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Nonlinear Manifold Learning for Financial Markets Integration
George Tzagkarakis1 & Thomas Dionysopoulos1,2
1 EONOS Investment Technologies, Paris (FR)2 Dalton Strategic Partnership, London (UK)
DLM International Workshop
Nice, 4 – 6 Sept 2017
Motivation
DLM International Workshop
Nice, 4 – 6 Sep 2017
Extract meaningful information
from financial data
Build smart trading
strategies
Quants Traders
Investment process
Risk premia vs Risk factors
Manifold learning for financial data
Detection of critical transitions in financial markets
Conclusions
Overview
DLM International Workshop
Nice, 4 – 6 Sep 2017
Investment process
Risk premia vs Risk factors
Manifold learning for financial data
Detection of critical transitions in financial markets
Conclusions
Overview
DLM International Workshop
Nice, 4 – 6 Sep 2017
Investment process
DLM International Workshop
Nice, 4 – 6 Sep 2017
Views(How do I produce my market signals)
Portfolio Construction(How do I aggregate my views together to
positions)
Dynamic Budgeting(How does my portfolio evolve in time)
• MVP [MinVar Portfolio]: lowest possible risk (volatility); concentration in low volatility asset classes
• MCP [MinCorr Portfolio]: lowest volatility weighted average corrcoeff between asset classes; asset classes with low corr and volatility relative to other asset classes within the portfolio receive higher weight
• RPP [Risk Parity Portfolio]: asset classes contribute the same amount of risk (volatility) to the overall portfolio; assets with lower risk (e.g. bonds) get a larger part of the portfolio than risky ones
• Driver-based models: identify mathematical relationship between business drivers and study their financial outcomes under certain operational decisions
• Rolling forecasting: continuous look forward on N-month basis updated monthly/quarterly to adjust positions in order to reach the financial goals
Repeated pattern
Statistics vs ThresholdMovAvg, MovVol, …
{-1, 1}
Investments for the long haul
Buy-and-Hold mentality
Resist the temptation to react or predict the stock market’s every next move
Successful passive investors keep their eye on the prize (returns) and ignore short-term setbacks, even sharp downturns
Passive vs Active investing
DLM International Workshop
Nice, 4 – 6 Sep 2017
Beat the stock market’s average returns and take full advantage of short-term price fluctuations
Involves a much deeper analysis and expertise to decide when to pivot into or out of a particular asset
Accurate determination of when and where prices change will be critical
Passive Active
Understand the true drivers of returns (From assets to small number of common factors)
Importance of true diversification/stability as opposed to decorrelation
Time-varying risk budgeting
Challenges of active investing
DLM International Workshop
Nice, 4 – 6 Sep 2017
Investment process
Risk premia vs Risk factors
Manifold learning for financial data
Detection of critical transitions in financial markets
Conclusions
Overview
DLM International Workshop
Nice, 4 – 6 Sep 2017
Commodities
Equities
Forex
Bonds
Interest rates
Assets Systematic Exposure
From assets to risk premia
DLM International Workshop
Nice, 4 – 6 Sep 2017
…
Momentum
Mean Reversion
Value
Volatility
Low Risk
Liquidity
…
RISK PREMIA
[A market segmentation]
Investors treat markets as a purely economic system
Investment decisions are made empirically based on economic interpretations of the markets
Premia vs Factors
DLM International Workshop
Nice, 4 – 6 Sep 2017
Segment market without relying on economic interpretations Exploit the power of mathematical and signal processing tools
Concept: Focus on achieving orthogonality
Total risk = Sum of marginal risks Risk Decomposition
Segmentation of market lacks economic interpretation BUT we are able to extract hidden information
Green World
(e.g. equities)Investors
Blue
Yellow
Quants
Commodities
Equities
Forex
Bonds
Interest rates
Assets Dimensionality Reduction
From assets to risk factors
DLM International Workshop
Nice, 4 – 6 Sep 2017
…
PCA, PPCA
Fourier Transform
Wavelet Transform
Machine Learning
Pattern Analysis
Manifold Learning
…
RISK FACTORS
[Market portfolios]
We treat markets as a mathematical system
Investment decisions are made based on market information extracted via DSP/ML/… methods
Chasing the decorrelation
DLM International Workshop
Nice, 4 – 6 Sep 2017
[Invariance]
Time Seriesto
Returns
From ReturnsTo
Smooth Manifolds
The desired portfolio is NOT the solution
to some optimization problem but an
invariant of a dynamical system
Investment process
Risk premia vs Risk factors
Manifold learning for financial data
Detection of critical transitions in financial markets
Conclusions
Overview
DLM International Workshop
Nice, 4 – 6 Sep 2017
Optimal portfolio as an invariant of a dynamical system
Optimal portfolio as an invariant
DLM International Workshop
Nice, 4 – 6 Sep 2017
Traditional view(optimization-based)
Utility function: max𝐄 𝒓𝒆𝒕𝒖𝒓𝒏𝒔 +min 𝑹𝒊𝒔𝒌Solution: straight line
Modern view(invariant-based)
Invariance with respect to the 1st (returns) and 2nd
(Var) derivatives(changes of these two derivatives affect portfolio performance)
Allow more features of the portfolio to remain invariant (e.g. correlations (= angles) between assets)
Financial data as dynamical systems
DLM International Workshop
Nice, 4 – 6 Sep 2017
Time-Delay Embedding
Financial Time Series
Manifold Learning
Early Warning
• Financial markets are highly complex,nonlinear dynamical systems
• Financial time series are comprehensivereflections of market condition/operationsand provide the ground for market analysis
• Dynamic nature of original system is oftencorrupted by irrelevant components thatdisturb useful intrinsic features
• Extract underlying manifold structure thatgoverns the dynamical system; embeddingin a more stable/smooth low-dimensionalspace
Taken’s theorem: complete information about the hidden state of dynamical systems can be preserved in observed time series
Phase space reconstruction
DLM International Workshop
Nice, 4 – 6 Sep 2017
Critical Parameters
τ Delay
m Embedding dimension
N ( = n - (m - 1)τ ) Number of states
1st min of Average Mutual Information
1st min of False Nearest Neighbors (%)
AM
I
Time lag
FN
N (
%)
Dimension
Information-based manifold learning
DLM International Workshop
Nice, 4 – 6 Sep 2017
Obtain the attractor manifolds by preserving geodesic distances between points in the state space
Classical Manifold Learning
Data points in state space
Data representation via probability distributions (e.g. risk quantification)
Considering only the geometric structure of a data space hides essential characteristics of the data and destroys the proximity relations (topology) of the original data space
Financial Practice
PDFs of data points in state space
Measure informationchange between data points
Information-based manifold learning
DLM International Workshop
Nice, 4 – 6 Sep 2017
Sta
te (p
has
e) s
pac
e
Kernel Density Estimator
K: Gaussian kernelh: plug-in bandwidth selection
Information SimilarityGlobal Relationship
Matrix
Extract an extended local linear structure (adjacency doesnot entirely depend on the states’ geometric relations)while retaining the global topological characteristics in theinherent low-dimensional manifold
Employ locally linear embedding (LLE)Maps its inputs into a single global coordinate system of lower dimensionality
Optimizations do not involve local minima
Recovers global nonlinear structure from locally linear fits; local geometry of locally linear patches characterized by linear coefficients that reconstruct each data point from its neighbors
Information-based manifold learning
DLM International Workshop
Nice, 4 – 6 Sep 2017
Embedding Cost FunctionTranslation-free embedding
Rotation/Scaling-free embedding
Solution: d eigenvectorscorresponding to the smallest
d eigenvalues of A
Investment process
Risk premia vs Risk factors
Manifold learning for financial data
Detection of critical transitions in financial markets
Conclusions
Overview
DLM International Workshop
Nice, 4 – 6 Sep 2017
Hard to predict accurately the market shifting points
Detect gradual increase of transition points’ likelihood
HMM classifier on the learned manifold of probability distributions 3 states (classes) [High, Medium, Low risk]
Early warning for market transitions
DLM International Workshop
Nice, 4 – 6 Sep 2017
Learned Manifold
Initial State Probability Distribution (GMMs)
Construct State Transition Matrix P
Compute Posterior Probability
Classification of the corresponding time series point (end of phase space
vectors)
S&P500 Index: daily closing prices in the period 2005-2016
Estimated phase space parameters: m = 8, τ = 23
Manifold learning/Early warning over sliding windows: length = 250, step = 25
3 warning states (posterior thresholds): 50%-70% Low, 70%-90% Medium, 90%-100% High
Early warning for market transitions
DLM International Workshop
Nice, 4 – 6 Sep 2017
Manifold of S&P500
Early warning signals
Pre-crisis period Crisis period
• High posteriors concentratedin 2007-2009
• Medium posteriors (earlysigns) in 2005-2006 (US realestate bubble)
Investment process
Risk premia vs Risk factors
Manifold learning for financial data
Detection of critical transitions in financial markets
Conclusions
Overview
DLM International Workshop
Nice, 4 – 6 Sep 2017
Take home messages
Manifold learning (ML) is an efficient framework for unveiling intrinsic dynamic structure of financial systems
Traditional geometry-based ML methods are not proper to handle investors’ probabilistic (risk-based) view
Information distance-based ML measures more complex relationships between financial data in a phase space
ML coupled with a conventional HMM enabled accurate identification of critical market transitions, providing reliable early warnings for investors
Future work: Study the effect of differential curvature of a financial system through its attractor manifold
as an indicator of market resilience against external disturbances Examine alternative manifold learning techniques and distance measures adapted to
financial data
DLM International Workshop
Nice, 4 – 6 Sep 2017
DLM International Workshop
Nice, 4 – 6 Sep 2017