meta-heuristics in finance · 2021. 3. 5. · 3.4 di˘erential evolution. . . . . . . . . . . . . ....

Meta-Heuristics in FinanceDietmar MaringerComputational Economics and FinanceBusiness and Economics Faculty, University of Basel

SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual WorkshopMarch 8-12, 2021

Contents

1 where’s the problem? 3

2 modeling price processes and risk 4

2.1 example: modeling risk . . . . . . . . . . . . . . . . . . . . . . 4

2.2 further examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Trading and Asset Management 63.1 portfolio: combination of assets . . . . . . . . . . . . . . . . . 63.2 portfolio optimization . . . . . . . . . . . . . . . . . . . . . . . 73.3 sample-based portfolio optimization . . . . . . . . . . . . . . . 113.4 Dierential Evolution . . . . . . . . . . . . . . . . . . . . . . . 12

4 Algorithmic and Autotrading 174.1 some types of algorithmic trading systems . . . . . . . . . . . . 174.2 machine learning and meta-heuristics techniques . . . . . . . . 18

5 Conclusion 21

Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual WorkshopMarch 8-12, 2021

2

1. where’s the problem?

• typical tasks

⊲ investment decisions

⊲ describing, modelling, (and predicting) risky assets

⊲ dealing with requirements (regulators, investors)

• data: “weather forecast” versus “climate change”

⊲ (ultra-)high frequency: highly liquid assets

» latency (under colocation): < 1 microsecond» relevant for certain types of algo-trading strategies

⊲ low frequency: daily prices or lower

» aggregate over longer stretches of time to reduce noise» thin trading: funds, bonds, ...

⊲ challenges

» statistical / stylized facts» scaling laws» choice of model

• need for meta-heuristics

⊲ non-convex models, challenging contraints


3

2. modeling price processes and risk

2.1. example: modeling risk

• stylized fact: volatility clustering in returns

• Engle (1982): ARCH(𝑞), Bollerslev (1986): GARCH(𝑞, 𝑝)

𝑟𝑡 = ` + 𝑒𝑡 where 𝑒𝑡 ∼ 𝑁 (0, 𝜎2𝑡 )and 𝜎 2𝑡 = 𝛼0 +

∑𝑞

𝛼𝑞Y2𝑡−𝑞 +

∑𝑝

𝛽𝑝𝜎2𝑡−𝑝

• estimating the parameters 𝝍 = [`, 𝜶, 𝜷] with log-likelihood

𝝍∗ = argmax𝝍

L = −𝑇2ln(2𝜋) − 1

2

𝑇∑𝑡=1

(ln(�� 2𝑡 ) +

��2𝑡

�� 2𝑡

)• Chris Brooks, Simon P. Burke, and Gita Persand (2001). “Benchmarks and the Accuracy of GARCHModel Estimation”. In: International Journal of Forecasting 17.1, pp. 45–56

⊲ estimate GARCH(1,1) parameters [`, 𝛼0, 𝛼1, 𝛽1] with nine standard software packages⊲ results dier noticeably

• heuristic optimization can easily solve this problem


4

2.2. further examples• model estimation e.g., Nelson-Siegel-Svensson for yield of bond with time-to-maturity 𝜏 at 𝑡:min𝜷,𝝀

𝐸(( 𝑦𝑡 − ��𝑡)2)

��𝑡 (𝜏) = 𝛽1,𝑡 + 𝛽2,𝑡𝐹 (𝜏/_1) + 𝛽3,𝑡

(𝐹 (𝜏/_1) − 𝑒−

𝜏/_1)+ 𝛽4,𝑡

(𝐹 (𝜏/_2) − 𝑒−

𝜏/_2)where 𝐹 (𝛾) = 1−𝑒−𝛾

𝛾

• model selection challenging combinatorial problems, e.g., factor (or asset) selection:minF ,𝒃

𝐸((𝑟 − ��)2):

�� =∑𝑓∈F

𝛽𝑓 𝑟𝑓 s.t. F ⊂ M , 𝜷 ∈ D , and ♯F ≤ 𝑘 � ♯M

• joint estimation and selection problem: Vector Error Correction model (VECM) for interest rates:

Δ𝒀 𝑡 = 𝒄 + 𝚷𝒀 𝑡−1 +𝑘∑

�=1𝚪�Δ𝒀 𝑡−� + 𝝐𝑡 with Δ𝒀 𝑡−� = 𝒀 𝑡−� − 𝒀 𝑡−�−1

=

[ 𝑐1...𝑐𝑛

]+[ 𝜋11 ··· 𝜋1𝑛.... . .

...𝜋𝑛1 ··· 𝜋𝑛𝑛

] [ 𝑦1,𝑡−1...

𝑦𝑛,𝑡−1

]+

𝑘∑�=1

[ 𝛾11� ··· 𝛾1𝑛�.... . .

...𝛾𝑛1� ··· 𝛾𝑛𝑛�

] [ Δ𝑦1,𝑡−1−�...

Δ𝑦𝑛,𝑡−1−�

]+[ 𝜖1𝑡...𝜖𝑛𝑡

]


5

3. Trading and Asset Management

3.1. portfolio: combination of assets• 𝑁 assets; return of asset are normally distributed

⊲ stock price: 𝑆𝑇 = 𝑆0 · (1 + 𝑟𝑇 )⊲ returns: expectations `𝑖 = 𝐸(𝑟𝑖), standard deviations (“volatility”) 𝜎𝑖 and covariances 𝜎𝑖 𝑗⊲ estimates based on past observations [𝑟𝑡𝑖] and/ or models

• portfolio

⊲ 𝑃0, value at beginning, 𝑡 = 0: 𝑛𝑖 stocks 𝑖 =⇒𝑃0 =∑

𝑖 𝑛𝑖𝑆0,𝑖

⊲ 𝑃𝑇 , value at end of investment period:

𝑃𝑇 =∑𝑖

𝑛𝑖𝑆𝑇,𝑖 =∑𝑖

𝑛𝑖𝑆0𝑖(1 + 𝑟𝑇,𝑖)𝑃𝑇

𝑃0= (1 + 𝑟𝑇,𝑝) =

∑𝑖

𝑛𝑖𝑆0𝑖

𝑃0(1 + 𝑟𝑇,𝑖) = 1 +

∑𝑖

𝑤𝑖𝑟𝑖

hence,

𝐸(𝑟𝑝) =∑𝑖

𝑤𝑖𝐸(𝑟𝑖) = 𝝁𝒘 𝜎𝑝 =

√∑𝑖

𝑤2𝑖𝜎 2𝑖+∑𝑖≠𝑗

𝑤𝑖𝑤𝑗𝜎𝑖 𝑗 =√𝒘′𝚺𝒘

⊲ asset weight: fraction of wealth invested in asset 𝑖, 𝑤𝑖 =𝑛𝑖𝑆0,𝑖

𝑃0


6

3.2. portfolio optimization

3.2.1. the problem• objective: find composition that meets investors’ and regulators’ requirements the best

⊲ decision variables: 𝑤𝑖

⊲ risk-adjusted performance

⊲ limitations on composition of portfolio

3.2.2. traditional approach: mean-variance portfolios

• Harry Markowitz: achieve target return with minimum risk:

min𝒘

𝜎 2𝑝 = 𝒘′𝚺𝒘 s.t. 𝐸(𝑟𝑝) = 𝝁𝒘 ≥ 𝑟∗, 𝑤𝑖 ≥ 0,∑𝑖

𝑤𝑖 = 1

max𝒘

(𝝁𝒘 − 𝛾 ·𝒘′𝚺𝒘) s.t. 1𝒘 = 1 and𝒘 ≥ 0

for given 𝛾 ≥ 0 =⇒quadratic programming problem

• James Tobin, William Sharpe: include safe asset (return 𝑟𝑓 ) and maximize “Sharpe ratio”

max 𝑆𝑅 =𝐸(𝑟𝑝) − 𝑟𝑓

𝜎𝑝

• negative positions, 𝑤𝑖 < 0: “short selling”


7

3.2.3. extensions

limitations on selection of assets, 𝑤𝑖

• integer contraints on 𝑛𝑖 ∈ {0, 1, 2, ...} =⇒discrete weights 𝑤𝑖 = 𝑛𝑖𝑆0𝑖/𝑃0

• avoid tiny positions:

{𝑤𝑖 = 0 asset 𝑖 not included

𝑤𝑖 ≥ 𝑤� asset 𝑖 included with (more than) minimum amount

• (explicit) cardinality constraint:∑

𝑖 1𝑤𝑖≠0 ≤ 𝑘

additional costs and eects on performance

• additional transaction costs, management fees, bid-ask spreads: 𝑐𝑣 · |𝑛𝑖 | · 𝑆0𝑖 + 𝑐𝑓 · 1𝑛𝑖≠0

• market impact: large order moves price (in non-linear fashion and by uncertain amount)

investors’ preferences

• risk aversion: higher wealth has more utility, but at decreasing rate

⊲ quadratic utility, (e.g., in Markowitz), 𝑤 − 𝑏𝑤2: justifies mean-variance

⊲ more realistically: e.g., power utility with 𝛾 ≥ 0, 𝑢(𝑤, 𝛾) ={ln(𝑤) for 𝛾 = 1(𝑤1−𝛾 − 1)/(1 − 𝛾) otherwise

• loss aversion (Kahnemann & Tversky): not just level of wealth, but also relative to initialendowment


8

distribution of returns

• stock returns typically exhibit negative skewness, excess kurtosis

• non-linear dependencies and correlations

additionale / alternative risk- and performance measures

• tail-risk, focusing on losses, 𝐿 = −(𝑃𝑇 − 𝑃0):

⊲ “Value-at-Risk” (VaR): loss that will be exceeded only with probability 𝛼,prob(𝐿 > 𝑉𝑎𝑅𝛼) = 𝛼

⊲ “conditional Value-at-Risk” (cVaR), ≈ “Expected Shortfall” (ES): 𝐸(𝐿|𝐿 > 𝑉𝑎𝑅𝛼)⊲ in portfolio optimization:

» constraint: equity requirements based VaR (regulations)» objective: minimize 𝛼 for giving VaR, minimize VaR for given 𝛼

• Keating and Shadwick: Ω𝜏 =

∫ ∞𝜏

(1−𝐹 (𝑟))d∫ 𝜏

−∞ 𝐹 (𝑟)d =⇒ max𝒘𝐸(max(𝑟𝑝−𝜏,0))−𝐸(min(𝑟𝑝−𝜏,0))


9

3.2.4. a quick reality check• Value-at-Risk under normal and empirical distribution

• three assets, daily returns

• constraints: 𝑤𝑖 ≥ 0 for 𝑖 = 1, 2∑

𝑖 𝑤𝑖 = 1 ⇔ 𝑥3 = 1 − 𝑥1 − 𝑥2


10

3.3. sample-based portfolio optimization

3.3.1. concept• back-testing, historical simulation:

⊲ get sample based on assets’ past observations

⊲ compute how portfolio would have performed in the past

⊲ evaluate strategy on past observations

• sample-based optimization:

⊲ create (large) sample for assets’ returns that exhibits properties assumed for investmentperiod

⊲ find portfolio / strategy that works best given criteria / objectives / constraints

3.3.2. how to solve it• data: 𝑆 samples for 𝑁 assets, 𝑹

𝑆×𝑁; objective function: 𝑉𝑎𝑅(𝒘|𝑹, 𝛼)

• decision variables: 𝒘 =⇒(in population-based method with 𝑃 individuals): 𝒘𝑁×𝑃

• constraints:

⊲ ideally: repair function,𝒘𝑓 = R(𝒘) s.t. 𝒘𝑓 ∈ D⊲ occasionally: punishment function


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3.4. Dierential Evolution

3.4.1. basic idea• suggested by Storn and Price (1995)

• continuous, 𝑛 dimensional search space

• population size of 𝑃 vectors, representing positions within search space

𝒙𝑝 = 𝑥𝑝 [𝑖] , 𝑖 = 1 . . . 𝑛, 𝑝 = 1 . . . 𝑃

• evolution: repeat creation + tournament:

⊲ generate another 𝑃 solutions (“ospring”) 𝒚𝑞, 𝑞 = 1 . . . 𝑃

⊲ for each current solution 𝒙𝑝, pick one ospring 𝒚𝑞

⊲ compare fitness values of current and new solutions: 𝑓 (𝒚𝑞)?≷ 𝑓 (𝒙𝑝)

⊲ the one with higher fitness survives

𝑓 (𝒙𝑝) < 𝑓 (𝒚𝑞) =⇒ 𝒙𝑝 := 𝒚𝑞


12

• creating new candidates

⊲ take one existing vector, 𝒙𝑚1 , and modify it

⊲ perturbations: vector weighted dierence of two othersolutions, 𝒙𝑚2 − 𝒙𝑚3

⊲ cross-over: elements of 4th solution 𝑝

𝑦𝑞 [𝑖] :={��𝑞 [𝑖] = 𝑥𝑚1 [𝑖] + 𝐹 · (𝑥𝑚2 [𝑖] − 𝑥𝑚3 [𝑖]) with prob. 𝜋

𝑥𝑝 [𝑖] otherwise

Minimum

current population (candidate solutions)

x𝑚3x𝑚2 x𝑚1

(x𝑚2 − x𝑚3 )

y𝑞 = x𝑚1 + 𝐹 · (x𝑚2 − x𝑚3 )

new candidate solution

Algorithm 1: DE pseudo-code for maximization and continuous search space1 set: population size 𝑃, scaling factor 𝐹, cross-over probability 𝜋 , halting criterion, etc.2 initialize: 𝒙𝑝 for 𝑝 = 1..𝑃;3 repeat4 foreach parent 𝑝 do5 randomly select individuals 𝑚1, 𝑚2, 𝑚36 create linear combination ��𝑝 := 𝒙𝑚1 + 𝐹 · (𝒙𝑚2 − 𝒙𝑚3 )7 create new ospring: 𝒚𝑝 := cross-over( ��𝑝, 𝒙𝑝, 𝜋 )

8 foreach ospring 𝑝 do9 if 𝑓 (𝒚𝑝) > 𝑓 (𝒙𝑝) then10 𝒙𝑝 := 𝒚𝑝

11 until halting criterion met;12 report elitist 𝒙∗ = argmax

𝑝=1:𝑃𝑓 (𝒙𝑝)


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3.4.2. extended versions• add extra noise with 𝐸(𝒛) = 0:

⊲ jitter: extra noise added to 𝐹, 𝑧1 [𝑖]⊲ extra noise added to dierence vector, 𝑧2 [𝑖]

𝑦𝑞 [𝑖] :={𝑥𝑚1 [𝑖] + (𝐹+𝑧1 [𝑖]) · (𝑥𝑚2 [𝑖] − 𝑥𝑚3 [𝑖]+𝑧2 [𝑖]) with prob. 𝜋

𝑥𝑝 [𝑖] otherwise

• additional dierence vectors

⊲ e.g., using two current solutions, one current member and the elitist, ...

• use elitist as base-vector 𝒙𝑚1

• for more details, see, e.g., Price, Storn, and Lampinen 2005 or Brabazon, O’Neill, and McGarraghy2015, §6 and the literature quoted therein

3.4.3. constraints• repair function to ensure candidate is feasible

• map candidate to feasible space during function evaluation

• punishment term deteriorates OF when candidate is infeasible


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3.4.4. (very) basic version of DE for maximization

1 import numpy as np2 import numpy.random as rd34 def DEmax(f, x0, repair=lambda x:x,5 popSize=30, FE=10000, F=0.7, probXO=.5):6 # initialize7 D = x0.shape[0] # provide x0 with shape (D,)8 xC = rd.randn(popSize,D)910 xC = repair(xC)11 fC = np.array( [f(xC[p]) for p in range(popSize)] )12 fE = np.min(fC)1314 # evolve15 for gen in range(int(FE/popSize)):16 # linear combination17 p = [rd.permutation(popSize) for i in range(3)]18 xN = xC[p[0]] + F * (xC[p[1]] − xC[p[2]])1920 #cross−over21 XO = rd.rand(popSize,D)<probXO22 xN[XO] = +xC[XO]2324 # evaluation and population update25 xN = repair(xN)26 fN = np.array( [f(xN[p]) for p in range(popSize)] )27 improved = fN>=fC28 fC[improved] = +fN[improved]29 xC[improved] = +xN[improved]3031 i = np.argmax(fC)32 return xC[i]


15

3.4.5. VaR optimization• constraint on asset weights:

⊲ repair function for 𝑤𝑖 ≥ 𝑤� ,∑

𝑖 𝑤𝑖 = 1

1 def weight_repair(w,wL=0.0) −> np.array: # lower bound for weights: wL2 w = w.T3 while True:4 w /= np.sum(w,0)5 below = w<wL6 if not(np.any(below)):7 break8 w[below] = wL +.5*(wL−w[below])9 w[np.abs(w)<1e−7] = 010 return w.T

• objective function: empirical VaR for simulated returns r and shortfall probability a

1 def VaR(w,r,a):2 rP = r@w3 v = np.percentile(rP,a)4 return v


16

4. Algorithmic and Autotrading

4.1. some types of algorithmic trading systems• buy and sell orders are placed by algorithm / machine

• arbitrage

⊲ self-financing portfolio that makes loss in no situation,but profit in some

⊲ prices are out of equilibrium, e.g., currencies

⊲ dierent prices in dierent markets for same asset USD, $

EUR, €

GBP, £

• executing rules

⊲ pre-specified trading rules

⊲ adjustment to market situations, e.g., iceberg orders

• statistical arbitrage: probability of loss converges to 0, prob of profit to 1

⊲ e.g., pairs trading

• self-adapting / learning systems

⊲ selection and/or combination of pre-specified rules, suitable for current situation

⊲ generation of new rules (algorithms)


17

4.2. machine learning and meta-heuristics techniques

4.2.1. setup• data: prices, factors, indices, news, etc.

⊲ suciently large sample of historical / generated data

⊲ online learning

• objective: typically some risk-adjusted performance measure

• model estimation and selection

⊲ e.g., genetic algorithms to find combination of indicators

• pattern detection

⊲ e.g., neural network (convolutional, recurrent, long short termmem-ory, anfis, ...) to get trading signal

• rule generation

⊲ e.g., genetic programming, grammatical evolution to generate trading and / or forecastingrules


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4.2.2. example: Genetic Programming• concept: + Koza 1990

⊲ “population” of (valid) lines of code / equations / rules / . . .⊲ evolve over generations by cross-over and mutation

𝑓𝑝𝑖 (𝑥) = sin(𝑥) + 𝑥4

+

sin

𝑥

÷𝑥 4

𝑓𝑝𝑗 (𝑥) = 3 ∗ (2 + 𝑥)*

3 +2 𝑥

𝑓𝑜𝑘 (𝑥) =𝑥4 ∗(2 + 𝑥)*

÷𝑥 4

+2 𝑥

⊲ supervised learning: prefer solutions based on loss function

⊲ unsupervised learning: preference based on some quality criterion

• examples in finance:

⊲ supervised: find pricing model (risk estimation model) that fit observations

⊲ unsupervised: find trading strategy that maximizes utility, Sharpe ratio, etc.ITE

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19

http://www.genetic-programming.com/jkpdf/tr1314.pdf

4.2.3. example: Grammatical Evolution• concept: + Ryan, Collins, and O’Neill 1998

• distinction genotype (list of integers) vs. phenotype (interpretation via grammar)

• formal description of valid statements, e.g., Backus-Naur form:

⊲ “grammar” G with G = {N ,T ,S,P}⊲ sets: N = non terminals, T = terminals, S = starting symbol, P = production rules

• sequence of numbers, interpreted according to grammar (“lookup table” of functions andterminals)

⊲ grammar:<expr> ::= <var> 0 | ( <expr> <op> <expr> ) 1

<var> ::= x 0 | y 1

<op> ::= + 0 | - 1 | * 2 | ÷ 3

⊲ genotype [0 0 1 . . .]<expr>

[0 0 1] <var>

[0 0 1] x

⊲ genotype [1 0 0 2 0 3 1 . . .]<expr>

[1 0 0 2 0 3 1 . . .] ( <expr> <op> <expr>)

[1 0 0 2 0 3 1 . . .] ( <var> <op> <expr>)

[1 0 0 2 0 3 1 . . .] ( x <op> <expr>)

[1 0 0 2 0 3 1 . . .] ( x * <expr>)

[1 0 0 2 0 3 1 . . .] ( x * <var>)

[1 0 0 2 0 3 1 . . .] ( x * y)

• examples in finance:

⊲ create models for pricing and price impact, trading rules, etc.


20

http://www.grammatical-evolution.org/papers/eurogp98.ps

5. Conclusion

• some literature to get started

⊲ Manfred Gilli, Dietmar Maringer, and Enrico Schumann (2019). Numerical Methods andOptimization in Finance. 2nd edition. Academic Press. url: https://www.elsevier.com/books/numerical-methods-and-optimization-in-finance/gilli/978-0-12-815065-8

⊲ Anthony Brabazon, Michael O’Neill, and Séan McGarraghy (2015). Natural ComputingAlgorithms. Springer

⊲ Dietmar Maringer (2005). Portfolio Management with Heuristic Optimization. Advances inComputational Management Science. Springer, Boston, MA. doi:https://doi.org/10.1007/b136219. url: https://doi.org/10.1007/b136219


21

https://www.elsevier.com/books/numerical-methods-and-optimization-in-finance/gilli/978-0-12-815065-8


https://doi.org/https://doi.org/10.1007/b136219

https://doi.org/10.1007/b136219

Bibliography

Brabazon, Anthony, Michael O’Neill, and Séan McGarraghy(2015). Natural Computing Algorithms. Springer.

Brooks, Chris, Simon P. Burke, and Gita Persand (2001).“Benchmarks and the Accuracy of GARCH ModelEstimation”. In: International Journal of Forecasting 17.1,pp. 45–56.

Gilli, Manfred, Dietmar Maringer, and Enrico Schumann (2019).Numerical Methods and Optimization in Finance. 2ndedition. Academic Press. url:https://www.elsevier.com/books/numerical-methods-and-

optimization-in-finance/gilli/978-0-12-815065-8.Koza, John R. (1990). Genetic Programming: A Paradigm for

Genetically Breeding Populations of Computer Programsto Solve Problems. Tech. rep. STAN-CS-90-1314. StanfordUniversity Computer Science Department. url:http://www.genetic-programming.com/jkpdf/tr1314.pdf.

Maringer, Dietmar (2005). Portfolio Management withHeuristic Optimization. Advances in ComputationalManagement Science. Springer, Boston, MA. doi:https://doi.org/10.1007/b136219. url:https://doi.org/10.1007/b136219.

Poli, Riccardo, William B. Langdon, and Nicholas F. McPhee(2008). A Field Guide to Genetic Programming.http://www.gp-field-guide.org.uk/. url:http://www.gp-field-guide.org.uk/.

Price, Kenneth V., Rainer M. Storn, and Jouni A. Lampinen(2005). Dierential Evolution: A Practical Approach toGlobal Optimization. Springer.

Ryan, Conor, J.J. Collins, and Michael O’Neill (1998).“Grammatical Evolution: Evolving Programs for anArbitrary Language”. In: Lecture Notes in ComputerScience 1391. First European Workshop on GeneticProgramming 1998. url:http://www.grammatical-evolution.org/papers/eurogp98.ps.


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http://www.genetic-programming.com/jkpdf/tr1314.pdf

https://doi.org/https://doi.org/10.1007/b136219

https://doi.org/10.1007/b136219

http://www.gp-field-guide.org.uk/

http://www.gp-field-guide.org.uk/

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meta-heuristics in finance · 2021. 3. 5. · 3.4 di˘erential evolution. . . . . . . . . . . . . ....

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