[Draft]Adjustment to the PCA Approach

arbitragelab/network/imports.py

@@ -69,6 +69,7 @@ def __import_libraries():
         import arbitragelab.cointegration_approach as cointegration_approach
         import arbitragelab.copula_approach as copula_approach
         import arbitragelab.distance_approach as distance_approach
+        import arbitragelab.pca_approach as pca_approach
         import arbitragelab.other_approaches as other_approaches
         import arbitragelab.util as util
         import arbitragelab.optimal_mean_reversion as optimal_mean_reversion

arbitragelab/other_approaches/__init__.py

@@ -2,4 +2,3 @@
 This module implements other Statistical Arbitrage strategies.
 """
 from arbitragelab.other_approaches.kalman_filter import KalmanFilterStrategy
-from arbitragelab.other_approaches.pca_approach import PCAStrategy

arbitragelab/pca_approach/__init__.py

@@ -0,0 +1,5 @@
+"""
+This module implements other PCA Approach strategies.
+"""
+from arbitragelab.pca_approach.pca_approach import PCAStrategy
+from arbitragelab.pca_approach.etf_approach import ETFStrategy

docs/source/index.rst

@@ -132,7 +132,15 @@ This project is licensed under an all rights reserved licence.
     :hidden:
 
     other_approaches/kalman_filter
-    other_approaches/pca_approach
+
+
+.. toctree::
+    :maxdepth: 2
+    :caption: PCA Approach
+    :hidden:
+
+    pca_approach/pca_approach
+
 
 .. toctree::
     :maxdepth: 2

docs/source/pca_approach/pca_approach.rst

@@ -0,0 +1,330 @@
+.. _other_approaches-pca_approach:
+
+.. note::
+   The following documentation closely follows a paper by Marco Avellaneda and Jeong-Hyun Lee:
+   `Statistical Arbitrage in the U.S. Equities Market <https://math.nyu.edu/faculty/avellane/AvellanedaLeeStatArb20090616.pdf>`__.
+
+============
+PCA Approach
+============
+
+Introduction
+############
+
+This module shows how the Principal Component Analysis can be used to create mean-reverting portfolios
+and generate trading signals. It's done by considering residuals or idiosyncratic components
+of returns and modeling them as mean-reverting processes.
+
+The original paper presents the following description:
+
+The returns for different stocks are denoted as :math:`\{ R_{i} \}^{N}_{i=1}`. The :math:`F` represents
+the return of a "market portfolio" over the same period. For each stock in the universe:
+
+.. math::
+
+    R_{i} = \beta_{i} F + \widetilde{R_{i}}
+
+which is a regression, decomposing stock returns into a systematic component :math:`\beta_{i} F` and
+an (uncorrelated) idiosyncratic component :math:`\widetilde{R_{i}}`.
+
+This can also be extended to a multi-factor model with :math:`m` systematic factors:
+
+.. math::
+
+    R_{i} = \sum^{m}_{j=1} \beta_{ij} F_{j} + \widetilde{R_{i}}
+
+A trading portfolio is a market-neutral one if the amounts :math:`\{ Q_{i} \}^{N}_{i=1}` invested in
+each of the stocks are such that:
+
+.. math::
+
+    \bar{\beta}_{j} = \sum^{N}_{i=1} \beta_{ij} Q_{i} = 0, j = 1, 2,, ..., m.
+
+where :math:`\bar{\beta}_{j}` correspond to the portfolio betas - projections of the
+portfolio returns on different factors.
+
+As derived in the original paper,
+
+.. math::
+
+    \sum^{N}_{i=1} Q_{i} R_{i} = \sum^{N}_{i=1} Q_{i} \widetilde{R_{i}}
+
+So, a market-neutral portfolio is only affected by idiosyncratic returns.
+
+PCA Approach
+############
+
+This approach was originally proposed by Jolliffe (2002). It is using a historical share price data
+on a cross-section of :math:`N` stocks going back :math:`M` days in history. The stocks return data
+on a date :math:`t_{0}` going back :math:`M + 1` days can be represented as a matrix:
+
+.. math::
+
+    R_{ik} = \frac{S_{i(t_{0} - (k - 1) \Delta t)} - S_{i(t_{0} - k \Delta t)}}{S_{i(t_{0} - k \Delta t)}}; k = 1, ..., M; i = 1, ..., N.
+
+where :math:`S_{it}` is the price of stock :math:`i` at time :math:`t` adjusted for dividends. For
+daily observations :math:`\Delta t = 1 / 252`.
+
+Returns are standardized, as some assets may have greater volatility than others:
+
+.. math::
+
+    Y_{ik} = \frac{R_{ik} - \bar{R_{i}}}{\bar{\sigma_{i}}}
+
+where
+
+.. math::
+
+   \bar{R_{i}} = \frac{1}{M} \sum^{M}_{k=1}R_{ik}
+
+and
+
+.. math::
+
+   \bar{\sigma_{i}}^{2} = \frac{1}{M-1} \sum^{M}_{k=1} (R_{ik} - \bar{R_{i}})^{2}
+
+And the empirical correlation matrix is defined by
+
+.. math::
+
+   \rho_{ij} = \frac{1}{M-1} \sum^{M}_{k=1} Y_{ik} Y_{jk}
+
+.. Note::
+
+    It's important to standardize data before inputting it to PCA, as the PCA seeks to maximize the
+    variance of each component. Using unstandardized input data will result in worse results.
+    The *get_signals()* function in this module automatically standardizes input returns before
+    feeding them to PCA.
+
+The original paper mentions that picking long estimation windows for the correlation matrix
+(:math:`M \gg N`, :math:`M` is the estimation window, :math:`N` is the number of assets in a portfolio)
+don't make sense because they take into account the distant past which is economically irrelevant.
+The estimation windows used by the authors is fixed at 1 year (252 trading days) prior to the trading date.
+
+The eigenvalues of the correlation matrix are ranked in the decreasing order:
+
+.. math::
+
+   N \ge \lambda_{1} \ge \lambda_{2} \ge \lambda_{3} \ge ... \ge \lambda_{N} \ge 0.
+
+And the corresponding eigenvectors:
+
+.. math::
+
+   v^{(j)} = ( v^{(j)}_{1}, ..., v^{(j)}_{N} ); j = 1, ..., N.
+
+Now, for each index :math:`j` we consider a corresponding "eigen portfolio", in which we
+invest the respective amounts invested in each of the stocks as:
+
+.. math::
+
+    Q^{(j)}_{i} = \frac{v^{(j)}_{i}}{\bar{\sigma_{i}}}
+
+And the eigen portfolio returns are:
+
+.. math::
+
+   F_{jk} = \sum^{N}_{i=1} \frac{v^{(j)}_{i}}{\bar{\sigma_{i}}} R_{ik}; j = 1, 2, ..., m.
+
+.. figure:: images/pca_approach_portfolio.png
+    :scale: 80 %
+    :align: center
+
+    Performance of a portfolio composed using the PCA approach in comparison to the market cap portfolio.
+    An example from `Statistical Arbitrage in the U.S. Equities Market <https://math.nyu.edu/faculty/avellane/AvellanedaLeeStatArb20090616.pdf>`__.
+    by Marco Avellaneda and Jeong-Hyun Lee.
+
+In a multi-factor model we assume that stock returns satisfy the system of stochastic
+differential equations:
+
+.. math::
+
+   \frac{dS_{i}(t)}{S_{i}(t)} = \alpha_{i} dt + \sum^{N}_{j=1} \beta_{ij} \frac{dI_{j}(t)}{I_{j}(t)} + dX_{i}(t),
+
+where :math:`\beta_{ij}` are the factor loadings.
+
+The idiosyncratic component of the return with drift :math:`\alpha_{i}` is:
+
+.. math::
+
+   d \widetilde{X_{i}}(t) = \alpha_{i} dt + d X_{i} (t).
+
+Based on the previous descriptions, a model for :math:`X_{i}(t)` is estimated as the Ornstein-Uhlenbeck
+process:
+
+.. math::
+
+   dX_{i}(t) = \kappa_{i} (m_{i} - X_{i}(t))dt + \sigma_{i} dW_{i}(t), \kappa_{i} > 0.
+
+which is stationary and auto-regressive with lag 1.
+
+.. Note::
+
+   To find out more about the Ornstein-Uhlenbeck model and optimal trading under this model
+   please check out our section on Trading Under the Ornstein-Uhlenbeck Model.
+
+The parameters :math:`\alpha_{i}, \kappa_{i}, m_{i}, \sigma_{i}` are specific for each stock.
+They are assumed to *de facto* vary slowly in relation to Brownian motion increments :math:`dW_{i}(t)`,
+in the chosen time-window. The authors of the paper were using a 60-day window to estimate the residual
+processes for each stock and assumed that these parameters were constant over the window.
+
+However, the hypothesis of parameters being constant over the time-window is being accepted
+for stocks which mean reversion (the estimate of :math:`\kappa`) is sufficiently high and is
+rejected for stocks with a slow speed of mean-reversion.
+
+An investment in a market long-short portfolio is being constructed by going long $1 on the stock and
+short :math:`\beta_{ij}` dollars on the :math:`j` -th factor. Expected 1-day return of such portfolio
+is:
+
+.. math::
+
+   \alpha_{i} dt + \kappa_{i} (m_{i} - X_{i}(t))dt
+
+The parameter :math:`\kappa_{i}` is called the speed of mean-reversion. If :math:`\kappa \gg 1` the
+stock reverts quickly to its means and the effect of drift is negligible. As we are assuming that
+the parameters of our model are constant, we are interested in stocks with fast mean-reversion,
+such that:
+
+.. math::
+
+   \frac{1}{\kappa_{i}} \ll T_{1}
+
+where :math:`T_{1}` is the estimation window to estimate residuals in years.
+
+Implementation
+**************
+.. py:currentmodule:: arbitragelab.other_approaches.pca_approach
+
+.. autoclass:: PCAStrategy
+    :noindex:
+    :members: __init__, standardize_data, get_factorweights, get_sscores
+
+PCA Trading Strategy
+####################
+
+The strategy implemented sets a default estimation window for the correlation matrix as 252 days, a window for residuals
+estimation of 60 days (:math:`T_{1} = 60/252`) and the threshold for the mean reversion speed of an eigen portfolio for
+it to be traded so that the reversion time is less than :math:`1/2` period (:math:`\kappa > 252/30 = 8.4`).
+
+For the process :math:`X_{i}(t)` the equilibrium variance is defined as:
+
+.. math::
+
+   \sigma_{eq,i} = \frac{\sigma_{i}}{\sqrt{2 \kappa_{i}}}
+
+And the following variable is defined:
+
+.. math::
+
+   s_{i} = \frac{X_{i}(t)-m_{i}}{\sigma_{eq,i}}
+
+This variable is called the S-score. The S-score measures the distance to the equilibrium of the
+cointegrated residual in units standard deviations, i.e. how far away a given asset eigen portfolio
+is from the theoretical equilibrium value associated with the model.
+
+.. figure:: images/pca_approach_s_score.png
+    :scale: 80 %
+    :align: center
+
+    Evolution of the S-score of JPM ( vs. XLF ) from January 2006 to December 2007.
+    An example from `Statistical Arbitrage in the U.S. Equities Market <https://math.nyu.edu/faculty/avellane/AvellanedaLeeStatArb20090616.pdf>`__.
+    by Marco Avellaneda and Jeong-Hyun Lee.
+
+If the eigen portfolio shows a mean reversion speed above the set threshold (:math:`\kappa`), the
+S-score based on the values from the residual estimation window is being calculated.
+
+The trading signals are generated from the S-scores using the following rules:
+
+- Open a long position if :math:`s_{i} < - \bar{s_{bo}}`
+
+- Close a long position if :math:`s_{i} < + \bar{s_{bc}}`
+
+- Open a short position if :math:`s_{i} > + \bar{s_{so}}`
+
+- Close a short position if :math:`s_{i} > - \bar{s_{sc}}`
+
+Opening a long position means buying $1 of the corresponding stock (of the asset eigen portfolio)
+and selling :math:`\beta_{i1}` dollars of assets from the first scaled eigenvector (:math:`Q^{(1)}_{i}`),
+:math:`\beta_{i2}` from the second scaled eigenvector (:math:`Q^{(2)}_{i}`) and so on.
+
+Opening a short position, on the other hand, means selling $1 of the corresponding stock and buying
+respective beta values of stocks from scaled eigenvectors.
+
+Authors of the paper, based on empirical analysis chose the following cutoffs. They were selected
+based on simulating strategies from 2000 to 2004 in the case of ETF factors:
+
+- :math:`\bar{s_{bo}} = \bar{s_{so}} = 1.25`
+
+- :math:`\bar{s_{bc}} = 0.75`, :math:`\bar{s_{sc}} = 0.50`
+
+The rationale behind this strategy is that we open trades when the eigen portfolio shows good mean
+reversion speed and its S-score is far from the equilibrium, as we think that we detected an anomalous
+excursion of the co-integration residual. We expect most of the assets in our portfolio to be near
+equilibrium most of the time, so we are closing trades at values close to zero.
+
+The signal generating function implemented in the ArbitrageLab package outputs target weights for each
+asset in our portfolio for each observation time - target weights here are the sum of weights of all
+eigen portfolios that show high mean reversion speed and have needed S-score value at a given time.
+
+Implementation
+**************
+
+.. autoclass:: PCAStrategy
+    :noindex:
+    :members: __init__, get_sscores, get_signals
+
+Examples
+********
+
+.. code-block::
+
+   # Importing packages
+   import pandas as pd
+   import numpy as np
+   from arbitragelab.other_approaches.pca_approach import PCAStrategy
+
+   # Getting the dataframe with time series of asset returns
+   data = pd.read_csv('X_FILE_PATH.csv', index_col=0, parse_dates = [0])
+
+   # The PCA Strategy class that contains all needed methods
+   pca_strategy = PCAStrategy()
+
+   # Simply applying the PCAStrategy with standard parameters
+   target_weights = pca_strategy.get_signals(data, k=8.4, corr_window=252,
+                                             residual_window=60, sbo=1.25,
+                                             sso=1.25, ssc=0.5, sbc=0.75,
+                                             size=1)
+
+   # Or we can do individual actions from the PCA approach
+   # Standardizing the dataset
+   data_standardized, data_std = pca_strategy.standardize_data(data)
+
+   # Getting factor weights using the first 252 observations
+   data_252days = data[:252]
+   factorweights = pca_strategy.get_factorweights(data_252days)
+
+   # Calculating factor returns for a 60-day window from our factor weights
+   data_60days = data[(252-60):252]
+   factorret = pd.DataFrame(np.dot(data_60days, factorweights.transpose()),
+                            index=data_60days.index)
+
+   # Calculating residuals for a set 60-day window
+   residual, coefficient = pca_strategy.get_residuals(data_60days, factorret)
+
+   # Calculating S-scores for each eigen portfolio for a set 60-day window
+   s_scores = pca_strategy.get_sscores(residual, k=8)
+
+Research Notebooks
+##################
+
+The following research notebook can be used to better understand the PCA approach described above.
+
+* `PCA Approach`_
+
+.. _`PCA Approach`: https://github.com/Hudson-and-Thames-Clients/arbitrage_research/blob/master/Other%20Approaches/pca_approach.ipynb
+
+References
+##########
+
+* `Avellaneda, M. and Lee, J.H., 2010. Statistical arbitrage in the US equities market. Quantitative Finance, 10(7), pp.761-782. <https://math.cims.nyu.edu/faculty/avellane/AvellanedaLeeStatArb071108.pdf>`__
+* `Jolliffe, I. T., Principal Components Analysis, Springer Series in Statistics, Springer-Verlag, Heidelberg, 2002. <https://www.springer.com/gp/book/9780387954424>`__