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181Test type: trace statistic , with linear trendEigenvalues (lambda):[1] 0.29311420 0.24149750 0.04308716Values of teststatistic and critical values of test:test 10pct 5pct 1pctr <= 2 | 11.01 6.50 8.18 11.65r <= 1 | 80.11 15.66 17.95 23.52r = 0 | 166.83 28.71 31.52 37.22Eigenvectors, normalised to first column:(These are the cointegration relations)SPY.Adjusted.l2 IVV.Adjusted.l2 VOO.Adjusted.l2SPY.Adjusted.l2 1.0000000 1.000000 1.0000000IVV.Adjusted.l2 -0.3669563 -4.649203 -0.5170538VOO.Adjusted.l2 -0.6783666 4.042139 -0.2426406Weights W:(This is the loading matrix)SPY.Adjusted.l2 IVV.Adjusted.l2 VOO.Adjusted.l2SPY.Adjusted.d -1.8692017 0.2875776 -0.3086708IVV.Adjusted.d -1.2062706 0.3965064 -0.3160481VOO.Adjusted.d -0.9414142 0.2182340 -0.2871731As before we sequentially carry out the hypothesis tests beginning with the null hypoothesisof r = 0 versus the alternative hypothesis of r > 0. There is clear evidence to reject the nullhypothesis at the 1% level and we can likely conclude that r > 0.Similarly when we carry out the r ≤ 1 null hypothesis versus the r > 1 alternative hypothesiswe have sufficient evidence to reject the null hypothesis at the 1% level and can conclude r > 1.However, for the r ≤ 2 hypothesis we can only reject the null hypothesis at the 5% level.This is weaker evidence than the previous hypotheses and, although it suggests we can reject thenull at this level, we should be careful that r might equal two, rather than exceed two. Whatthis means is that it may be possible to form a linear combination with only two assets ratherthan requiring all three to form a cointegrating portfolio.In addition we should be extremely cautious of interpreting these results as I have only usedone years worth of data, which is approximately 250 trading days. Such a small sample is unlikelyto provide a true representation of the underlying relationships. Hence, we must always be carefulin interpreting statistical tests!12.9.3 Full Code
182library("quantmod")library("tseries")library("urca")set.seed(123)## Simulated cointegrated seriesz <- rep(0, 10000)for (i in 2:10000) z[i] <- z[i-1] + rnorm(1)p <- q <- r <- rep(0, 10000)p <- 0.3*z + rnorm(10000)q <- 0.6*z + rnorm(10000)r <- 0.8*z + rnorm(10000)jotest=ca.jo(data.frame(p,q,r), type="trace", K=2,ecdet="none", spec="longrun")summary(jotest)s = 1.000*p + 1.791324*q - 1.717271*rplot(s, type="l")adf.test(s)## EWA, EWC and IGEgetSymbols("EWA", from="2006-04-26", to="2012-04-09")getSymbols("EWC", from="2006-04-26", to="2012-04-09")getSymbols("IGE", from="2006-04-26", to="2012-04-09")ewaAdj = unclass(EWA$EWA.Adjusted)ewcAdj = unclass(EWC$EWC.Adjusted)igeAdj = unclass(IGE$IGE.Adjusted)jotest=ca.jo(data.frame(ewaAdj,ewcAdj,igeAdj), type="trace",K=2, ecdet="none", spec="longrun")summary(jotest)## SPY, IVV and VOOgetSymbols("SPY", from="2015-01-01", to="2015-12-31")getSymbols("IVV", from="2015-01-01", to="2015-12-31")getSymbols("VOO", from="2015-01-01", to="2015-12-31")
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ContentsI Introduction 11 Introduct
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38.2 Correlation . . . . . . . . .
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513.2 The Kalman Filter . . . . . .
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719 Support Vector Machines . . . .
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928 Kalman Filter-Based Pairs Tradi
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Limit of Liability/Disclaimer ofWar
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Part IIntroduction1
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4The aim of this book is to provide
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61.3 How Is The Book Laid Out?The b
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81.5 How Does This Book Differ From
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101.7.1 AlternativesThere are many
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Chapter 2Introduction to Bayesian S
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15Table 2.1: Comparison of Frequent
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172.2 Applying Bayes’ Rule for Ba
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19the next chapter. At this stage,
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21# Create and plot a Beta distribu
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Chapter 3Bayesian Inference of a Bi
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25• The fairness of the coin does
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273.4.3 Multiple Flips of the CoinN
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29x = np.linspace(0, 1, 100)params
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31Hence, all we need to do is re-ar
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33Figure 3.3: The prior and posteri
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Chapter 4Markov Chain Monte CarloIn
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374.2.1 Markov Chain Monte Carlo Al
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39We discussed the fact that we cou
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41Finally we specify the Metropolis
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43we do not need to compute 100,000
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45x, stats.beta.pdf(x, alpha, beta)
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Chapter 5Bayesian Linear Regression
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49non-informative priors.• Poster
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51# Use a linear model (y ~ beta_0
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53# Use the No-U-Turn Samplerstep =
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55plt.show()We can see the sampled
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57"""Calculates the Markov Chain Mo
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Chapter 6Bayesian Stochastic Volati
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61plt.show()Figure 6.1: Different r
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63( )yilog ∼ t(ν, 0, exp(−2s i
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656.3.2 Model Specification in PyMC
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67Figure 6.5: Overlay of samples fr
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69plt.xlabel(’Time’)plt.ylabel(
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Part IIITime Series Analysishttps:/
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74• Seasonal Variation - Many tim
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767.5 How Does This Relate to Other
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78Definition 8.1.1. Expectation. Th
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808.2 CorrelationCorrelation is a d
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82Definition 8.3.4. Stationary in t
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84Figure 8.2: Correlogram plotted i
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868.6 Next StepsNow that we’ve di
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88• Use our knowledge of time ser
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909.3.2 CorrelogramWe can also plot
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929.4.2 CorrelogramThe autocorrelat
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94created the difference series, we
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96there are a few that are marginal
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9810.1 How Will We Proceed?In this
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10010.4.1 RationaleIn the previous
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10210.4.4 Simulations and Correlogr
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104Figure 10.2: Realisation of AR(1
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10610.4.5 Financial DataAmazon Inc.
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108Figure 10.6: Correlogram of Firs
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110> plot(gspcrt)Figure 10.8: First
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112model, as well as the conditiona
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114Figure 10.10: Realisation of MA(
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116Coefficients:ma1 intercept-0.729
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118> require(quantmod)> getSymbols(
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120> amznrt.maCall:arima(x = amznrt
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122Figure 10.16: Residuals of MA(1)
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124Figure 10.18: Residuals of MA(3)
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12610.6.3 RationaleTo date we have
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128Figure 10.20: Correlogram of an
Page 143 and 144: 130Figure 10.22: Correlogram of an
Page 145 and 146: 132Figure 10.23: Correlogram of the
Page 147 and 148: 134Notice that there are some signi
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Page 151 and 152: 138Figure 11.1: Plot of simulated A
Page 153 and 154: 140> amzn = diff(log(Cl(AMZN)))As i
Page 155 and 156: 142We fit an ARIMA model by looping
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Page 159 and 160: 14611.5.2 Why Does This Model Volat
Page 161 and 162: 148ɛ t = σ t w t (11.17)σ 2 t =
Page 163 and 164: 1502.5 % 97.5 %a0 0.1786255 0.21726
Page 165 and 166: 152Figure 11.10: Residuals of an AR
Page 167 and 168: Figure 11.13: Squared residuals of
Page 169 and 170: 156mean, but due to its stationarit
Page 171 and 172: 158x t = pz t + w x,t (12.1)y t = q
Page 173 and 174: 160Figure 12.3: Plot of x t and y t
Page 175 and 176: 162> layout(1:2)> plot(badcomb, typ
Page 177 and 178: 164We then created two subsequent s
Page 179 and 180: 166Figure 12.6: Backward-adjusted c
Page 181 and 182: 168Coefficients:(Intercept) ewaAdj3
Page 183 and 184: 170> plot(rdsaAdj, rdsbAdj,xlab="RD
Page 185 and 186: 172set.seed(123)## SIMULATED DATA##
Page 187 and 188: 174plot(comb1$residuals, type="l",x
Page 189 and 190: 176> q <- 0.6*z + rnorm(10000)> r <
Page 191 and 192: 178Figure 12.12: Plot of s t , the
Page 193: 180EWA.Adjusted.l2 EWC.Adjusted.l2
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Page 199 and 200: 186• Smoothing - Estimating the p
Page 201 and 202: 18813.2.1 A Bayesian ApproachRecall
Page 203 and 204: 190E[y t+1 |D t ] = E[Ft+1θ T t +
Page 205 and 206: 192of our linear regression. The ne
Page 207 and 208: 194Figure 13.1: Scatterplot of the
Page 209 and 210: 196Figure 13.2: Time-varying slope
Page 211 and 212: 198delta = 1e-5trans_cov = delta /
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Page 215 and 216: 202"risk manager". They will be use
Page 217 and 218: 204Figure 14.1: Two-state Markov Ch
Page 219 and 220: 206Figure 14.2: Hidden Markov Model
Page 221 and 222: 208> bull_var <- 0.1> bear_mean <-
Page 223 and 224: 210historical financial data.14.3.3
Page 225 and 226: 212The same process will now be car
Page 227 and 228: 214install.packages(’quantmod’)
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Page 232 and 233: Chapter 15Introduction to Machine L
Page 234 and 235: 22115.3.1 Linear RegressionAn eleme
Page 236 and 237: 223While seemingly not a traditiona
Page 238 and 239: Chapter 16Supervised LearningSuperv
Page 240 and 241: 227ŷ = ˆf(x) = argmax z∈R p(y =
Page 242 and 243: Chapter 17Linear RegressionIn this
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231if __name__ == "__main__":# Set
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23317.3 Maximum Likelihood Estimati
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235To simplify the notation the lat
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237Once the full dataset is created
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239if __name__ == "__main__":# Set
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241))lr_model.coef_[0]# Create a sc
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Chapter 18Tree-Based MethodsIn this
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245Figure 18.2: The resulting parti
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24718.3 Decision Trees for Classifi
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249In quantitative finance applicat
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251(a) Grow a tree ˆf b with k spl
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253tslag["Lag%s" % str(i+1)] = ts["
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255The same approach is carried out
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257seen whether it is feasible to p
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259amzn = create_lagged_series("AMZ
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Chapter 19Support Vector MachinesIn
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263• Non-Probabilistic - Since th
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265The key point here is that it is
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267classification. Overfitting mean
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269Figure 19.6: Addition of a singl
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27119.8 Support Vector MachinesThe
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273p∑K(x i , x k ) = (1 + x ij x
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Chapter 20Model Selection andCross-
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277This motivates the concept of a
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279Figure 20.1: Various estimates o
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281error for a particular model est
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283Figure 20.3: Amazon, Inc. - Pric
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28520.2.5 Python ImplementationWe a
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287At this stage we have the necess
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289..import pylab as plt....def plo
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291fold = 0# Loop over the k-foldsf
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293Figure 20.5: Test MSE curves for
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295return tsretdef validation_set_p
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297model = Pipeline([("polynomial_f
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299X = lags[["Lag1", "Lag2", "Lag3"
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Chapter 21Unsupervised LearningThe
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303parameters of the model, θ.Conv
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Chapter 22Clustering MethodsIn this
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3072. Iterate the following until c
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309]# Generate a list of the 2D clu
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311clustered using K-Means and then
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313ax.set_axis_bgcolor((1,1,0.9))ax
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315ClusterTomorrow to contain tomor
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317Figure 22.4: 3D scatterplot of n
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319# Apply the K-Means Algorithm fo
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321# up days and red for down daysc
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323sp500["ClusterMatrix"] = list(zi
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Chapter 23Natural Language Processi
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327The Reuters 21578 dataset can be
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329for export as shippers are now e
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331tungtung-oilveg-oilwheatwoolwpiy
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333in order to minimise memory usag
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335’The Tokyo Grain Exchange said
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337for t in d[0]:if t in topics:d_t
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339This would mean that we are givi
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341partitions into the training and
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3430.660194174757[[21 0 0 0 2 3 0 0
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345self.encoding = encodingdef _res
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347"""topics = open("data/all-topic
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Part VQuantitative Trading Techniqu
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352While vectorised backtests are s
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354• Backtest - Encapsulates the
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356The second strategy provides a 3
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35825.4.1 MonthlyLiquidateRebalance
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360"""Size the order to reflect the
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362Figure 25.1: US Equities/Bonds 6
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364five years relative to US equiti
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366# Use the monthly liquidate and
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368import datetimefrom qstrader imp
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37026.2 Strategy ImplementationTo i
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372final.order[1],final.order[3]),
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374> spIntersect = merge( spArimaGa
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376enters what looks to be more a s
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378# Output the CSV file to "foreca
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380
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382natural gas.The hypothesis prese
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384## Carry out linear regression o
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38627.5 Python QSTrader Implementat
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388that arrive out of order in the
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390if self.invested is not None:if
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392take into account commission dif
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394adf.test(comb$residuals, k=1)# c
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396def go_short_units(self):"""Go s
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398from qstrader.trading_session.ba
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400default=’’,help=’Pickle (.
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402Hence we can "long the spread" i
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404class KalmanPairsTradingStrategy
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406if all(self.latest_prices > -1.0
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408It also changes the FixedPositio
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410’--testing/--no-testing’, de
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412• Asset Selection - Choosing a
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414if self.R is not None:self.R = s
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416)self.events_queue.put(SignalEve
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418@click.option(’--testing/--no-
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420• Short-Term Trend - Predict m
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422It takes the CSV file path as an
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424produce a final truth column whe
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426bias-variance trade-off of the m
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428used by the machine learning mod
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430IQFeedIntradayCsvBarPriceHandler
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432’--filename’,default=’’,
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Figure 29.2: Tearsheet for Random F
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436):lookforward_minutes=5,up_down_
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438#ts["UpDown"] = down_tot & up_to
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440events_queue - A handle to the s
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442from intraday_ml_strategy import
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444if __name__ == "__main__":main()
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446sentiment.In recent years there
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448Ticker Name Period LinkMSFT Micr
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450self.statistics.update(event.tim
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452if self.end_date is not None:sen
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454a sent_sell corresponding exit t
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456sentiment_handler = SentdexSenti
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458Figure 30.2:volatile, posting mo
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460fication by adding many more sto
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462events_queue = queue.Queue()csv_
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464def main(config, testing, ticker
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466In quantitative trading this pro
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468self.tickers[ticker]["timestamp"
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470df["Adj Close"][mask],".", lines
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472from qstrader.event import (Sign
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474class RegimeHMMRiskManager(Abstr
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476# If in the undesirable regime,
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47831.5 Strategy Results31.5.1 Tran
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480Figure 31.3: Trend Following Reg
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482ax.grid(True)plt.show()if __name
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484elif short_sma < long_sma and se
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486elif action == "SLD":if self.inv
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488backtest = Backtest(price_handle
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490holding an instrument representi
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492rolling_sharpe_s = np.sqrt(self.
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494In order to use the annualised r
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496Figure 32.2: Aluminum Smelting S
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498
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500[18] Wikipedia: Viterbi algorith
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502[56] Investor, S. Regime detecti
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504[92] Sinclair, E. Volatility Tra
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