3 Generate new candidate solutions from the

initial one4 Evaluate new candidate solutions minimizing

LMS5 Repeat until a very good solution is found

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM usingLMS estimators

1 Generate random set values α and β (initialsolutions)

2 Evaluate initial solutions minimizing LMS3 Generate new candidate solutions from the

initial one4 Evaluate new candidate solutions minimizing

LMS5 Repeat until a very good solution is found

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM usingLMS estimators

1 Generate random set values α and β (initialsolutions)

2 Evaluate initial solutions minimizing LMS3 Generate new candidate solutions from the

initial one4 Evaluate new candidate solutions minimizing

LMS5 Repeat until a very good solution is found

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM usingLMS estimators

1 Generate random set values α and β (initialsolutions)

2 Evaluate initial solutions minimizing LMS3 Generate new candidate solutions from the

initial one4 Evaluate new candidate solutions minimizing

LMS5 Repeat until a very good solution is found

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM usingLMS estimators

1 Generate random set values α and β (initialsolutions)

2 Evaluate initial solutions minimizing LMS3 Generate new candidate solutions from the

initial one4 Evaluate new candidate solutions minimizing

LMS5 Repeat until a very good solution is found

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM using LMS estimators

InitializationStep 1 & 2: Generate and evaluate random set valuesα[−1, 2] and β[0, 1] (initial solutions)

P(0)=

d

12

np

minα,β(med(ε2

i,t))

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM using LMS estimators

InitializationStep 1 & 2: Generate and evaluate random set valuesα[−1, 2] and β[0, 1] (initial solutions)

P(0)=

d

12

np

minα,β(med(ε2

i,t))

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM using LMS estimators

Generate new candidate solutions from the initial oneStep 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 crossoverStep 3b: Crossover elements fromP(0)

1,i and P(υ)1,i

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

2 Determine the crossoverprobability, 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 EvolutionOptimal parameter estimation of CAPM using LMS estimators

Generate new candidate solutions from the initial oneStep 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

...u1

ud

P(u)1,i

. . . P(u)d ,i

Random crossoverStep 3b: Crossover elements fromP(0)

d ,i and P(υ)d ,i

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

2 Determine the crossoverprobability, CR ∈ [0, 1]

3 Only if u < CR, P(u)d ,i = P(υ)

d ,i

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

d ,i

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM using LMS estimators

InitializationStep 4: Evaluate new candidate solutions minimizing LMS

if f (P(u).,i ) < f (P(0)

.,i ) then f (P(0).,i ) = f (P(u)

.,i )

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM using LMS estimators

InitializationStep 4: Evaluate new candidate solutions minimizing LMS

if f (P(u).,i ) < f (P(0)

.,i ) then f (P(0).,i ) = f (P(u)

.,i )

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM using LMS estimators

InitializationStep 5: Repeat steps 3 & 4 until a good solution is found orfor a predefined number

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Differential Evolution

Differential EvolutionOptimal parameter estimation of CAPM using LMS estimators

InitializationStep 5: Repeat steps 3 & 4 until a good solution is found orfor a predefined number

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

1 IntroductionFinancial WorldComplexity

2 Heuristic Optimization TechniquesClassical conceptDifferential Evolution

3 Financial ApplicationsOptimization heuristic strategies in finance

4 ConclusionHave in mind

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

VAR

Transaction Costs

Cardinality constraints

Index tracking

Model estimation

Model selectionFinancial forecasting

Portfolio improvement

Mutual funds style

Portfolio OptimizationRobust methods

Clustering

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

VAR

Transaction Costs

Cardinality constraints

Index tracking

Heuristics

Model estimation

Model selectionFinancial forecasting

Portfolio improvement

Mutual funds style

Portfolio OptimizationRobust methods

Clustering

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Portfolio SelectionDueck and Winker (1992) appliedThreshold Accepting

1 Portfolio optimization usingvarious risk measures

2 Index tracking to mutual fundreplication

3 Currency portfolio optimization

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Portfolio SelectionDueck and Winker (1992) appliedThreshold Accepting

1 Portfolio optimization usingvarious risk measures

2 Index tracking to mutual fundreplication

3 Currency portfolio optimization

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Portfolio SelectionDueck and Winker (1992) appliedThreshold Accepting

1 Portfolio optimization usingvarious risk measures

2 Index tracking to mutual fundreplication

3 Currency portfolio optimization

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Model estimation1 Risk estimation and GARCH models2 Indirect estimation and Agent Based Models3 Yield curve estimation

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Model estimation1 Risk estimation and GARCH models2 Indirect estimation and Agent Based Models3 Yield curve estimation

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Model estimation1 Risk estimation and GARCH models2 Indirect estimation and Agent Based Models3 Yield curve estimation

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Yield curve estimation

Fig. 6. Error in the estimation of the interest rates. Comparison of the traditional non-linear least squares and the GA for the Nelson and Siegel (1987) on

the left and for the Svensson (1994) function on the right for 1 year (top graph) and 10 years (bottom graph) government bonds.

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Model selection1 Risk factor selection (Asset Pricing Theory model (APT))

ri,t = α +k∑

f=1

βf rf ,t + εi,t (3)

2 Selection of bankruptcy predictors

BMW assets = 0.8German factor + 0.2US factor+0.9automotive factor + 0.1finance factor

+BMW non − systematic risk (4)

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Model selection1 Risk factor selection (Asset Pricing Theory model (APT))

ri,t = α +k∑

f=1

βf rf ,t + εi,t (3)

2 Selection of bankruptcy predictors

BMW assets = 0.8German factor + 0.2US factor+0.9automotive factor + 0.1finance factor

+BMW non − systematic risk (4)

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Clustering1 Bankruptcy prediction2 Credit risk rating3 Portfolio performance improvement4 Identify optimal number of clusters

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Clustering1 Bankruptcy prediction2 Credit risk rating3 Portfolio performance improvement4 Identify optimal number of clusters

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Clustering1 Bankruptcy prediction2 Credit risk rating3 Portfolio performance improvement4 Identify optimal number of clusters

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Optimization heuristic strategies in finance

Clustering1 Bankruptcy prediction2 Credit risk rating3 Portfolio performance improvement4 Identify optimal number of clusters

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

1 IntroductionFinancial WorldComplexity

2 Heuristic Optimization TechniquesClassical conceptDifferential Evolution

3 Financial ApplicationsOptimization heuristic strategies in finance

4 ConclusionHave in mind

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Have in mind

Heuristic optimization techniquesFlexible to tackle many complex optimization problems

Flexibility costMight be computationally more demanding than traditionalmethods (not a limitation nowadays)Results carefully interpreted

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Introduction Heuristic Optimization Techniques Financial Applications Conclusion

Have in mind

SummaryApply Optimization Heuristics