EARL_Boston_2015.Rnw

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% Title page setup
\title[Quantitative Portfolio Management with High Frequency Data]{Quantitative Portfolio Management with High Frequency Data}
\subtitle{EARL Boston 2015}
% \subject{Getting Started With R}
\institute[NYU Polytechnic]{NYU Tandon School of Engineering}
\titlegraphic{\includegraphics[scale=0.2]{/Users/jerzy/Develop/R/FRE6871/image/tandon_long_color}}
\author[Jerzy Pawlowski]{Jerzy Pawlowski \emph{\href{mailto:jp3900@nyu.edu}{jp3900@nyu.edu}}}
% \email{jp3900@poly.edu}
\date{November 4, 2015}
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\begin{document}


%%%%%%%%%%%%%%%
\maketitle



%%%%%%%%%%%%%%%
\section{Introduction}


%%%%%%%%%%%%%%%
\subsection{Investor \protect\emph{Utility}, \protect\emph{Risk Aversion}, \protect\emph{Prudence} and \protect\emph{Temperance}}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      The \emph{expected utility} hypothesis states that investor risk and return preferences are based on the expected value of the \emph{utility} of their wealth, instead of on the level of wealth, 
      \vskip1ex
      Investor risk and return preferences depend on the signs of the derivatives of the \emph{utility} function, 
      \vskip1ex
      A \emph{utility} function with a positive first derivative implies a preference for higher \emph{returns} (first moment), while a negative second derivative implies risk aversion and a preference for lower \emph{volatility} (second moment), 
      \vskip1ex
      A positive third derivative implies \emph{prudence}, or a preference for higher \emph{skewness} (third moment), while a negative fourth derivative implies \emph{temperance}, or a preference for lower \emph{kurtosis} (fourth moment), 
      \vskip1ex
      Investors with a logarithmic \emph{utility} of wealth base their preferences on the percentage change of wealth, instead of the absolute change, and have a preference for larger odd moments and smaller even moments, 
    \column{0.5\textwidth}
      \begin{multline*}
        u(w) = u(w_0) + \mathrm{d}w \frac{\mathrm{d}u}{\mathrm{d}w} + {\mathrm{d}w}^2 \frac{1}{2} \frac{\mathrm{d}^2u}{\mathrm{d}w^2} + \\
        {\mathrm{d}w}^3 \frac{1}{3!} \frac{\mathrm{d}^3u}{\mathrm{d}w^3} + {\mathrm{d}w}^4 \frac{1}{4!} \frac{\mathrm{d}^4u}{\mathrm{d}w^4} + \ldots
      \end{multline*}
      \begin{multline*}
        \mathrm{d}E[u] = \alpha1 mean + \alpha1 variance + \\
        \alpha1 skew + \alpha4 kurtosis + \ldots
      \end{multline*}
      mean: $\bar{w}=\frac{1}{k} \sum_{i=1}^{k} w_i$
      \vskip1ex
      variance: $\hat{\sigma}^2=\frac{1}{k-1} \sum_{i=1}^{k} (w_i-\bar{w})^2$
      \vskip1ex
      skew: $\hat{s}=\frac{1}{k-1} \sum_{i=1}^{k} (\frac{w_i-\bar{w}}{\hat{\sigma}})^3$
      \vskip1ex
      kurtosis: $\hat{k}=\frac{1}{k-1} \sum_{i=1}^{k} (\frac{w_i-\bar{w}}{\hat{\sigma}})^4$
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Investor Preferences and Empirical Return Distributions}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      Investor preference for higher \emph{returns} and for lower \emph{volatility} is expressed by maximizing the \emph{Sharpe} ratio, 
      \vskip1ex
      Stock price \emph{momentum} refers to the fact that stocks with high past returns tend to have high future returnv, and vice versa, 
      \vskip1ex
      Stock price \emph{momentum} has been observed, that is driven by approximately one-year of past returns:\\
Eugene Fama and Kenneth French, \href{http://papers.ssrn.com/sol3/papers.cfm?abstract_id=911960}{\emph{Dissecting Anomalies}}\\
      Asness et al., \href{http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2435323}{\emph{Fact, Fiction and Momentum Investing}}
      \vskip1ex
      The question then is can momentum also work on shorter, intraday time scales?
      \vskip1ex
      Also, do higher moments (skew, kurtosis) have predictive power as well?\\
      Amaya et al., \href{http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1898735}{\emph{Does Realized Skewness Predict the Cross-Section of Equity Returns?}}\\
      \vskip1ex
      Stocks typically have negative skew and excess kurtosis, the opposite of what investors prefer, 
      \vskip1ex
      Higher moments are hard to estimate from low frequency (daily) returnv, 
    \column{0.5\textwidth}
      \vspace{-3em}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_ret_hist-1}
      \vspace{-3em}
      <<echo=(-(1:6)),eval=TRUE>>=
# load package "HighFreq"
library(HighFreq)
symbol <- "SPY"  # define symbol
# load OHLC data
output_dir <- "/Users/jerzy/Develop/data/hfreq/scrub/"
symbol <- load(file.path(output_dir, paste0(symbol, ".RData")))
returnv <- calc_rets(xts_data=to.daily(get(symbol)))
len_rets <- nrow(returnv)  # number of observations
mean_rets <- mean(returnv[, 1])  # calculate mean
sd_rets <- sd(returnv[, 1])  # calculate standard deviation
# calculate skew and kurtosis
(sum(((returnv[, 1] - mean_rets)/sd_rets)^3))/len_rets
(sum(((returnv[, 1] - mean_rets)/sd_rets)^4))/len_rets
      @
      \vspace{-3em}
      <<earl_ret_hist,echo=TRUE,eval=FALSE,fig.width=7,fig.height=6,fig.show='hide'>>=
library(PerformanceAnalytics)
chart.Histogram(returnv[, 1], main="", 
  xlim=c(-6e-5, 6e-5), 
  methods = c("add.density", "add.normal"))
# add title
title(main=paste(symbol, 
         "density"), line=-1)
      @
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Package \texttt{HighFreq} for Managing High Frequency Data}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      Package \texttt{HighFreq} contains functions for managing high frequency \emph{TAQ} and \emph{OHLC} market data:
      \begin{itemize}
        \item reading and writing data from files, 
        \item managing time zones and alligning indices, 
        \item chaining and joining time series, 
        \item scrubbing bad data points, 
        \item converting \emph{TAQ} data to \emph{OHLC} format, 
        \item aggregating data to lower frequency,
      \end{itemize}
      \texttt{HighFreq} is inspired by the package \texttt{highfrequency}, and follows many of its conventions,
      \vskip1ex
      \texttt{HighFreq} depends on packages \texttt{xts}, \texttt{quantmod}, \texttt{lubridate}, and \texttt{caTools},
      \vskip1ex
      The function \texttt{scrub\_agg()} scrubs a single day of \emph{TAQ} data, aggregates it, and converts it to \emph{OHLC} format,
      \vskip1ex
      The function \texttt{save\_scrub\_agg()} loads, scrubs, aggregates, and binds multiple days of \emph{TAQ} data for a single symbol, and saves the \emph{OHLC} time series to a single \texttt{*.RData} file,
    \column{0.5\textwidth}
      <<echo=TRUE,eval=FALSE>>=
# install package "HighFreq" from github
install.packages("devtools")
library(devtools)
install_github(repo="algoquant/HighFreq")
# load package "HighFreq"
library(HighFreq)
# set data directories
data_dir <- "/Users/jerzy/Develop/data/hfreq/src/"
output_dir <- "/Users/jerzy/Develop/data/hfreq/scrub/"
# define symbol
symbol <- "SPY"
# load a single day of TAQ data
symbol <- load(
  file.path(data_dir, 
            paste0(symbol, "/2014.05.02.", 
                   symbol, ".RData")))
# scrub, aggregate single day of TAQ data to OHLC
ohlc_data <- scrub_agg(taq_data=get(symbol))
# aggregate TAQ data for symbol, save to file
save_scrub_agg(symbol, 
               data_dir=data_dir, 
               output_dir=output_dir, 
               period="minutes")
      @
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{High Frequency \protect\emph{OHLC} Data}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      Aggregating high frequency \emph{TAQ} data into \emph{OHLC} format with lower periodicity allows for data compression while maintaining some information about volatility,
      <<earl_ohlc_chart,echo=TRUE,eval=FALSE,fig.width=7,fig.height=6,fig.show='hide'>>=
# load package "HighFreq"
library(HighFreq)
# define symbol
symbol <- "SPY"
# load OHLC data
output_dir <- "/Users/jerzy/Develop/data/hfreq/scrub/"
symbol <- load(
  file.path(output_dir, 
            paste0(symbol, ".RData")))
dates <- 
  "2013-11-11 09:30:00/2013-11-11 10:30:00"
chart_Series(get(symbol)[dates], 
            name=symbol)
      @
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_ohlc_chart-1}
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Estimating Volatility From \protect\emph{OHLC} Data}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      Package \texttt{TTR} contains statistical estimators and technical indicators implemented in fast \texttt{C} code,
      \vskip1ex
      The function \texttt{volatility()} from package \texttt{TTR} estimates the volatility from \protect\emph{OHLC} data,
      \vskip1ex
      \texttt{volatility()} includes the \emph{Garman-Klass} estimator:\\
\vspace{-2em}
      \begin{displaymath}
        {\sigma}^2=\frac{1}{n} \sum_{i=1}^{n} (0.5(H_{i}-L_{i})^2 - (2\log2-1)(C_{i}-O_{i})^2)
      \end{displaymath}
\vspace{-1em}
      \vskip1ex
      and the \emph{Rogers-Satchell} estimator:\\
\vspace{-2em}
      \begin{displaymath}
        {\sigma}^2=\frac{1}{n} \sum_{i=1}^{n} ((H_{i}-O_{i})(H_{i}-C_{i}) + (L_{i}-O_{i})(L_{i}-C_{i}))
      \end{displaymath}
      \vspace{-1em}
      <<earl_ttr_runvol,eval=FALSE,echo=(-(1:1)),fig.width=7,fig.height=8,fig.show='hide'>>=
par(mfrow=c(2,1))  # set plot panels
library(quantmod)
library(TTR)
dates <- "2013-11-11/2013-11-15"
volat <- volatility(OHLC=get(symbol), 
                     calc="yang.zhang", n=20)
chart_Series(volat[dates], 
            name=paste(symbol, "vol w/ ON spikes"))
volat <- volatility(OHLC=get(symbol), 
                     calc="rogers.satchell", n=20)
chart_Series(volat[dates], 
            name=paste(symbol, "vol w/o ON spikes"))
      @
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_ttr_runvol-1}
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Estimating Skew From \protect\emph{OHLC} Data}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      The function \texttt{skew\_ohlc()} from package \texttt{HighFreq} calculates a skew-like indicator:\\
\vspace{-2em}
      \begin{multline*}
        \hskip-1em {s}^3=\frac{1}{n} \sum_{i=1}^{n} ((H_{i}-O_{i})(H_{i}-C_{i})(H_{i}-0.5(O_{i}+C_{i})) +\\ (L_{i}-O_{i})(L_{i}-C_{i})(L_{i}-0.5(O_{i}+C_{i}))
      \end{multline*}
      The function \texttt{run\moment\_ohlc()} calculates running, volume weighted moment estimators,
      <<earl_highfreq_runskew,eval=FALSE,echo=(-(1:1)),fig.width=7,fig.height=8,fig.show='hide'>>=
par(mfrow=c(2,1))  # set plot panels
library(HighFreq)  # load package "HighFreq"
# running volatility
volat <- runmoment_ohlc(ohlc=get(symbol))
# running skew
skew <- runmoment_ohlc(ohlc=get(symbol), 
                         mom_fun="skew_ohlc")
skew <- skew/(volat)^(1.5)
skew[1, ] <- 0
skew <- na.locf(skew)
dates <- "2013-11-11/2013-11-15"
chart_Series(volat[dates], 
            name=paste(symbol, "volatility"))
chart_Series(skew[dates],
            name=paste(symbol, "Skew"), 
            ylim=c(-1, 1))
      @
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_highfreq_runskew-1}
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Daily Volatility and Skew From \protect\emph{OHLC} Data}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      The function \texttt{moment\_ohlc()} calculates the volume weighted moment of a \emph{OHLC} time series,
      <<earl_dailyskew,eval=FALSE,echo=(-(1:6)),fig.width=7,fig.height=8,fig.show='hide'>>=
par(mfrow=c(2,1))  # set plot panels
# define symbol
symbol <- "SPY"
# load OHLC data
output_dir <- "/Users/jerzy/Develop/data/hfreq/scrub/"
symbol <- load(file.path(output_dir, paste0(symbol, ".RData")))
library(HighFreq)  # load package "HighFreq"
# daily volatility and skew
dailyvar <- apply.daily(x=get(symbol), 
                         FUN=moment_ohlc)
colnames(dailyvar) <- 
  paste(symbol, "vol", sep=".")
daily_skew <- apply.daily(x=get(symbol), 
              FUN=moment_ohlc, mom_fun="skew_ohlc")
daily_skew <- daily_skew/(dailyvar)^(1.5)
colnames(daily_skew) <- 
  paste(symbol, "skew", sep=".")
dates <- "2013-06-01/"
chart_Series(dailyvar[dates], 
             name=paste(symbol, "volatility"))
chart_Series(daily_skew[dates],
             name=paste(symbol, "skew"))
      @
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_dailyskew-1}
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Market Crash Prediction Using High Frequency Data}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      Recent research suggests that market crashes are conditioned by declining market liquidity:\\
      Donier et al., \href{http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2583743}{\emph{Why Do Markets Crash?  Bitcoin Data Offers Unprecedented Insights}}
      \vskip1ex
      Market illiquidity is defined as the market price impact resulting from supply-demand imbalance, 
      \vskip1ex
      Market illiquidity is proportional to the market price volatility divided by the square root of trading volume: $\mathcal{L}^{-1} \sim \frac{\sigma}{\sqrt{v}}$
      \vspace{-1em}
      <<earl_crash,eval=FALSE,echo=(-(1:8)),fig.width=7,fig.height=9,fig.show='hide'>>=
par(mfrow=c(3,1))  # set plot panels
# define symbol
symbol <- "SPY"
# load OHLC data
output_dir <- "/Users/jerzy/Develop/data/hfreq/scrub/"
symbol <- load(file.path(output_dir, paste0(symbol, ".RData")))
library(HighFreq)  # load package "HighFreq"
# daily volatility and volume
dailyvar <- apply.daily(x=get(symbol), 
                         FUN=moment_ohlc)
colnames(dailyvar) <- 
  paste(symbol, "vol", sep=".")
daily_volu <- apply.daily(x=get(symbol), 
              FUN=function(datav) sum(Vo(datav)))
colnames(daily_volu) <- 
  paste(symbol, "volume", sep=".")
dates <- "2010"
chart_Series(sqrt(dailyvar[dates]), 
             name=paste(symbol, "std dev"))
chart_Series(daily_volu[dates],
             name=paste(symbol, "volume"))
chart_Series(
  sqrt(dailyvar[dates]/daily_volu[dates]),
  name=paste(symbol, "illiquidity"))
@
    \column{0.5\textwidth}
      \vspace{-1em}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_crash-1}
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Regression of Skews Versus Returns}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      A regression of lagged skews versus returns appears to be statistically significant, especially in periods of high volatility during the financial crisis of 2008-09, 
      \vspace{-1em}
      <<eval=TRUE,echo=(-(1:1))>>=
# skew scatterplot
returnv <- calc_rets(xts_data=get(symbol))
skew <- skew_ohlc(log_ohlc=log(get(symbol)[, -5]))
colnames(skew) <- paste(symbol, "skew", sep=".")
lag_skew <- lag(skew)
lag_skew[1, ] <- 0
datav <- cbind(returnv[, 1], sign(lag_skew))
formulav <- as.formula(paste(colnames(datav)[1], 
          paste(paste(colnames(datav)[-1], 
            collapse=" + "), "- 1"), sep="~"))
formulav
regmod <- lm(formulav, data=as.data.frame(datav))
summary(regmod)$coef
summary(lm(formulav, data=
             as.data.frame(datav["/2011-01-01"])))$coef
summary(lm(formulav, data=
             as.data.frame(datav["2011-01-01/"])))$coef
      @
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_skewscatter-1}
      <<earl_skewscatter,eval=FALSE,echo=TRUE,fig.width=7,fig.height=6,fig.show='hide'>>=
dates <- "2013-12-01/"
plot(formulav, data=datav[dates], 
     xlim=c(-2e-09, 2e-09), 
     cex=0.6, xlab="skew", ylab="rets")
abline(regmod, col="blue", lwd=2)
      @
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Contrarian Strategy Using Skew Oscillator}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      The contrarian skew trading strategy involves long or short positions in a single unit of stock, that is opposite to the sign of the skew, 
      \vskip1ex
      Skew is calculated over one-minute bars, and trades are executed in the following period, 
      \vskip1ex
      The contrarian strategy shows good hypothetical performance before transaction costs, and since it's a liquidity providing strategy, should have very low transaction costs, 
      \vskip1ex
      The contrarian strategy is hyperactive, trading almost 46\% of the time in each period, 
      <<eval=TRUE,echo=(-(1:1))>>=
# contrarian skew trading strategy
# lag the skew to get positions
posv <- -sign(lag_skew)
posv[1, ] <- 0
# cumulative PnL
cumu_pnl <- cumsum(posv*returnv[, 1])
# calculate frequency of trades
50*sum(abs(sign(skew)-sign(lag_skew)))/nrow(skew)
# calculate transaction costs
bid_offer <- 0.001  # 10 bps for liquid ETFs
bid_offer*sum(abs(sign(skew)-sign(lag_skew)))
@
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_skewstrat-1}
      \vspace{-1em}
      <<earl_skewstrat,eval=FALSE,echo=TRUE,fig.width=7,fig.height=6,fig.show='hide'>>=
chart_Series(
  cumu_pnl[endpoints(cumu_pnl, on="hours"), ], 
  name=paste(symbol, "contrarian skew strategy pnl"))
@
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Volume-Weighted Average Price Indicator}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      The Volume-Weighted Average Price (\emph{VWAP}) is an indicator used for momentum strategies, 
      \vskip1ex
      The momentum reverses when the fast-moving \emph{VWAP} crosses the slow-moving one, 
      \vspace{-1em}
      <<earl_vwap_plot,echo=(-(1:1)),eval=FALSE,fig.width=7,fig.height=6,fig.show='hide'>>=
# vwap plot
vwap_short <- 
  vwapv(xtes=get(symbol), lookb=70)
vwap_long <- 
  vwapv(xtes=get(symbol), lookb=225)
vwap_diff <- vwap_short - vwap_long
colnames(vwap_diff) <- 
  paste(symbol, "vwap", sep=".")
dates <- "2010-05-05/2010-05-07"
invisible(
  chart_Series(x=Cl(get(symbol)[dates]), 
               name=paste(symbol, "plus VWAP")))
invisible(
  add_TA(vwap_short[dates], 
         on=1, col="red", lwd=2))
invisible(
  add_TA(vwap_long[dates], 
         on=1, col="blue", lwd=2))
invisible(
  add_TA(vwap_diff[dates] > 0, on=-1, 
         col="lightgreen", border="lightgreen"))
add_TA(vwap_diff[dates] < 0, on=-1, 
       col="lightgrey", border="lightgrey")
      @
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_vwap_plot-1}
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Regression of \protect\emph{VWAP} Versus Returns}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      A regression of the \emph{VWAP} indicator versus returns appears to be statistically significant, 
      \vspace{-1em}
      <<eval=TRUE,echo=(-(1:6))>>=
# vwap scatterplot
returnv <- calc_rets(xts_data=get(symbol))
vwap_short <- vwapv(xtes=get(symbol), lookb=70)
vwap_long <- vwapv(xtes=get(symbol), lookb=225)
vwap_diff <- vwap_short - vwap_long
colnames(vwap_diff) <- paste(symbol, "vwap", sep=".")
lag_vwap <- lag(vwap_diff)
lag_vwap[1, ] <- 0
datav <- cbind(returnv[, 1], sign(lag_vwap))
formulav <- as.formula(paste(colnames(datav)[1], 
          paste(paste(colnames(datav)[-1], 
            collapse=" + "), "- 1"), sep="~"))
formulav
regmod <- lm(formulav, data=as.data.frame(datav))
summary(regmod)$coef
summary(lm(formulav, 
     data=as.data.frame(datav["/2011-01-01"])))$coef
summary(lm(formulav, 
     data=as.data.frame(datav["2011-01-01/"])))$coef
      @
    \column{0.5\textwidth}
    \vspace{-1em}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_vwap_scatter-1}
      <<earl_vwap_scatter,eval=FALSE,echo=TRUE,fig.width=7,fig.height=6,fig.show='hide'>>=
dates <- "2013-12-01/"
plot(formulav, data=cbind(returnv[, 1], lag_vwap)[dates], 
     cex=0.6, xlab="skew", ylab="rets")
abline(regmod, col="blue", lwd=2)
      @
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Momentum Strategy Using \protect\emph{VWAP}}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
  \begin{columns}[T]
    \column{0.5\textwidth}
      The momentum trading strategy involves long or short positions in a single unit of stock, that is equal to the sign of the \emph{VWAP} indicator, 
      \vskip1ex
      The \emph{VWAP} indicator is calculated over one-minute bars, and trades are executed in the following period, 
      \vskip1ex
      The momentum strategy shows good hypothetical performance before transaction costs, 
      \vskip1ex
      The momentum strategy is infrequent, trading only 0.56\% of the time in each period, 
      <<eval=TRUE,echo=(-(1:1))>>=
# momentum trading strategy
# cumulative PnL
cumu_pnl <- cumsum(sign(lag_vwap)*returnv[, 1])
# calculate frequency of trades
50*sum(abs(sign(vwap_diff)-sign(lag_vwap)))/nrow(vwap_diff)
# calculate transaction costs
bid_offer <- 0.001  # 10 bps for liquid ETFs
bid_offer*sum(abs(sign(vwap_diff)-sign(lag_vwap)))
@
    \column{0.5\textwidth}
      \includegraphics[width=0.5\paperwidth,valign=t]{figure/earl_vwap_strat-1}
      \vspace{-1em}
      <<earl_vwap_strat,eval=FALSE,echo=TRUE,fig.width=7,fig.height=6,fig.show='hide'>>=
chart_Series(
  cumu_pnl[endpoints(cumu_pnl, on="hours"), ], 
  name=paste(symbol, "VWAP momentum strategy pnl"))
@
  \end{columns}
\end{block}

\end{frame}


%%%%%%%%%%%%%%%
\subsection{Conclusion}
\begin{frame}[fragile,t]{\subsecname}
\vspace{-1em}
\begin{block}{}
      Open questions:
      \begin{itemize}
        \item is there an interaction between volatility, volume and skew? 
        \item does kurtosis also have predictive value for market direction? 
        \item how can higher moments (skew, kurtosis) help predict market crashes? 
        \item what is the persistence of market anomalies over time? 
        \item what is relationship between returns and cross-section of skew?
        \item This presentation is available on GitHub: \url{https://github.com/algoquant/presentations}
      \end{itemize}
      \vskip1ex
      Acknowledgements:
      \begin{itemize}
        \item \emph{Snowfall Systems} offers the \emph{PortfolioEffect} system for: 
        \includegraphics[scale=0.1]{figure/portfolioeffect-logo-full-200-950.png}
        \begin{itemize}
          \item real-time high frequency market data aggregations and risk metrics, 
          \item real-time portfolio analytics and optimization, 
          \item portfolio hosting, 
          \item \url{https://www.portfolioeffect.com/}
        \end{itemize}
        \item Brian Peterson for Thomson Reuters tick data,
        \item Brian Peterson, Joshua Ulrich, and Jeffrey Ryan for packages \texttt{xts}, \texttt{quantmod}, \texttt{PerformanceAnalytics}, and \texttt{TTR},
      \end{itemize}
\end{block}

\end{frame}



\end{document}