Meta-Heuristics in Finance - Dietmar Maringer Computational Economics and Finance Business and Economics Faculty, University of Basel SAMSI ...
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Meta-Heuristics in Finance Dietmar Maringer Computational Economics and Finance Business and Economics Faculty, University of Basel SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop March 8-12, 2021
Contents 3 Trading and Asset Management 3.1 portfolio: combination of assets . . . . . . . . . . . . . . . . . 6 6 3.2 portfolio optimization . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 sample-based portfolio optimization . . . . . . . . . . . . . . . 11 3.4 Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . 12 1 where’s the problem? 3 4 Algorithmic and Autotrading 17 2 modeling price processes and risk 4 4.1 some types of algorithmic trading systems . . . . . . . . . . . . 17 4.2 machine learning and meta-heuristics techniques . . . . . . . . 18 2.1 example: modeling risk . . . . . . . . . . . . . . . . . . . . . . 4 2.2 further examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 Conclusion 21 Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 2 March 8-12, 2021
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 Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 3 March 8-12, 2021
2. modeling price processes and risk 2.1. example: modeling risk • stylized fact: volatility clustering in returns • Engle (1982): ARCH( ), Bollerslev (1986): GARCH( , ) = + where ∼ (0, 2 ) ∑︁ ∑︁ 2 2 2 and = 0 + − + − • estimating the parameters = [ , , ] with log-likelihood 2 ∗ 1 ∑︁ 2 b = argmax L = − ln(2 ) − ) + 2 ln(b 2 2 =1 b • Chris Brooks, Simon P. Burke, and Gita Persand (2001). “Benchmarks and the Accuracy of GARCH Model Estimation”. In: International Journal of Forecasting 17.1, pp. 45–56 ⊲ estimate GARCH(1,1) parameters [ , 0 , 1 , 1 ] with nine standard software packages ⊲ results differ noticeably • heuristic optimization can easily solve this problem Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 4 March 8-12, 2021
2.2. further examples
• model estimation e.g., Nelson-Siegel-Svensson for yield of bond with time-to-maturity at :
) 2 )
min (( − b
,
1− −
( ) = 1, + 2, ( / 1 ) + 3, ( / 1 ) − − / 1 + 4, ( / 2 ) − − / 2 where ( ) =
b
• model selection challenging combinatorial problems, e.g., factor (or asset) selection:
) 2 ) :
min (( − b
F ,
∑︁
=
b s.t. F ⊂ M, ∈ D, and ♯F ≤ ♯M
∈F
• joint estimation and selection problem: Vector Error Correction model (VECM) for interest rates:
∑︁
Δ = + −1 + Δ − + with Δ − = − − −−1
=1
11 ··· 1 Δ 1, −1−
1 11 ··· 1 1, −1 ∑︁ " #
1
. .
= .. + .. . . ..
. . .
.. + . .
.. . . .. . .
.. + ...
1 ··· , −1 =1 1 ··· Δ , −1−
Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 5
March 8-12, 20213. Trading and Asset Management 3.1. portfolio: combination of assets • assets; return of asset are normally distributed ⊲ stock price: = 0 · (1 + ) ⊲ returns: expectations = ( ) , standard deviations (“volatility”) and covariances ⊲ estimates based on past observations [ ] and/ or models • portfolio Í ⊲ 0 , value at beginning, = 0: stocks =⇒ 0 = 0, ⊲ , value at end of investment period: ∑︁ ∑︁ ∑︁ 0 ∑︁ = , = 0 (1 + , ) = (1 + , ) = (1 + , ) = 1 + 0 0 hence, ∑︁ √︄∑︁ ∑︁ √ ( ) = ( ) = = 2 2 + = 0 ≠ 0, ⊲ asset weight: fraction of wealth invested in asset , = 0 Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 6 March 8-12, 2021
3.2. portfolio optimization 3.2.1. the problem • objective: find composition that meets investors’ and regulators’ requirements the best ⊲ decision variables: ⊲ risk-adjusted performance ⊲ limitations on composition of portfolio 3.2.2. traditional approach: mean-variance portfolios • Harry Markowitz: achieve target return with minimum risk: ∑︁ min 2 = 0 s.t. ( ) = ≥ ∗ , ≥ 0, = 1 max ( − · 0 ) s.t. 1 = 1 and ≥ 0 for given ≥ 0 =⇒quadratic programming problem • James Tobin, William Sharpe: include safe asset (return ) and maximize “Sharpe ratio” ( ) − max = • negative positions, < 0: “short selling” Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 7 March 8-12, 2021
3.2.3. extensions
limitations on selection of assets,
• integer contraints on ∈ {0, 1, 2, ...} =⇒discrete weights = 0 / 0
(
= 0 asset not included
• avoid tiny positions:
≥ asset included with (more than) minimum amount
Í
• (explicit) cardinality constraint: 1 ≠0 ≤
additional costs and effects on performance
• additional transaction costs, management fees, bid-ask spreads: · | | · 0 + · 1 ≠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), − 2 : justifies mean-variance
(
ln( ) for = 1
⊲ more realistically: e.g., power utility with ≥ 0, ( , ) =
( 1− − 1)/(1 − ) otherwise
• loss aversion (Kahnemann & Tversky): not just level of wealth, but also relative to initial
endowment
Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 8
March 8-12, 2021distribution 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, = −( − 0 ) : ⊲ “Value-at-Risk” (VaR): loss that will be exceeded only with probability , prob( > ) = ⊲ “conditional Value-at-Risk” (cVaR), ≈ “Expected Shortfall” (ES): ( | > ) ⊲ in portfolio optimization: » constraint: equity requirements based VaR (regulations) » objective: minimize for giving VaR, minimize VaR for given ∫∞ (1− ( ))d (max( − ,0)) • Keating and Shadwick: Ω = ∫ =⇒ max − (min( − ,0)) −∞ ( )d Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 9 March 8-12, 2021
3.2.4. a quick reality check • Value-at-Risk under normal and empirical distribution • three assets, daily returns Í • constraints: ≥ 0 for = 1, 2 = 1 ⇔ 3 = 1 − 1 − 2 Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 10 March 8-12, 2021
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 investment period ⊲ find portfolio / strategy that works best given criteria / objectives / constraints 3.3.2. how to solve it • data: samples for assets, ; objective function: ( | , ) × • decision variables: =⇒(in population-based method with individuals): × • constraints: ⊲ ideally: repair function, = R ( ) s.t. ∈ D ⊲ occasionally: punishment function Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 11 March 8-12, 2021
3.4. Differential Evolution 3.4.1. basic idea • suggested by Storn and Price (1995) • continuous, dimensional search space • population size of vectors, representing positions within search space = [ ], = 1 . . . , = 1 . . . • evolution: repeat creation + tournament: ⊲ generate another solutions (“offspring”) , = 1 . . . ⊲ for each current solution , pick one offspring ? ⊲ compare fitness values of current and new solutions: ( ) ≷ ( ) ⊲ the one with higher fitness survives ( ) < ( ) =⇒ := Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 12 March 8-12, 2021
• creating new candidates current population (candidate solutions) ⊲ take one existing vector, 1 , and modify it new candidate solution Minimum ⊲ perturbations: vector weighted difference of two other (x 2 − x 3 ) solutions, 2 − 3 x 3 x 2 x 1 ⊲ cross-over: elements of 4th solution ỹ = x 1 + · (x 2 − x 3 ) ( ˜ [ ] = 1 [ ] + · ( 2 [ ] − 3 [ ]) with prob. [ ] := [ ] otherwise Algorithm 1: DE pseudo-code for maximization and continuous search space 1 set: population size , scaling factor , cross-over probability , halting criterion, etc. 2 initialize: for = 1.. ; 3 repeat 4 foreach parent do 5 randomly select individuals 1 , 2 , 3 6 ˜ := 1 + · ( 2 − 3 ) create linear combination 7 ˜ , , ) create new offspring: := cross-over( 8 foreach offspring do 9 if ( ) > ( ) then 10 := 11 until halting criterion met; 12 report elitist ∗ = argmax ( ) =1: Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 13 March 8-12, 2021
3.4.2. extended versions • add extra noise with ( ) = 0: ⊲ jitter: extra noise added to , ˜1 [ ] ⊲ extra noise added to difference vector, ˜2 [ ] ( 1 [ ] + ( +˜ 1 [ ]) · ( 2 [ ] − 3 [ ]+˜ 2 [ ]) with prob. [ ] := [ ] otherwise • additional difference vectors ⊲ e.g., using two current solutions, one current member and the elitist, ... • use elitist as base-vector 1 • for more details, see, e.g., Price, Storn, and Lampinen 2005 or Brabazon, O’Neill, and McGarraghy 2015, §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 Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 14 March 8-12, 2021
3.4.4. (very) basic version of DE for maximization 1 import numpy as np 2 import numpy.random as rd 3 4 def DEmax(f, x0, repair=lambda x:x, 5 popSize=30, FE=10000, F=0.7, probXO=.5): 6 # initialize 7 D = x0.shape[0] # provide x0 with shape (D,) 8 xC = rd.randn(popSize,D) 9 10 xC = repair(xC) 11 fC = np.array( [f(xC[p]) for p in range(popSize)] ) 12 fE = np.min(fC) 13 14 # evolve 15 for gen in range(int(FE/popSize)): 16 # linear combination 17 p = [rd.permutation(popSize) for i in range(3)] 18 xN = xC[p[0]] + F * (xC[p[1]] − xC[p[2]]) 19 20 #cross−over 21 XO = rd.rand(popSize,D)=fC 28 fC[improved] = +fN[improved] 29 xC[improved] = +xN[improved] 30 31 i = np.argmax(fC) 32 return xC[i] Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 15 March 8-12, 2021
3.4.5. VaR optimization • constraint on asset weights: Í ⊲ repair function for ≥ , = 1 1 def weight_repair(w,wL=0.0) −> np.array: # lower bound for weights: wL 2 w = w.T 3 while True: 4 w /= np.sum(w,0) 5 below = w
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, EUR, € but profit in some ⊲ prices are out of equilibrium, e.g., currencies USD, $ GBP, £ ⊲ different prices in different markets for same asset • 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) Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 17 March 8-12, 2021
4.2. machine learning and meta-heuristics techniques 4.2.1. setup • data: prices, factors, indices, news, etc. ⊲ sufficiently 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 term mem- ory, anfis, ...) to get trading signal • rule generation ⊲ e.g., genetic programming, grammatical evolution to generate trading and / or forecasting rules Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 18 March 8-12, 2021
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 ( ) = sin( ) + 4 ( ) = 3 ∗ (2 + ) ( ) = 4 ∗(2 + ) + * * sin ÷ 3 + ÷ + 4 2 4 2 ⊲ 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 > L ITE Single tree 1bp, in−sample cumulated profits Single tree 1bp, out−of−sample cumulated profits 0.8 0.3 MA + < S N Cumulated profits 0.6 0.2 Cumulated profits 0.4 0.1 C L1 MA K MA − 0.2 0 0 −0.1 C L2 C L1 MA K −0.2 −0.2 −0.4 −0.3 −0.4 C L2 08/07/03 02/23/04 09/10/04 03/29/05 10/15/05 05/03/06 08/11/06 11/19/06 02/27/07 06/07/07 Date Date average performance, results from individual runs, 95% and 99% confidence intervals © D Maringer, 4303110CR Numerical Optimisation and Simulation Learning (55) Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 19 March 8-12, 2021
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 and
terminals)
⊲ genotype [0 0 1 . . .]
[0 0 1]
[0 0 1] x
⊲ grammar:
::= 0 | ( ) 1 ⊲ genotype [1 0 0 2 0 3 1 . . .]
::= x 0 | y 1
[1 0 0 2 0 3 1 . . .] ( )
::= + 0 | - 1 | * 2 | ÷3 [1 0 0 2 0 3 1 . . .] ( )
[1 0 0 2 0 3 1 . . .] ( x )
[1 0 0 2 0 3 1 . . .] ( x * )
[1 0 0 2 0 3 1 . . .] ( x * )
[1 0 0 2 0 3 1 . . .] ( x * y)
• examples in finance:
⊲ create models for pricing and price impact, trading rules, etc.
Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 20
March 8-12, 20215. Conclusion • some literature to get started ⊲ Manfred Gilli, Dietmar Maringer, and Enrico Schumann (2019). Numerical Methods and Optimization 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 Computing Algorithms. Springer ⊲ Dietmar Maringer (2005). Portfolio Management with Heuristic Optimization. Advances in Computational Management Science. Springer, Boston, MA. doi: https://doi.org/10.1007/b136219. url: https://doi.org/10.1007/b136219 Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 21 March 8-12, 2021
Bibliography Maringer, Dietmar (2005). Portfolio Management with Heuristic Optimization. Advances in Computational Management Science. Springer, Boston, MA. doi: Brabazon, Anthony, Michael O’Neill, and Séan McGarraghy https://doi.org/10.1007/b136219. url: (2015). Natural Computing Algorithms. Springer. https://doi.org/10.1007/b136219. Brooks, Chris, Simon P. Burke, and Gita Persand (2001). Poli, Riccardo, William B. Langdon, and Nicholas F. McPhee “Benchmarks and the Accuracy of GARCH Model (2008). A Field Guide to Genetic Programming. Estimation”. In: International Journal of Forecasting 17.1, http://www.gp-field-guide.org.uk/. url: pp. 45–56. http://www.gp-field-guide.org.uk/. Gilli, Manfred, Dietmar Maringer, and Enrico Schumann (2019). Price, Kenneth V., Rainer M. Storn, and Jouni A. Lampinen Numerical Methods and Optimization in Finance. 2nd (2005). Differential Evolution: A Practical Approach to edition. Academic Press. url: Global Optimization. Springer. https://www.elsevier.com/books/numerical-methods-and- Ryan, Conor, J.J. Collins, and Michael O’Neill (1998). optimization-in-finance/gilli/978-0-12-815065-8. “Grammatical Evolution: Evolving Programs for an Koza, John R. (1990). Genetic Programming: A Paradigm for Arbitrary Language”. In: Lecture Notes in Computer Genetically Breeding Populations of Computer Programs Science 1391. First European Workshop on Genetic to Solve Problems. Tech. rep. STAN-CS-90-1314. Stanford Programming 1998. url: University Computer Science Department. url: http://www.grammatical-evolution.org/papers/eurogp98.ps. http://www.genetic-programming.com/jkpdf/tr1314.pdf. Dietmar Maringer, Meta-Heuristics in Finance, SAMSI Metaheuristic Optimization, Machine Learning and AI Virtual Workshop 22 March 8-12, 2021
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