quantstratTutorial_Basic.Rnw

\documentclass[fleqn,xcolor=pdftex,dvipsnames,table]{beamer}

\input{boilerPlateUW}

\usepackage{etoolbox}
\makeatletter
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\AtBeginSection[]
{
   \begin{frame}
       \frametitle{Outline}
       % sectionstyle=<style for current section>/<style for other sections>
       % subsectionstyle=<style for current subsection>/<style for other subsections in current section>/<style for subsections in other sections>
       \tableofcontents[sectionstyle=show/shaded,subsectionstyle=show/show/hide]
   \end{frame}
}
\AtBeginSubsection[]
{
   \begin{frame}
       \frametitle{Outline}
       % sectionstyle=<style for current section>/<style for other sections>
       % subsectionstyle=<style for current subsection>/<style for other subsections in current section>/<style for subsections in other sections>
       \tableofcontents[sectionstyle=show/shaded,subsectionstyle=show/shaded/hide]
   \end{frame}
}

\title{Getting started with quantstrat}
\author{\href{http://faculty.washington.edu/gyollin/}{Guy Yollin}}
\date{R/Finance 2015}
\institute[Copyright \copyright \ 2015]{}
%\logo{\includegraphics[width=0.25in]{logo}}

<<echo=FALSE>>=
#########################################################################
# Copyright (C) 2011-2015 Guy Yollin                                    #
# License: http://www.gnu.org/licenses/gpl.html GPL version 2 or higher #
#########################################################################
@
<<include=FALSE>>=
library(knitr)
opts_chunk$set(tidy=FALSE,cache=FALSE,size='scriptsize',
  fig.path='figures/',fig.show='hide',fig.keep='last',
  fig.align='center',	fig.width=7,	fig.height=5,
  message=FALSE,warning=FALSE)
@
<<echo=FALSE,cache=FALSE>>=
options(width=81,continue=" ",digits=8)
@

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\begin{document}
\normalem
\lstset{language=Pascal,basicstyle=\scriptsize,tabsize=2}

%\frame{\titlepage}
\begin{frame}
\vb
\frametitle{\textcolor{purple}{Computational Finance and Risk Management}}
\begin{center}
\includegraphics[width=0.9\textwidth,keepaspectratio]{images/CFRM_Logo.png} \\ \vspace{2.0\baselineskip}
\LARGE {\textbf{Getting started with quantstrat}} \\ \vspace{2.0\baselineskip}
\large{Guy Yollin} \\ \vq
\scriptsize{Applied Mathematics\\University of Washington}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Outline}
\tableofcontents
\end{frame}

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

\begin{frame}[fragile]
\frametitle{Download data}
<<>>=
library(PerformanceAnalytics)
library(quantmod)
library(lattice)
startDate <- '2010-01-01'  # start of data
endDate <-  '2015-05-01'   # end of data
Sys.setenv(TZ="UTC")       # set time zone
symbols = c("XLF", "XLP", "XLE", "XLY", "XLV", "XLI", "XLB", "XLK", "XLU")
@
\vm
<<echo=FALSE>>=
if(file.exists("XLX.RData"))
{
  load("XLX.RData")
} else {
  getSymbols(symbols, from=startDate, to=endDate, index.class="POSIXct")
  for(symbol in symbols) {
      x<-get(symbol)
      x<-adjustOHLC(x,symbol.name=symbol)
      x<-to.weekly(x,indexAt='lastof',drop.time=TRUE)
      indexFormat(x)<-'%Y-%m-%d'
      colnames(x)<-gsub("x",symbol,colnames(x))
      assign(symbol,x)
  }
  save(list=symbols,file="XLX.RData")
}
@
<<eval=FALSE>>=
  getSymbols(symbols, from=startDate, to=endDate, index.class="POSIXct")
  for(symbol in symbols) {
      x<-get(symbol)
      x<-adjustOHLC(x,symbol.name=symbol)
      x<-to.weekly(x,indexAt='lastof',drop.time=TRUE)
      indexFormat(x)<-'%Y-%m-%d'
      colnames(x)<-gsub("x",symbol,colnames(x))
      assign(symbol,x)
  }
@
\end{frame}

\begin{frame}[fragile]
\frametitle{Compute returns}
<<plotETF,echo=TRUE, fig.keep='all', tidy=FALSE>>=
prices <- NULL
for(i in 1:length(symbols))
  prices <- cbind(prices,Cl(get(symbols[i])))
colnames(prices) <- symbols
returns <- diff(log(prices))[-1, ]
num.ass <- ncol(returns)

xyplot(prices, xlab = "", layout = c(3, 3),type=c("l","g"))

stacked.df <- stack(as.data.frame(returns))
colnames(stacked.df) <- c("returns", "symbol")

densityplot(~returns | symbol, stacked.df, cex = 0.25, xlab="",type=c("l","g"))
@
\end{frame}

\begin{frame}[fragile]
\frametitle{Calculate and plot Bollinger bands}
<<XLFBB,cache=FALSE>>=
args(BBands)
b <- BBands(HLC=HLC(XLF["2013"]), n=20, sd=2)
tail(b)
myTheme<-chart_theme()
myTheme$col$dn.col<-'lightblue'
myTheme$col$dn.border <- 'lightgray'
myTheme$col$up.border <- 'lightgray'
chart_Series(XLF,TA='add_BBands(lwd=2)',theme=myTheme,name="XLF")
@
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic quantstrat strategy example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Quantstrat object model}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsObjectModel.png}
\end{center}
Generic Signal-Based Strategy Modeling:
\begin{itemize}
  \item Instruments contain market data
  \item Indicators are quantitative values derived from market data
  \item Interaction between indicators and market data are used to generate signals (e.g. crossovers, thresholds)
  \item Rules use market data, indicators, signals, and current account/portfolio characteristics to generate orders
  \item Interaction between orders and market data generates transactions
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Basic strategy backtesting workflow for quantstrat}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/QSflowchart2}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Key \texttt{blotter} functions} \vm
\begin{scriptsize}
\begin{center}
\begin{tabular}{ll}
  \multicolumn{2}{c}{\color{blue} \textbf{Initialization}} \\ [0.1em] \hline
  & \\ [-0.9em]
  initPortf & initializes a portfolio object \\ [0.1em]
  & \\ [-0.9em]
  initAcct & initializes an account object \\ [0.1em]
  & \\ [-0.9em] \\ [-0.6em]
  \multicolumn{2}{c}{\color{blue} \textbf{Processing}} \\ [0.1em] \hline
  & \\ [-0.9em]
  addTxn & add transactions to a portfolio \\ [0.1em]
  & \\ [-0.9em]
  updatePortf & calculate P\&L for each symbol for each period \\ [0.1em]
  & \\ [-0.9em]
  updateAcct & calculate equity from portfolio data \\ [0.1em]
  & \\ [-0.9em]
  updateEndEq & update ending equity for an account \\ [0.1em]
  & \\ [-0.9em]
  getEndEq & retrieves the most recent value of the capital account \\ [0.1em]
  & \\ [-0.9em]
  getPosQty & gets position at Date \\ [0.1em]
  & \\ [-0.9em] \\ [-0.6em]
  \multicolumn{2}{c}{\color{blue} \textbf{Analysis}} \\ [0.1em] \hline
  & \\ [-0.9em]
  chart.Posn & chart market data, position size, and cumulative P\&L \\ [0.1em]
  & \\ [-0.9em]
  PortfReturns & calculate portfolio instrument returns \\ [0.1em]
  & \\ [-0.9em]
  getAccount & get an account object from the .blotter environment \\ [0.1em]
  & \\ [-0.9em]
  getPortfolio & get a portfolio object from the .blotter environment \\ [0.1em]
  & \\ [-0.9em]
  getTxns & retrieve transactions from a portfolio \\ [0.1em]
  & \\ [-0.9em]
  tradeStats & calculate trade statistics  \\ [0.1em]
  & \\ [-0.9em]
  perTradeStats & calculate flat to flat per-trade statistics
\end{tabular}
\end{center}
\end{scriptsize}
\end{frame}

\begin{frame}
\frametitle{Key \texttt{quantstrat} functions}
\begin{scriptsize}
\begin{center}
\begin{tabular}{ll}
  \multicolumn{2}{c}{\color{blue} \textbf{Initialization}} \\ [0.1em] \hline
  & \\ [-0.9em]
  initOrders & initialize order container \\ [0.1em]
  & \\ [-0.9em]
  strategy & constructor for strategy object \\ [0.1em]
  & \\ [-0.9em] \\
  \multicolumn{2}{c}{\color{blue} \textbf{Strategy definition}} \\ [0.1em] \hline
  & \\ [-0.9em]
  add.indicator & add an indicator to a strategy \\ [0.1em]
  & \\ [-0.9em]
  add.signal & add a signal to a strategy \\ [0.1em]
  & \\ [-0.9em]
  add.rule & add a rule to a strategy \\ [0.1em]
  & \\ [-0.9em]
  add.distribution & add a distribution to a paramset in a strategy \\ [0.1em]
  & \\ [-0.9em]
  add.constraint & add a constraint on 2 distributions within a paramset \\ [0.1em]
  & \\ [-0.9em] \\
  \multicolumn{2}{c}{\color{blue} \textbf{Processing}} \\ [0.1em] \hline
  & \\ [-0.9em]
  applyStrategy & apply the strategy to arbitrary market data \\ [0.1em]
  & \\ [-0.9em]
  addPosLimit & add position and level limits at timestamp \\ [0.1em]
  & \\ [-0.9em]
  apply.paramset & apply a paramset to the strategy \\ [0.1em]
  & \\ [-0.9em]
  applyStrategy.rebalancing & apply the strategy to data with periodic rebalancing \\ [0.1em]
\end{tabular}
\end{center}
\end{scriptsize}
\footnotetext{The functions in quantstrat are used in conjunction with the functions in \texttt{blotter}}
\end{frame}

\begin{frame}[fragile]
\frametitle{Initialize instruments}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow1}
\end{center} \vspace{-1.5\baselineskip}
<<results='hide'>>=
library(quantstrat)
initDate <- '2009-12-31'
initEq <- 1e6
currency("USD")
stock(symbols, currency="USD", multiplier=1)
@
\begin{itemize}
  \item Initialize currency instrument first and then stock instrument
  \item Important that portfolio, account, and orderbook initialization date be before start of data
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{Initialize portfolio, account, and orders object}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow2}
\end{center} \vspace{-1.5\baselineskip}
<<results='hide'>>=
rm.strat("multiAsset.bb1") # remove portfolio, account, orderbook if re-run
initPortf(name="multiAsset.bb1", symbols, initDate=initDate)
initAcct(name="multiAsset.bb1", portfolios="multiAsset.bb1",
  initDate=initDate, initEq=initEq)
initOrders(portfolio="multiAsset.bb1", initDate=initDate)
@
<<results='hide'>>=
strategy("bbands", store=TRUE)
@
\begin{itemize}
  \item The function \texttt{rm.strat} removes any existing portfolio, account, or orderbook objects which facilitates re-running the code
  \item The function \texttt{strategy} initializes and new strategy object
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{The \texttt{add.indicator} function}
\begin{itemize}
  \item Indicators are typically standard technical or statistical analysis outputs, such as moving averages, bands, or pricing models
  \item Indicators are applied before signals and rules, and the output of indicators may be used as inputs to construct signals or fire rules
\end{itemize}
<<>>=
args(add.indicator)
@
\vq
\textmd{Main arguments:}
\begin{mydescription}{arguments}
  \item [strategy] strategy object
  \item [name] name of the indicator (must be an R function)
  \item [arguments] arguments to be passed to the indicator function
  \item [label] name to reference the indicator
\end{mydescription}
\end{frame}

\begin{frame}[fragile]
\frametitle{Define indicators}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow3}
\end{center} \vspace{-1.5\baselineskip}
<<>>=
args(BBands)
@
<<results='hide'>>=
add.indicator("bbands", name = "BBands",
  arguments = list(HLC = quote(HLC(mktdata)), maType='SMA'), label='bbInd')
@
\begin{itemize}
  \item \texttt{quote()} returns it's argument without evaluating
  \item \texttt{mktdata} is the time series object that holds the current symbols data during evaluation
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{The \texttt{add.signals} function}
quantstrat supports the following signal types:
\begin{mydescription}{sigComparison}
  \item [sigCrossover] crossover signal ("gt", "lt", "eq", "gte", "lte")
  \item [sigComparison] comparison signal ("gt", "lt", "eq", "gte", "lte")
  \item [sigThreshold] threshold signal ("gt", "lt", "eq", "gte", "lte")
  \item [sigPeak] peak/valley signals ("peak", "bottom")
  \item [sigFormula] signal calculated from a formula
\end{mydescription}
<<>>=
args(add.signal)
@
\vq
\textmd{Main arguments:}
\begin{mydescription}{arguments}
  \item [strategy] strategy object
  \item [name] name of the signal, must correspond to an R function
  \item [arguments] arguments to be passed to the signal function
\end{mydescription}
\end{frame}

\begin{frame}[fragile]
\frametitle{Define signals}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow3}
\end{center} \vspace{-1.5\baselineskip}
<<results='hide'>>=
add.signal("bbands", name="sigCrossover",
  arguments=list(columns=c("High","up"),relationship="gt"),
  label="H.gt.UpperBand")
@
<<results='hide'>>=
add.signal("bbands", name="sigCrossover",
  arguments=list(columns=c("Low","dn"),relationship="lt"),
  label="L.lt.LowerBand")
@
\end{frame}

\begin{frame}[fragile]
\frametitle{The \texttt{add.rules} function}
The function \texttt{add.rule} adds a rule to a strategy \\
<<>>=
args(add.rule)
@
\vh
\textmd{Main arguments:}
\begin{mydescription}{arguments}
  \item [strategy] strategy object
  \item [name] name of the rule (typically \texttt{ruleSignal})
  \item [arguments] arguments to be passed to the rule function
  \item [type] type of rule ("risk","order","rebalance","exit","enter")
  \item [label] user supplied text label for rule
\end{mydescription}
\end{frame}

\begin{frame}[fragile]
\frametitle{The \texttt{ruleSignal} function}
\texttt{ruleSignal} is the default rule to generate a trade order on a signal \\
<<>>=
args(ruleSignal)
@
\textmd{Main arguments:}
\begin{mydescription}{ordertype}
  \item [sigcol] column name to check for signal
  \item [sigval] signal value to match
  \item [orderqty] quantity for order or 'all', modified by osFUN
  \item [ordertype] "market","limit","stoplimit","stoptrailing","iceberg"
  \item [orderside] "long", "short", or NULL
  \item [osFUN] function or name of order sizing function (default is osNoOp)
\end{mydescription}
\end{frame}

\begin{frame}[fragile]
\frametitle{Add rules}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow3}
\end{center} \vspace{-1.5\baselineskip}
<<results='hide'>>=
add.rule("bbands", name='ruleSignal',
  arguments=list(sigcol="H.gt.UpperBand",sigval=TRUE,
    orderqty=+100, ordertype='market', orderside='long'),
  type='enter',
  label='LongEntry')
@
<<results='hide'>>=
add.rule("bbands", name='ruleSignal',
  arguments=list(sigcol="L.lt.LowerBand",sigval=TRUE,
    orderqty= 'all', ordertype='market', orderside='long'),
  type='exit',
  label='LongExit')
@
\begin{itemize}
  \item Long-only channel breakout system with pyramiding
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{The \texttt{applyStrategy} function}
The \texttt{applyStrategy} function applies the strategy to a portfolio and generates transactions according to the strategy rules and the market data \\
<<>>=
args(applyStrategy)
@
\vh
\textmd{Main arguments:}
\begin{mydescription}{parameters}
  \item [strategy] an object of type 'strategy'
  \item [portfolios] a list of portfolios to apply the strategy to
  \item [parameters] named list of parameters to be applied during evaluation of the strategy
\end{mydescription}
\end{frame}

\begin{frame}[fragile]
\frametitle{Applying strategy to a multi-asset portfolio}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow4}
\end{center} \vspace{-1.5\baselineskip}
<<results='hide'>>=
nSD = 2
nMA = 20
@
<<results='hide'>>=
out <- applyStrategy("bbands",
  portfolios="multiAsset.bb1",parameters=list(sd=nSD,n=nMA))
@
\begin{itemize}
  \item Indicator parameters can be passed when applying the strategy; for this run the length of the moving average is \Sexpr{nMA} and the standard deviation multiplier is \Sexpr{nSD}
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{Apply the strategy}
Calling \texttt{applyStrategy} generates transactions in the specified portfolio.
<<echo=F>>=
options(width=120)
@
<<size='tiny'>>=
getTxns(Portfolio="multiAsset.bb1", Symbol="XLK")
@
<<echo=F>>=
options(width=81)
@
\end{frame}

\begin{frame}[containsverbatim]
\frametitle{The \texttt{mktdata} object}
\texttt{mktdata} is a special variable constructed during the execution of \texttt{applyStrategy}.  It is a time series object which contains the historic price data for the current symbol being evaluated as well as the calculated indicators and signals:\\ \vq
<<echo=FALSE>>=
options(width=180,digits=6)
@
<<size='Tiny'>>=
mktdata["2015"]
@
<<echo=FALSE>>=
options(width=81,digits=8)
@
\begin{itemize}
  \item Inspecting \texttt{mktdata} can be very helpful in understanding strategy processing and debugging
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{Update portfolio and account}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow5}
\end{center} \vspace{-1.5\baselineskip}
<<results='hide'>>=
updatePortf("multiAsset.bb1")
updateAcct("multiAsset.bb1")
updateEndEq("multiAsset.bb1")
@
\end{frame}

\begin{frame}[fragile]
\frametitle{Data integrity check}
<<size='Tiny'>>=
checkBlotterUpdate <- function(port.st,account.st,verbose=TRUE)
{
  ok <- TRUE
  p <- getPortfolio(port.st)
  a <- getAccount(account.st)
  syms <- names(p$symbols)
  port.tot <- sum(sapply(syms,FUN = function(x) eval(parse(
    text=paste("sum(p$symbols",x,"posPL.USD$Net.Trading.PL)",sep="$")))))
  port.sum.tot <- sum(p$summary$Net.Trading.PL)
  if( !isTRUE(all.equal(port.tot,port.sum.tot)) ) {
    ok <- FALSE
    if( verbose )
      print("portfolio P&L doesn't match sum of symbols P&L")
  }
  initEq <- as.numeric(first(a$summary$End.Eq))
  endEq <- as.numeric(last(a$summary$End.Eq))
  if( !isTRUE(all.equal(port.tot,endEq-initEq)) ) {
    ok <- FALSE
    if( verbose )
      print("portfolio P&L doesn't match account P&L")
  }
  if( sum(duplicated(index(p$summary))) ) {
    ok <- FALSE
    if( verbose )
      print("duplicate timestamps in portfolio summary")
  }
  if( sum(duplicated(index(a$summary))) ) {
    ok <- FALSE
    if( verbose )
      print("duplicate timestamps in account summary")
  }
  return(ok)
}
checkBlotterUpdate("multiAsset.bb1","multiAsset.bb1")
@
\end{frame}

\begin{frame}
\frametitle{How the \texttt{blotter\_portfolio} object gets updated} \vm \vm
\begin{center}
\includegraphics[width=0.95\textwidth,keepaspectratio]{images/blotterClassDiagram2.pdf}
\end{center}
\end{frame}

\begin{frame}[fragile]
\frametitle{Generate position plots}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{images/qsFlow6}
\end{center} \vspace{-1.5\baselineskip}
<<CHARTPOSNPANEL,fig.width=24,fig.height=18>>=
par(mfrow=c(3,3))
for(symbol in symbols)
{
chart.Posn(Portfolio="multiAsset.bb1",Symbol=symbol,theme=myTheme,
	TA="add_BBands(n=20,sd=2)")
}
par(mfrow=c(1,1))
@
<<XLBCP,cache=FALSE>>=
chart.Posn("multiAsset.bb1","XLU",TA="add_BBands(n=20,sd=2)",theme=myTheme)
@
\end{frame}

\begin{frame}
\frametitle{Position plots} \vm
\begin{center}
\includegraphics[width=0.90\textwidth,keepaspectratio]{figures/CHARTPOSNPANEL-1}
\end{center}
\end{frame}

\begin{frame}
\frametitle{BBands strategy for XLU} \vm
\begin{center}
\includegraphics[width=0.95\textwidth,keepaspectratio]{figures/XLBCP-1}
\end{center}
\end{frame}

\begin{frame}[containsverbatim]
\frametitle{Trade stats by instrument}
<<MULTITRADESTATS,echo=TRUE,fig.width=12,fig.height=9>>=
textplot(t(tradeStats("multiAsset.bb1")))
@
\vspace{-2.25\baselineskip}
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{figures/MULTITRADESTATS-1}
\end{center}
\end{frame}

\begin{frame}[fragile]
\frametitle{Individual asset returns}
<<echo=FALSE>>=
options(width=78)
@
<<IASSRET,cache=FALSE>>=
rets.multi <- PortfReturns("multiAsset.bb1")
colnames(rets.multi) <- sort(symbols)
round(tail(rets.multi,5),6)
chart.CumReturns(rets.multi, colorset= rich10equal, legend.loc = "topleft",
  main="SPDR Cumulative Returns",minor.ticks=FALSE)
@
<<echo=FALSE>>=
options(width=81)
@
\end{frame}

\begin{frame}
\frametitle{Cumulative returns by asset} \vm
\begin{center}
\includegraphics[width=0.95\textwidth,keepaspectratio]{figures/IASSRET-1}
\end{center}
\end{frame}

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\section{Position sizing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Stop orders}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Parameter optimization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{\textcolor{purple}{Computational Finance and Risk Management}}
\begin{center}
\includegraphics[width=0.75\textwidth,keepaspectratio]{images/CFRM_Logo.png} \\ \vb
\url{http://depts.washington.edu/compfin}
\end{center}
\end{frame}

\end{document}