Nonlinear Manifold Learning for Financial Markets Integration

Transcription

1 Nonlinear Manifold Learning for Financial Markets Integration George Tzagkarakis 1 & Thomas Dionysopoulos 1,2 1 EONOS Investment Technologies, Paris (FR) 2 Dalton Strategic Partnership, London (UK) Nice, 4 6 Sept 2017

2 Motivation Extract meaningful information from financial data Build smart trading strategies Quants Traders

3 Overview Investment process Risk premia vs Risk factors Manifold learning for financial data Detection of critical transitions in financial markets Conclusions

4 Overview Investment process Risk premia vs Risk factors Manifold learning for financial data Detection of critical transitions in financial markets Conclusions

5 Investment process Views Repeated pattern Statistics vs Threshold MovAvg, MovVol, (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) {-1, 1} 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

6 Passive vs Active investing Passive Active 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 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

7 Challenges of active investing 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

8 Overview Investment process Risk premia vs Risk factors Manifold learning for financial data Detection of critical transitions in financial markets Conclusions

9 From assets to risk premia [A market segmentation] Assets Systematic Exposure Investors treat markets as a purely economic system Commodities Equities Momentum Mean Reversion Investment decisions are made empirically based on economic interpretations of the markets Forex Bonds Value Volatility RISK PREMIA Interest rates Low Risk Liquidity

10 Premia vs Factors Segment market without relying on economic interpretations Exploit the power of mathematical and signal processing tools Investors Green World (e.g. equities) Yellow Blue Quants 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

11 From assets to risk factors [Market portfolios] Assets Dimensionality Reduction We treat markets as a mathematical system Commodities Equities PCA, PPCA Fourier Transform Investment decisions are made based on market information extracted via DSP/ML/ methods Forex Bonds Wavelet Transform Machine Learning RISK FACTORS Interest rates Pattern Analysis Manifold Learning

12 Chasing the decorrelation [Invariance] Time Series to Returns From Returns To Smooth Manifolds The desired portfolio is NOT the solution to some optimization problem but an invariant of a dynamical system

13 Overview Investment process Risk premia vs Risk factors Manifold learning for financial data Detection of critical transitions in financial markets Conclusions

14 Optimal portfolio as an invariant Optimal portfolio as an invariant of a dynamical system Traditional view (optimization-based) Modern view (invariant-based) Utility function: max E returns + min Risk Solution: straight line Invariance with respect to the 1 st (returns) and 2 nd (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)

15 Financial data as dynamical systems Financial markets are highly complex, nonlinear dynamical systems Financial time series are comprehensive reflections of market condition/operations and provide the ground for market analysis Dynamic nature of original system is often corrupted by irrelevant components that disturb useful intrinsic features Extract underlying manifold structure that governs the dynamical system; embedding in a more stable/smooth low-dimensional space Financial Time Series Time-Delay Embedding Manifold Learning Early Warning

16 AMI FNN (%) Phase space reconstruction Taken s theorem: complete information about the hidden state of dynamical systems can be preserved in observed time series Time lag Dimension Critical Parameters τ m N ( = n - (m - 1)τ ) Delay Embedding dimension Number of states 1 st min of Average Mutual Information 1 st min of False Nearest Neighbors (%)

17 Information-based manifold learning Classical Manifold Learning Obtain the attractor manifolds by preserving geodesic distances between points in the state space Data points in state space Financial Practice 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 Measure information change between data points PDFs of data points in state space

18 State (phase) space Information-based manifold learning Kernel Density Estimator Information Similarity Global Relationship Matrix K: Gaussian kernel h: plug-in bandwidth selection Extract an extended local linear structure (adjacency does not entirely depend on the states geometric relations) while retaining the global topological characteristics in the inherent low-dimensional manifold

19 Information-based manifold learning 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 Embedding Cost Function Translation-free embedding Rotation/Scaling-free embedding Solution: d eigenvectors corresponding to the smallest d eigenvalues of A

20 Overview Investment process Risk premia vs Risk factors Manifold learning for financial data Detection of critical transitions in financial markets Conclusions

21 Early warning for market transitions 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] Initial State Probability Distribution (GMMs) Construct State Transition Matrix P Learned Manifold Compute Posterior Probability Classification of the corresponding time series point (end of phase space vectors)

22 Early warning for market transitions S&P500 Index: daily closing prices in the period 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 signals Pre-crisis period Crisis period Manifold of S&P500 High posteriors concentrated in Medium posteriors (early signs) in (US real estate bubble)

23 Overview Investment process Risk premia vs Risk factors Manifold learning for financial data Detection of critical transitions in financial markets Conclusions

24 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

25