An application of the method of moments to volatility estimation using daily high, low, opening and closing prices %\footnote{Portions of this article were contributed by C. Buescu during postdoctoral %studies at University of Missouri. %%This research was supported by NSF Grant DMS-0505435. %}
Fatoumata J. Kon\'{e} %\footnote{address, phone, email} %
%quote} %{We use the %density and expectation of the range of an arithmetic Brownian motion and the method of moments on the daily high, low, opening and closing prices to estimate the %intra-day volatility of the stock price. The daily price jump at the opening is %then %incorporated %in the process of annualizing intra-day volatility %by considering considered to be the result of the unobserved evolution of an after-hours virtual trading day. The annualized volatility is used to calculate Black-Scholes prices for European options, and a trading strategy is devised to profit when these prices differ flagrantly from the market prices. %With the estimated parameters we price European options and compare %them to market prices. We compute the profit to be made when the market %grossly misprices these options.% of the resulting arbitrage opportunity. %} %
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% \usepackage[final]{graphicx}
%       \special{pdf: docinfo << /Title    (Vol estimation using high, low, opening and closing prices)
%                                 /Subject  (volatility estimation, method of moments)
%                                 /Author   (C. Buescu, M. Taksar, F.J. Kone)
%                                 /Keywords (density and expectation of range of arithmetic BM)
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\fancyhead[RO]{Volatility estimation using high-low-open-close}
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\begin{document}
\title{An application of the method of moments to volatility estimation 
using daily high, low, opening and closing prices
%\footnote{Portions of this article were contributed by C. Buescu during postdoctoral
%studies at University of Missouri.
%%This research was supported by NSF Grant DMS-0505435.
%}
}
%%\author{Cristin Buescu
%\footnote{Department of Mathematics, King's College, Strand, London, UK, WC2R 2LS, +44-20-7848-2226, %cristin.buescu@kcl.ac.uk}
%\footnote{Significant contributions of C. Buescu were made during his
%postdoctoral studies at University of Missouri.}
%\and Michael Taksar
%\footnote{Mathematics Department, 202 Mathematical Sciences Bldg, University of Missouri,
%Columbia, MO 65211, USA, +1-573-882-7674, taksarm@missouri.edu
%}
% \and Fatoumata J. Kon\'{e}
%\footnote{address, phone, email}
%}
%%\institute{UNIV}


%\date{Sept. 10, 2010}


%\begin{titlepage}
\maketitle


%\tableofcontents
%\newpage


%\vspace*{0.5cm}
%%\begin{center}
%\noindent
%{\bf \Large
%Drift and volatility estimation using daily highs and lows
%DRIFT AND VOLATILITY ESTIMATION USING DAILY HIGHS AND LOWS
%Optimal portfolio management in markets with asymmetric taxation
%\footnote{This research was supported by NSF Grant DMS-0505435.}
%%PTIMAL PORTFOLIO MANAGEMENT IN MARKETS WITH ASYMMETRIC TAXATION$^{*}$
%}
%%\end{center}

%\bigskip

%\begin{center}
%This version: Sep. 5, 2008.
%\end{center}

\bigskip

\begin{center}
%{\bf Cristin Buescu$\dagger$, Michael Taksar$\ddagger$}
\noindent{\bf Cristin Buescu}

%\footnote{Please address all correspondence to: buescu\_cristin\_2004@yahoo.com} \\
%Department of Mathematics, King's College, Strand, London, UK, WC2R 2LS
Department of Mathematics, King's College, London
%\\ph: +44-20-7848-2226,  email: cristin.buescu@kcl.ac.uk

\medskip

\noindent{\bf Michael Taksar}\footnote{This work was supported by 
the Norwegian Research Council: Forskerprosjekt ES445026, ``Stochastic
Dynamics of Financial Markets."}\\
Mathematics Department, University of Missouri
%Mathematics Department, 202 Mathematical Sciences Bldg, University of Missouri,\\
%Columbia, MO 65211, USA %; ph: +1-573-882-7674, email: taksarm@missouri.edu

\medskip

\noindent{\bf  Fatoumata J. Kon\'{e}}\\
Citibank, London
                %, 33 Canada Square, London U.K. E14 5LB
%\bigskip
%Mathematics Department\\
%202 Mathematical Sciences Bldg.\\
%University of Missouri\\
%Columbia, MO 65211 USA
%United States%\\
%\{cristin,taksar\}@math.missouri.edu
%\end{center}

%\begin{center}
%\sl The Journal of Computational Finance 4(11), 2008
%Apr. 16, 2008 %Jan. 24, 2008 %May 31, 2006.
%FINAL revision
%revision based upon editor's comments
\end{center}


%\vspace*{0.2in}
%\vspace*{4.3in}
%\noindent
%\rule{110mm}{0.1mm}
%\footnotetext{testing}
%\noindent
%${\mbox{}}^{*}$
%\noindent{This research was supported by NSF Grant DMS-0505435.}\\
%$\dagger$ Corresponding Author, Post Doctoral Fellow: 
%%{buescu\_cristin\_2004@yahoo.com} 
%cristin@math.missouri.edu, (573) 882-4884\\
%$\ddagger$ Professor of Mathematics: taksar@math.missouri.edu, (573) 882-7674
%\newpage

%\begin{center}
%{\bf 
%OPTIMAL PORTFOLIO MANAGEMENT IN MARKETS WITH ASYMMETRIC TAXATION %$^{*}$
%}
%\end{center}

%\newpage

%\begin{center}
%{\bf \large An application of the method of moments to volatility estimation 
%using daily high, low, opening and closing prices}
%\end{center}

\medskip

\begin{center}
 {\bf Abstract}
\end{center}
%\noindent



%\begin{abstract}%quote}
%{\it 
We use the %density and 
expectation of the range of an arithmetic Brownian
motion and the method of moments on the daily high, low, opening and closing
prices to estimate the 
%intra-day 
volatility of the stock price. 
The daily price jump at the opening is 
%then 
%incorporated 
%in the process of annualizing intra-day volatility 
%by considering 
considered to be the result of the unobserved evolution of an after-hours 
virtual trading day.
The annualized volatility is used to calculate Black-Scholes prices for European options,
and a trading strategy is devised to profit when these prices differ flagrantly from
the market prices.
%With the estimated parameters we price European options and compare
%them to market prices. We compute the profit to be made when the market
%grossly misprices these options.% of the resulting arbitrage opportunity.
%}
%\end{abstract}%quote}

\bigskip

\noindent{\bf Key words}: Range-based volatility estimation, method of moments, 
daily high, low, opening and closing prices, density and expectation of the 
range of an arithmetic Brownian motion.
%Markov chain approximation technique to obtain
%numerical solutions.

%We consider a financial market with a riskless asset (bank account)
%and a risky asset (stock). For each transaction, brokerage fees are paid as
%a fixed proportion of the portfolio value. Profits are taxed at a fixed rate, 
%while for losses the investor obtains a tax credit. The objective is to 
%maximize the long-run growth rate of the cash value of the wealth. 
%
%We investigate the case when there are two tax rates in effect: one for 
%short-term investments
%that are held less than one year, and another one, smaller, for long-term 
%investments.

\bigskip


%{KEY WORDS}: Portfolio management, asymmetric taxation, ergodic impulse control,
%%Markov chain approximation,
%viscosity solution
%%, HJB equation, Markov chain approximation. %taxes, transaction costs.

%%{\em Journal of Economic
%%Literature} Classification Numbers:
%%C61, D81, F31, G15.

%%{JEL classification:}  G11, H20, C63.
%
%%{\bf AMS (MOS) subject classifications:} 90A09, 93E20, 60H30, 60G44, 90A16.
%

%%{AMS (MSC2000) subject classifications:} 91B28, 91B70, 60G40, 60G44, 62L15, 62L20\\
%%
%%%91B28=Finance, portfolios,investment/91B70=Stochastic Models/60G40=Stopping Times:
%%%optimal stopping problems/60G44=Martingales with continuous parameter/62L15=Optimal
%%%Stopping/62L20=Stochastic Approximation
%
%%{\bf Abbreviated title:} Taxes and Transaction Costs.
%


%%\vspace*{0.2in}
%%\vspace*{2.3in}
%%\noindent
%%\rule{110mm}{0.1mm}

%%\noindent
%${\mbox{}}^{*}$
%\noindent{This research was supported by NSF Grant DMS-0505435.}

%\end{titlepage}



%\newpage








%{\em Journal of Economic
%Literature} Classification Numbers:
%C61, D81, F31, G15.

%{JEL classification:}  G11, H20, C63.
%
%%{\bf AMS (MOS) subject classifications:} 90A09, 93E20, 60H30, 60G44, 90A16.
%

{\bf AMS subject classifications:} 91G20, 60J65, 62F10, 62G05, 62P05%, 91B70, 60G40, 60G44, 62L15, 62L20\\
%
%%91B28=Finance, portfolios,investment/91B70=Stochastic Models/60G40=Stopping Times:
%%optimal stopping problems/60G44=Martingales with continuous parameter/62L15=Optimal
%%Stopping/62L20=Stochastic Approximation 62f10 point estimation 62g05 estimation 
%62p05 applications to actuarial sciences and financial mathematics
%91g20 derivative securities 60j65 brownian motion
%
%{\bf Abbreviated title:} Taxes and Transaction Costs.
%

{\bf JEL classification:}  G12, G13, C13, C46.
%c13 estimation:general g12 asset pricing
%C46 specific distributions; specific statistics
%\vspace*{0.2in}
%\vspace*{2.3in}
%\noindent
%\rule{110mm}{0.1mm}

%\noindent
%${\mbox{}}^{*}$
%\noindent{This research was supported by NSF Grant DMS-0505435.}

\noindent

\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
%\end{titlepage}



%\newpage

\section{Introduction}
\recount
\recoup
\setcounter{equation}{0}

%In recent years there has been an increase in the use of the
%maximum drawdown and range-based volatility as measures of risk, of special
%relevance to hedge funds. An important ingredient of these measures is
%the range of an arithmetic Brownian motion, whose density and expectation 
%were obtained relatively recently.

This article is a modified version of what has been studied in the Ph.D. thesis
of Kon\'{e} (1996). It concerns the application of the method of moments to 
range-based volatility estimation using daily high, low, opening and closing 
stock prices. 
%The Black-Scholes framework with constant volatility and the daily data are 
%chosen for reasons not unlike those of Rogers and Zhou (2008).
Aiming to estimate volatility and not to measure it, we assume a Black-Scholes
framework with constant volatility and use daily data to achieve it (see 
Rogers and Zhou (2008) for further motivation for this choice).

Incidental to this is the derivation of the density and expectation of the
range of an arithmetic Brownian motion. Subsequent to this thesis, portions of 
it have been studied for different purposes (see, for instance, Sutrick et al
(1997) for the use of the density of the range of an arithmetic Brownian motion
in the {\em do-nothing option}, or Magdon-Ismail et al (2000, 2004) for the use of 
the expectation of the range of an arithmetic Brownian motion in different
contexts). In particular, expressing the density of the range in the context
of Sutrick et al (1997) corrects their expression.
%We refer the reader to additional literature, although not 
%directly related to the methods of the current article, in Domin\'{e} (1996)
%and
%Douady et al (2000). 

The literature on range-based volatility estimation includes classic work by
Garman and Klass (1980), Parkinson (1980), Rogers and Satchell (1991) and 
Rogers et al (1994), whose estimators are reviewed in Yang and Zhang (2000). 
Of these, the latter paper is most related to the current one because it
considers after-hours price jumps in addition to drift. However, the methods 
presented here are different\footnote{see
Remarks \ref{important} and \ref{comparison}}, 
and perhaps more practical (see the remarks of 
Chan and Lien (2003) on the empirical availability of certain parameters in 
Yang and Zhang (2000)).

Starting from the joint density of the running maximum and the current value of
an arithmetic Brownian motion, the density of their difference (referred to as
half-range) is obtained, and its expectation computed. This allows the 
computation of the expectation of the full range (defined as maximum minus 
minimum), which will then be used in the method of moments for intra-day
volatility estimation.

After-hours arrival of information results in price jumps at the opening, and
we model this as a virtual trading day which is unobservable, but which,
when succeeding the trading day, gives on average the complete statistical
representation of one day.
%ose average
%contribution to overall annual volatility is a fraction of that of an average
%trading day. This fraction allows adjusting the number of trading days to be
%considered when annualizing the intra-day volatility.

Black-Scholes option prices are computed using the parameter estimates, and
when they differ the most from the observed market prices a profit is made by
an appropriately devised trading strategy.


%We note that the probabilistic results used as starting points by Kon\'{e}
%(1996) and Sutrick et al (1997) are very close
%%not only related, but are in fact equivalent 
%(see Remark \ref{equiv}). Specifically, a result presented by 
%Karatzas and Shreve (1988) and traced to Feller (1951) is equivalent to that
%of Billingsley (1968).

The paper is organized as follows. In Section 2 we derive the expectation and
the density function of an arithmetic Brownian motion. In Section 3 the
method of moments is used to estimate the parameters of the stock price based on
daily high, low, opening and closing data. The estimated parameters are then used in Section 4
to price European options on the stock, which are then compared to market
prices to identify instances of flagrant differences.
The effect of the mispricing is estimated by computing the 
profit to be made in these opportunities. We conclude in Section 5 with 
some comments on the efficiency of the method of moments in volatility 
estimation.
% the volatility of the stock price.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The range of an arithmetic Brownian motion: expectation and density }
\recount
\recoup
Let $\{\Omega,{\cal F},P\}$ be a probability space 
endowed with a filtration $\{{\cal F}_{t}\}_{t\geq 0}$,
% that is the $P$-augmentation of the
%filtration generated by 
and let $\{W_{t}\}_{t\geq 0}$ be a one-dimensional standard Brownian motion
adapted to $\{{\cal F}_{t}\}_{t\geq 0}$. %with $W_0=0$.
For $t\geq 0$ let $X_t$ denote a standard arithmetic Brownian motion with 
drift $\mu$ and volatility $\sigma>0$:
\begin{equation}
X_t =  \mu \;t +\sigma \;W_t ,\;\;X_0=0,
          \label{equation1}
\end{equation}
and let $\{M_{t}\}_{t\geq 0}$, $\{m_{t}\}_{t\geq 0}$ and $\{R_{t}\}_{t\geq 0}$
denote its running maximum, minimum, and range, respectively:
\begin{equation}
M_t:=\sup_{0\leq s\leq t}{X_s},\;\;m_t=\inf_{0\leq s\leq t}{X_s},\;\;
R_t:=M_t-m_t.
\label{equation2}
\end{equation}

\medskip

First we derive the expectation $E[R_t]$ of the range of the arithmetic Brownian
motion $X_t$. This is achieved by computing the density and expectation of the
half-range $M_t-X_t$ from the joint density of $X_t$ and $M_t$.
 
\begin{Lemma} \label{lemma1}
%(3.2.3 of Kon\'{e}) 
The joint density function of an arithmetic Brownian motion and its
running maximum can be expressed as:
\begin{equation}
P(X_t\in da,M_t\in db)=\frac{2(2b-a)}{\sqrt{2\pi t^3}\sigma^3}\exp{
\left\{-\frac{(2b-a)^2}{2t\sigma^2}+\frac{\mu}{\sigma^2}a-\frac 12
\frac{\mu^2}{\sigma^2}t\right\}}\;da\;db. \label{jointmM}
\end{equation}
\end{Lemma}

\begin{proof} The proof is standard. 
%follows the idea of Theorem 3.2.3 of Kon\'{e} (1996).
Using the martingale $Z_t(Y)=\exp\{W_t\mu/\sigma-t\mu^2/(2\sigma^2)\}$, 
Girsanov's change of measure defines a new probability measure $\tilde{P}$ for
any measurable set $A$ by $\tilde{P}(A)=E[Z_t(Y)1_A]$.
Theorem 3.2.2 of Karatzas and Shreve (1988) with $Y_t=\mu/\sigma$ and
$N_t=\sigma W_t$ gives that $\tilde{N}_t=\sigma W_t-\mu t$ is a local
martingale.
%Under the new probability measure $\tilde {P}$ 
The process $\tilde{W}_t=W_t-t\mu/\sigma$ is 
a Brownian motion under the new probability measure $\tilde{P}$. Equivalently, 
$\sigma W_t=\mu t+\sigma \tilde{W}_t$ is a Brownian
motion with drift under $\tilde{P}$, and we can write:
\begin{eqnarray}
P(X_t\leq a, {M}_t\leq b)&=&\tilde{P}(\sigma W_t\leq a, \sigma \bar{M}_t\leq b)
\nonumber\\
&=&\int_{-\infty}^{a} \exp\Big{(}\frac{\mu}{\sigma^2}x
-\frac 12 \frac{\mu^2}{\sigma^2}t\Big{)}
\;P(\sigma W_t\in dx, \sigma \bar{M}_t\leq b).\label{jtildeP}
\end{eqnarray}
%where $X_t$ is the original Brownian motion with drift, $M_t$ its running
%maximum, and $\bar{M}_t$ is the running maximum of $W_t$.

An application of the reflection principle gives:
%(recall that $\bar{M}_t$ is the running maximum of the Brownian motion $W_t$)
\begin{eqnarray*}
P(\sigma W_t\leq a, \sigma \bar{M}_t\leq b)&=&
%P\Big{(}W_t\leq \frac{a}{\sigma}, 
%\bar{M}_t\leq \frac{b}{\sigma}\Big{)}
P\Big{(}W_t\leq \frac{a}{\sigma}\Big{)}
-P\Big{(}W_t\leq \frac{a}{\sigma},\sigma \bar{M}_t> b\Big{)}\\
&=&P\Big{(}W_t\leq \frac{a}{\sigma}\Big{)}-P\Big{(}W_t>\frac{2b-a}{\sigma}\Big{)}
\\
&=&\Phi\Big{(}\frac{a}{\sigma\sqrt{t}}\Big{)}-1+
\Phi\Big{(}\frac{2b-a}{\sigma\sqrt{t}}\Big{)},
\end{eqnarray*}
where we denote by $\phi(\cdot)$ and $\Phi(\cdot)$ the standard normal density 
and cumulative distribution functions, respectively. 

Differentiating the formula above with respect to $a$ gives:
\begin{equation}
P(\sigma W_t\in da, \sigma \bar{M}_t\leq b )=\frac{1}{\sigma \sqrt{t}}
\Bigg{(}\phi\Big{(}\frac{a}{\sigma\sqrt{t}}\Big{)}-
\phi\Big{(}\frac{2b-a}{\sigma\sqrt{t}}\Big{)}\Bigg{)}.\label{jP}
\end{equation}

%Under the new measure $\tilde{P}$ we have:
%\begin{eqnarray*}
%\tilde{P}(\sigma W_t\leq a, \sigma \bar{M}_t\leq b)
%=\int_{-\infty}^{a} \exp\Big{(}\frac{\mu}{\sigma^2}x
%-\frac 12 \frac{\mu^2}{\sigma^2}t\Big{)}
%\;P(\sigma W_t\in dx, \sigma \bar{M}_t\leq b)\\
%=\int_{-\infty}^{a} \exp\Big{(}\frac{\mu}{\sigma^2}x
%-\frac 12 \frac{\mu^2}{\sigma^2}t\Big{)}
%\frac{1}{\sigma\sqrt{t}}\Bigg{(}\phi\Big{(}\frac{x}{\sigma\sqrt{t}}\Big{)}-
%\phi\Big{(}\frac{2b-x}{\sigma\sqrt{t}}\Big{)}\Bigg{)}dx.
%\end{eqnarray*}
Replacing (\ref{jP}) in (\ref{jtildeP}) and differentiating first with respect to 
$a$ gives:
\begin{eqnarray*}
%\tilde{P}(\sigma W_t\in da, \sigma \bar{M}_t\leq b)
P(X_t\in da, M_t\leq b)=\frac{1}{\sigma\sqrt{t}}
\exp\Big{(}\frac{\mu}{\sigma^2}a-\frac 12 \frac{\mu^2}{\sigma^2}t\Big{)}
\Bigg{(}\phi\Big{(}\frac{a}{\sigma\sqrt{t}}\Big{)}-
\phi\Big{(}\frac{2b-a}{\sigma\sqrt{t}}\Big{)}\Bigg{)}da,
\end{eqnarray*}
and differentiating then with respect to $b$ gives:
\begin{eqnarray*}
%\tilde{P}(\sigma W_t\in da, \sigma \bar{M}_t\in db)
P(X_t\in da, M_t \in db)=\frac{1}{\sigma\sqrt{t}}
\exp\Big{(}\frac{\mu}{\sigma^2}a-\frac 12 \frac{\mu^2}{\sigma^2}t\Big{)}
\Big{(}-\frac{2}{\sigma\sqrt{t}}\Big{)}\phi'\Big{(}\frac{2b-a}{\sigma\sqrt{t}}
\Big{)}\;da\;db\\
=\Big{(}-\frac{2}{t\sigma^2}\Big{)}\exp\Big{(}\frac{\mu}{\sigma^2}a-
\frac 12 \frac{\mu^2}{\sigma^2}t\Big{)}\frac{1}{\sqrt{2\pi}}
\exp\Big{(}-\frac 12\Big{(}\frac{2b-a}{\sigma\sqrt{t}}\Big{)}^2\Big{)}
\Big{(}-\frac 12\Big{)}2\frac{2b-a}{\sigma\sqrt{t}}\;da\;db\\
=\frac{2(2b-a)}{\sqrt{2\pi t^3}\sigma^3}\exp{
\left\{-\frac{(2b-a)^2}{2t\sigma^2}+\frac{\mu}{\sigma^2}a-\frac 12
\frac{\mu^2}{\sigma^2}t\right\}}\;da\;db.
\end{eqnarray*}
The joint density $f_{X_t,M_t}(a,b)$ is given by the term multiplying $da\;db$ 
above.
\end{proof}

\begin{Remar}
This is one of those results that seemed to be always at hand (it can be
obtained from equation (1.8.8) of Harrison (1985)), but never derived.
Note the typo in Yang and Zhang (2000), whose expression (B1) has a plus for the
first fraction in the exponential.  For $\sigma=1$ this was used in Example E5
of Karatzas and Shreve (1998) in relationship to Clark's formula to obtain
explicitly the hedging portfolio.
\end{Remar}


\medskip

The density of the half-range $M_t-X_t$ can be obtained using a standard
two-dimensional transformation of the above joint density.

\begin{Lemma} \label{lemma2}
The density of the half-range $M_t-X_t$ is given by:
\begin{eqnarray}
f_{M_t-X_t}(c)=2\frac{\mu}{\sigma^2}\;\Phi\Big{(}
\frac{\mu t-c}{\sigma\sqrt{t}}\Big{)}
\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}
+\frac{2}{\sigma\sqrt{2t\pi}}
\exp\left\{-\frac{(\mu t +c)^2}{2t\sigma^2}\right\}.
%\underline{
%\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}
\end{eqnarray}
\end{Lemma}
\begin{proof}
%If $X_1$ and $X_2$ are two random variables, and
For $Y_1=X_1+X_2$ and $Y_2=X_1-X_2$ 
%is a two-dimensional transformation, 
the joint density is
%ies are related through the formula:
%\begin{equation}
$
f_{Y_1,Y_2}(y_1,y_2)=\frac 12 f_{X_1,X_2}\Big{(}\frac{y_1+y_2}{2},\frac{y_1-y_2}{2}\Big{)}.
$
%\end{equation}
Taking $X_1=M_t$ and $X_2=X_t$ and using Lemma \ref{lemma1} gives: % $Y_2=M_t-X_t$ and:
\begin{eqnarray*}
f_{Y_1,Y_2}(y_1,y_2)=\frac 12 f_{M_t,X_t}\Big{(}\frac{y_1+y_2}{2},\frac{y_1-y_2}{2}\Big{)}
=\frac 12 \;f_{X_t,M_t}\Big{(}\frac{y_1-y_2}{2},\frac{y_1+y_2}{2}\Big{)}\\
=\frac 12 \;\frac{2y_1+2y_2-y_1+y_2}{\sqrt{2\pi t^3}\sigma^3}
\exp{\left\{-\frac{(2y_1+2y_2-y_1+y_2)^2}{8t\sigma^2}+\frac{\mu}{\sigma^2}
\frac{y_1-y_2}{2}-\frac 12 \frac{\mu^2}{\sigma^2}t\right\}}\\
=\frac 12 \;\frac{y_1+3y_2}{\sqrt{2\pi t^3}\sigma^3}
\exp{\left\{-\frac{(y_1+3y_2)^2}{8t\sigma^2}+\frac{\mu}{\sigma^2}
\frac{y_1-y_2}{2}-\frac 12 \frac{\mu^2}{\sigma^2}t\right\}}\\
=\frac 12 \;\frac{y_1+3y_2}{\sqrt{2\pi t^3}\sigma^3}
\exp{\left\{-\frac 12 \frac{(y_1+3y_2-2\mu t)^2}{4t\sigma^2}
-2 \frac{\mu}{\sigma^2}y_2\right\}}
\end{eqnarray*}
Note that $M_t\geq 0$ implies %$X_t+M_t\geq X_t-M_t$, or 
$Y_1\geq -Y_2$, thus the marginal density of $Y_2=M_t-X_t$ is:
\begin{eqnarray*}
f_{Y_2}(y_2)=\int_{-y_2}^{\infty}f_{Y_1,Y_2}(y_1,y_2)dy_1.
\end{eqnarray*}
%, so $Y_1+3Y_2\geq 2Y_2$.

A change of variable $z=(y_1+3y_2-2\mu t)/(2\sigma\sqrt{t})$ gives: 
\[
z>z_0:=\frac{y_2-\mu t}{\sigma\sqrt{t}},\;\;
dy_1=2\sigma\sqrt{t}dz,
\]
therefore:
\begin{eqnarray*}
&&f_{Y_2}(y_2)=\int_{z_0}^{\infty}\frac 12 \frac{2z\sigma\sqrt{t}+2\mu t}
{\sqrt{2\pi t^3}\sigma^3}\exp{\left\{-\frac 12 {z^2}
-2 \frac{\mu}{\sigma^2}y_2\right\}}2\sigma\sqrt{t}dz\\
&=&\frac{2}{\sigma\sqrt{t}}\int_{z_0}^{\infty}z\frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}dz
\exp\left\{-2\frac{\mu}{\sigma^2}y_2\right\}+2\frac{\mu}{\sigma^2}\int_{z_0}^\infty 
\frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}dz\exp\left\{-2\frac{\mu}{\sigma^2}y_2\right\}\\
&=&\frac{2}{\sigma\sqrt{t}}\int_{z_0}^{\infty}-\Big{(}\frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}\Big{)}'dz
\exp\left\{-2\frac{\mu}{\sigma^2}y_2\right\}+2\frac{\mu}{\sigma^2}\Big{(}1-\Phi(z_0)\Big{)}
\exp\left\{-2\frac{\mu}{\sigma^2}y_2\right\}\\
&=&\frac{2}{\sigma\sqrt{t}}\frac{1}{\sqrt{2\pi}}e^{-\frac{z_0^2}{2}}
\exp\left\{-2\frac{\mu}{\sigma^2}y_2\right\}+2\frac{\mu}{\sigma^2}\Phi(-z_0)
\exp\left\{-2\frac{\mu}{\sigma^2}y_2\right\}.
\end{eqnarray*}
This can be rewritten as:
\begin{eqnarray}
P(M_t-X_t\in dc)&=&\Bigg{(}2\frac{\mu}{\sigma^2}\Phi\Big{(}
\frac{\mu t-c}{\sigma\sqrt{t}}\Big{)}
\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}\nonumber\\
&+&\frac{2}{\sigma\sqrt{2t\pi}}
\exp\left\{-\frac{(\mu t -c)^2}{2t\sigma^2}\right\}
%\underline{
\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}
%}
\Bigg{)}dc, \label{halfdensity}
\end{eqnarray}
and the result follows.
%The density of the half-range $M_t-X_t$ is then the term multiplying $dc$:
%\begin{eqnarray}
%f_{M_t-X_t}(c)=2\frac{\mu}{\sigma^2}\;\Phi\Big{(}
%\frac{\mu t-c}{\sigma\sqrt{t}}\Big{)}
%\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}
%+\frac{2}{\sigma\sqrt{2t\pi}}
%\exp\left\{-\frac{(\mu t +c)^2}{2t\sigma^2}\right\}.
%%\underline{
%%\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}
%\end{eqnarray}
\end{proof}

\begin{Propo} \label{propo1} %((3.31) of Kon\'{e})
The expectation of the half-range is given by:
\begin{eqnarray}
E(M_t-X_t)&=&\frac{\sigma^2}{2\mu}\;\Phi\Big{(}\frac{\mu}{\sigma} \sqrt{t}\Big{)}
-\Big{(}\mu t+\frac{\sigma^2}{2\mu} \Big{)}\Big{(}1-\Phi\Big{(}\frac \mu \sigma
 \sqrt(t)\Big{)}\Big{)}\nonumber\\
&+&\frac{\sigma\sqrt{t}}{\sqrt{2\pi}}
\exp{\Big{(}-\frac{t \mu^2}{2\sigma^2}\Big{)}}.  \label{halfexpectation}
\end{eqnarray}
\end{Propo}

\begin{proof} 
A simple calculation yields:
%This proof follows that of Proposition 3.31 of Kon\'{e} (1996)
\begin{eqnarray*}
E(M_t-X_t)&=&\int_0^\infty c\;\frac{2\mu}{\sigma^2} \;\Phi\Big{(}
\frac{\mu t-c}{\sigma\sqrt{t}}\Big{)}\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}
dc\\
&&+\int_0^\infty c \frac{2}{\sigma\sqrt{2t\pi}}
\exp\left\{-\frac{(\mu t +c)^2}{2t\sigma^2}\right\}dc\\
&=&\int_0^\infty 
\left\{
-(c+\frac{\sigma^2}{2\mu})
\exp\Big{(}-2\frac{\mu}{\sigma^2}c\Big{)}
\right\}'
\;\Phi\Big{(}\frac{\mu t-c}{\sigma\sqrt{t}}\Big{)}dc\\
&&+\int_0^\infty c \frac{2}{\sigma\sqrt{2t\pi}}
\exp\left\{-\frac{(\mu t +c)^2}{2t\sigma^2}\right\}dc\\
=\frac{\sigma^2}{2\mu}\;\Phi\Big{(}\frac\mu\sigma\sqrt{t}\Big{)}
&+&\frac{1}{\sigma\sqrt{t}}\frac{1}{\sqrt{2\pi}}\int_0^\infty 
\Big{(}-c-\frac{2\sigma^2}{\mu}+2c\Big{)}
\exp\left\{-\frac{(\mu t +c)^2}{2t\sigma^2}\right\}dc.
\end{eqnarray*}
A change of variable $c=z\sigma\sqrt{t}-\mu t$ gives the result.
\end{proof}

\medskip

Consider now $E[X_t-m_t]$. For each path of the Brownian motion $X_t$ 
with drift $\mu$ consider a symmetric path of a Brownian motion $\tilde X_t$ 
having drift $-\mu$. Then $X_t-m_t=-(\tilde X_t-\tilde M_t)$ and $E[X_t-m_t]$ 
can be calculated using the equation (\ref{halfexpectation}) with $\mu$ 
replaced by $-\mu$. Whereas the formula for the expectation of the range 
follows.


\begin{Theor}\label{theor1}
The expectation of the range of the arithmetic Brownian motion $X_t$ defined in
(\ref{equation1}) is given by:
\begin{equation}
E[R_t]=\Big{(}\mu t+\frac{\sigma^2}{\mu}\Big{)}
\Bigg{(}1-2\Phi\Big{(}-\sqrt{t}\frac{\mu}{\sigma}\Big{)}\Bigg{)}
+2\frac{\sigma\sqrt{t}}{\sqrt{2\pi}}
\exp\Big{(}-\frac{t\mu^2}{2\sigma^2}\Big{)}. \label{equation5}
\end{equation}
\end{Theor}
Let us denote this
expected range function by $ER(\mu,\sigma,t)$. On closer inspection this can
be further simplified as a function of just two quantities:
\begin{equation}
E[R_t]=ER(\mu,\sigma,t)=h\Big{(}\frac{\mu t}{\sigma \sqrt{t}},
\frac{\sigma^2 }{\mu }\Big{)},\label{er}
\end{equation}
where the function $h$ is defined by:
\begin{equation}
h(x,y):=\left\{(x^2+1)(2\Phi(x)-1)+\frac{2x}{\sqrt{2\pi}}
\exp\Big{(}-\frac{x^2}{2}\Big{)}\right\}y. \label{er2}
\end{equation}
Note that $ER(\mu,\sigma,t)=ER(\mu t,\sigma\sqrt{t},1)$ as it should (the range
over a time interval $(0,t)$ of an arithmetic Brownian motion with parameters
$\mu$ and $\sigma$ is the same as that over $(0,1)$ when the parameters change
to $\mu t$ and $\sigma\sqrt{t}$).

\bigskip

In the remainder of this section we derive the density of the range $R_t$ of
the arithmetic Brownian motion $X_t$. This is achieved by the use of the joint
density of the minimum and the maximum of $X_t$, a result with its own merit,
that we could not find published prior to Kon\'{e} (1996) (Borodin and Salminen
(1996, 1.15.4) published in the same year the joint cumulative distribution 
function only in terms of some definite integrals).

A version of this result is used in Sutrick et al (1997) for the same purpose,
but there it seems to have incorporated an error.


\medskip


%We start with the joint density of the
%maximum and the minimum. 
%(see Theorem 3.4.1 of Kon\'{e} (1996)).


%\begin{Remar}
%The term underlined above is missing from formula (3.12) of Corollary 3.3.1 of
%Kon\'{e}.
%\end{Remar}


%so we have $a=(y_1-y_2)/2$ and $b=(y_1+y_2)/2$.

To obtain the joint density $F(a,b)$ of the maximum and the minimum we start 
with a lemma.
\begin{Lemma}\label{lemma_main}
We can write:
\begin{eqnarray}
F(a,b)=\int_a^bh(a,b,x)\exp{\Big{(}\frac{\mu}{\sigma^2}x-\frac 12
\frac{\mu^2}{\sigma^2}t\Big{)}}\;dx,\label{aux28}
\end{eqnarray}
where
\begin{eqnarray}
h(a,b,x)&=&h_1(a,b,x)-h_2(a,b,x),\\
h_1(a,b,x)&=&\sum_{k=-\infty}^\infty\frac{2k(2k-2)}{\sigma^3 t\sqrt{2\pi t}}
\Bigg{[}
 1-\frac{[2k(b-a)-2b+x]^2}{t\sigma^2}
\Bigg{]}\nonumber\\
&&\times\exp{\Big{(}-\frac{[2k(b-a)-2b+x]^2}{2t\sigma^2}\Big{)}},
\nonumber\\
h_2(a,b,x)&=&\sum_{k=-\infty}^\infty\frac{4k^2}{\sigma^3 t\sqrt{2\pi t}}
\Bigg{[}
 1-\frac{[2k(b-a)-x]^2}{t\sigma^2}
\Bigg{]}
\exp{\Big{(}-\frac{[2k(b-a)-x]^2}{2t\sigma^2}\Big{)}}.\nonumber%\label{aux29}
%\hspace*{2cm}
\end{eqnarray}
\end{Lemma}

\begin{proof}
%TO BE TYPED IN.
Using the change of measure of the proof of Lemma \ref{lemma1} and 
Girsanov's theorem we have:
\begin{equation}
P(a<m_t<M_t<b)=\int_a^b p_t(x;a,b)\exp{\Big{(}\frac{\mu}{\sigma^2}x-\frac 12
\frac{\mu^2}{\sigma^2}t\Big{)}}dx,\label{aux21}
\end{equation}
which gives $F(a,b)$ via:
\begin{equation}
F(a,b)=-\frac{\partial^2 P(a<m_t<M_t<b)}{\partial a \;\partial b}.\label{aux22}
\end{equation}
Leibniz rule of differentiation:
\begin{equation}
\frac{\partial}{\partial z}\int_{a(z)}^{b(z)}f(x,z)\;dx=
\int_{a(z)}^{b(z)}\frac{\partial f(x,z)}{\partial z}dx
+f(b(z),z)\frac{\partial b}{\partial z}-f(a(z),z)
\frac{\partial a}{\partial z}
\label{Leibniz}
\end{equation}
gives the partial derivative of (\ref{Feller}) wrt $b$:
\begin{eqnarray}
%\frac{\partial P(a<m_t<M_t<b)}{\partial b}=
\int_a^b\frac{1}{\sigma\sqrt{t}}
\sum_{k=-\infty}^{\infty}\Bigg{[}\frac{\partial\phi}{\partial b}
\Big{(}\frac{2k(b-a)-x}{\sigma\sqrt{t}}\Big{)}
-\frac{\partial\phi}{\partial b}
\Big{(}\frac{2k(b-a)-2b+x}{\sigma\sqrt{t}}\Big{)}
\Bigg{]}
\nonumber\\
\times \exp{\Big{(}\frac{\mu}{\sigma^2}x-\frac 12
\frac{\mu^2}{\sigma^2}t\Big{)}}dx.\label{aux24}
\end{eqnarray}
Differentiating wrt $a$ this last equation gives:
\begin{eqnarray}
F(a,b)=-\int_a^b\frac{1}{\sigma\sqrt{t}}
\sum_{k=-\infty}^{\infty}\Bigg{[}\frac{\partial^2\phi}{\partial a\;\partial b}
\Big{(}\frac{2k(b-a)-x}{\sigma\sqrt{t}}\Big{)}
-\frac{\partial^2\phi}{\partial a\;\partial b}
\Big{(}\frac{2k(b-a)-2b+x}{\sigma\sqrt{t}}\Big{)}
\Bigg{]}
\nonumber\\
\times 
\exp{\Big{(}\frac{\mu}{\sigma^2}x-\frac 12
\frac{\mu^2}{\sigma^2}t\Big{)}}\;dx.\hspace*{0.4cm}\label{aux25}
\end{eqnarray}
Direct calculation gives:
\begin{eqnarray}
\frac{\partial^2\phi}{\partial a\;\partial b}
\Big{(}
  \frac{2k(b-a)-x}{\sigma\sqrt{t}}
\Big{)}
=
\frac{\partial}{\partial a}
\Bigg{[}
  \frac{(-2k)[2k(b-a)-x]}{\sqrt{2\pi}t\sigma^2} 
  \exp{\Big{(}-\frac{[2k(b-a)-x]^2}{2t\sigma^2}\Big{)}}
\Bigg{]}
\nonumber\\
=
\frac{4k^2}{\sqrt{2\pi}t\sigma^2}
\exp{\Big{(}-\frac{[2k(b-a)-x]^2}{2t\sigma^2}\Big{)}}
\Big{(}
  1-\frac{[2k(b-a)-x]^2}{t\sigma^2}
\Big{)},\hspace*{0.3cm}\label{aux26}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{\partial^2\phi}{\partial a\;\partial b}
\Big{(}
 \frac{2k(b-a)-2b+x}{\sigma\sqrt{t}}
\Big{)}\hspace*{9cm}
\nonumber\\
=
\frac{\partial}{\partial a}
\Bigg{[}
 \frac{-(2k-2)[2k(b-a)-2b+x]}{\sqrt{2\pi}t\sigma^2} 
\exp{\Big{(}-\frac{[2k(b-a)-2b+x]^2}{2t\sigma^2}\Big{)}} 
\Bigg{]}\hspace*{1cm}\label{aux27}
\nonumber\\
=
\frac{4k(k-1)}{\sqrt{2\pi}t\sigma^2}
\exp{\Big{(}-\frac{[2k(b-a)-2b+x]^2}{2t\sigma^2}\Big{)}} 
\Big{(}
 1-\frac{[2k(b-a)-2b+x]^2}{t\sigma^2}
\Big{)}.\hspace*{0.7cm}
\end{eqnarray}
Substituting (\ref{aux26})-(\ref{aux27}) in (\ref{aux25}) gives
the result.
\end{proof}


\begin{Propo} \label{propo_joint}%(3.4.1 of Kon\'{e})
The joint density function $F(a,b)$ of $M_t$ and $m_t$ can be represented as:
\begin{eqnarray}
F(a,b)&=&F_1(a,b)-F_2(a,b)-F_3(a,b)+F_4(a,b)\\
&-&F_5(a,b)+F_6(a,b)+F_7(a,b)-F_8(a,b),\nonumber
\end{eqnarray}
with
\begin{eqnarray}
\lefteqn{F_1(a,b)=\sum_{k=-\infty}^{\infty}\frac{4k(k-1)}{t\sigma^3\sqrt{2\pi t}}
\;\;[(2k-1)b-2ka+\mu t]}\hspace*{2cm}\nonumber
\\
&&\times\exp\left\{
-\frac{\mu}{\sigma^2}[2(k-1)b-2ka]-\frac{[(2k-1)b-2ka-\mu t]^2}{2t\sigma^2}
\right\},\hspace*{0.8cm}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{F_2(a,b)=\sum_{k=-\infty}^{\infty}\frac{4k(k-1)}{t\sigma^3\sqrt{2\pi t}}
\;\;[2(k-1)b-(2k-1)a+\mu t]}\hspace*{1cm}\nonumber
\\
&&\times\exp\left\{
-\frac{\mu}{\sigma^2}[2(k-1)b-2ka]-\frac{[2(k-1)b-(2k-1)a-\mu t]^2}{2t\sigma^2}
\right\},\hspace*{0.9cm}
\end{eqnarray}
\begin{eqnarray}
F_3(a,b)=\sum_{k=-\infty}^{\infty}\frac{4k(k-1)\mu^2}{2\sigma^4}
\exp\left\{
-\frac{\mu}{\sigma^2}[2(k-1)b-2ka]
\right\}\hspace*{3cm}\nonumber
\\ 
\times\operatorname{erf}\Bigg{(}\frac{(2k-1)b-2ka-\mu t}{\sigma\sqrt{2t}}\Bigg{)},
\hspace*{0.6cm}
\end{eqnarray}
\begin{eqnarray}
F_4(a,b)=\sum_{k=-\infty}^{\infty}\frac{4k(k-1)\mu^2}{2\sigma^4}
\exp\left\{
-\frac{\mu}{\sigma^2}[2(k-1)b-2ka]
\right\}\hspace*{3cm}\nonumber 
\\
\times\operatorname{erf}\Bigg{(}\frac{2(k-1)b-(2k-1)a-\mu t}{\sigma\sqrt{2t}}\Bigg{)},
\hspace*{0.6cm}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{F_5(a,b)
=\sum_{k=-\infty}^{\infty}\frac{4k^2}{t\sigma^3\sqrt{2\pi t}}
\;\;[(2k+1)b-2ka+\mu t]}\hspace*{4cm}\nonumber
\\
&&\times\exp\left\{
-\frac{\mu}{\sigma^2}2k(b-a)-\frac{[(2k+1)b-2ka-\mu t]^2}{2t\sigma^2}
\right\},\hspace*{1cm}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{F_6(a,b)
=\sum_{k=-\infty}^{\infty}\frac{4k^2}{t\sigma^3\sqrt{2\pi t}}
\;\;[2kb-(2k-1)a+\mu t]}\hspace*{5cm}
\\
&&\times\exp\left\{
-\frac{\mu}{\sigma^2}2k(b-a)-\frac{[2kb-(2k-1)a-\mu t]^2}{2t\sigma^2}
\right\},\hspace*{0.3cm}\nonumber
\end{eqnarray}
\begin{eqnarray}
F_7(a,b)=\sum_{k=-\infty}^{\infty}\frac{4k^2\mu^2}{2\sigma^4}
\exp\left\{
-\frac{\mu}{\sigma^2}2k(b-a)
\right\}\; 
\operatorname{erf}\Bigg{(}\frac{(2k+1)b-2ka-\mu t}{\sigma\sqrt{2t}}\Bigg{)},
\end{eqnarray}
\begin{eqnarray}
F_8(a,b)=\sum_{k=-\infty}^{\infty}\frac{4k^2\mu^2}{2\sigma^4}
\exp\left\{
-\frac{\mu}{\sigma^2}2k(b-a)
\right\}\; 
\operatorname{erf}\Bigg{(}\frac{2kb-(2k-1)a-\mu t}{\sigma\sqrt{2t}}\Bigg{)},
\end{eqnarray}
where
\begin{equation}
\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt.\label{aux33}
\end{equation}
\end{Propo}

\begin{proof}
%INSERT
From Lemma \ref{lemma_main} we write $F(a,b)$ as the difference of two terms,
which we denote $I_1$ and $I_2$:

\begin{eqnarray}
I_1:=\int_a^b h_1(a,b,x)\exp{\Big{(}\frac{\mu}{\sigma^2}x-\frac 12
\frac{\mu^2}{\sigma^2}t\Big{)}}\;dx,\nonumber\\
I_2:=\int_a^b h_2(a,b,x)\exp{\Big{(}\frac{\mu}{\sigma^2}x-\frac 12
\frac{\mu^2}{\sigma^2}t\Big{)}}\;dx.\label{i1}
\end{eqnarray}
In each $I_1$ and $I_2$ we combine the exponents and then use, respectively, 
a change of variable:      
\begin{equation}
z=\frac{x+2k(b-a)-2b-\mu t}{\sigma\sqrt{t}},\;\;
z=\frac{x+2k(b-a)-\mu t}{\sigma\sqrt{t}}, \label{change12}
\end{equation}
followed by an integration by parts for $\int z^2\exp(-z^2/2)dz$ and replacement
of $\Phi(x)$ by $\operatorname{erf}(x)$ via $\Phi(x)=0.5\operatorname{erf}
(x/\sqrt{2})+0.5$. The resulting eight terms are then then denoted $F_i(a,b)$,
$i=1,\dots, 8$.
\end{proof}
\medskip


We use the above expression to derive the density of the range. To make it
suitable for comparison with that obtained by Sutrick et al (1997) in their 
Proposition 1, we change $k\rightarrow k+1$ in the summations of 
$F_1, F_2, F_3$ and $F_4$.

\begin{Propo}
%(1 of Sutrick (1997) JFE)
The density function $f_{R_t}(r)$ for the range of an arithmetic Brownian motion 
can be written as:
\begin{equation}
f_{R_t}(r)=\frac{1}{\sigma\sqrt{t}}\sum_{k=-\infty}^{\infty} 4k^2 I(k)
+\frac{1}{\sigma\sqrt{t}}\sum_{k=-\infty}^{\infty}4k(k+1)J(k),
\end{equation}
where
\begin{eqnarray}
I(k)&=&e^{-\frac{2\mu kr}{\sigma^2}}(1+c^2)
(\phi(K_1-c)-2\phi(K_0-c)+\phi(K_{-1}-c))\nonumber\\
&+&e^{-\frac{2\mu kr}{\sigma^2}}
[(c^2K_1-2c-c^3)\Phi(K_1-c)-2(c^2K_0-2c-c^3)\Phi(K_0-c)\nonumber\\
&+&(c^2K_{-1}-2c-c^3)\Phi(K_{-1}-c)], 
\end{eqnarray}
and
\begin{eqnarray}
J(k)&=&e^{\frac{2\mu kr}{\sigma^2}}(\phi(K_1+c)-\phi(K_0+c))
-e^{\frac{2\mu(k+1)r}{\sigma^2}}(\phi(K_2+c)-\phi(K_1+c))\nonumber\\
&&+e^{-\frac{2\mu kr}{\sigma^2}}\Big{(}-\frac c2 \Phi(K_1-c)
+\frac c2 \Phi(K_0-c)\Big{)}\nonumber\\
&&-e^{-\frac{2\mu (k+1)r}{\sigma^2}}\Big{(}-\frac c2 \Phi(K_2-c)
+\frac c2 \Phi(K_1-c)\Big{)}
\nonumber\\
&&+e^{\frac{2\mu kr}{\sigma^2}}(\Phi(K_1+c)-\Phi(K_0+c))\nonumber\\
&&-e^{\frac{2\mu(k+1)r}{\sigma^2}}(\Phi(K_2+c)-\Phi(K_1+c)),
\end{eqnarray}
with 
%$\phi(x), \Phi(x)$ the density and distribution functions of a standard 
%normal variable, and
\begin{eqnarray*}
K_2=\frac{(2k+2)r}{\sigma\sqrt{t}},\;K_1=\frac{(2k+1)r}{\sigma\sqrt{t}},\;
K_0=\frac{2kr}{\sigma\sqrt{t}},\;K_{-1}=\frac{(2k-1)r}{\sigma\sqrt{t}},\;
c=\frac{\mu\sqrt{t}}{\sigma}.
\end{eqnarray*}
\end{Propo}

\begin{proof}
After replacing $k$ by $k+1$ in $F_i$, ${i=1,\dots,4}$ of Proposition 
\ref{propo_joint},
% (for comparison with the result of Sutrick (1997)).
a two-dimensional transformation $a=u-v, b=u$, gives, via Jacobian, the density
of the range and the running maximum. Its marginal density is the one we seek:
$$
f(r)=\int_0^rF(u-r,u)du.
$$
Applying a change of variable and integration by parts gives the result.
\end{proof}

\begin{Remar}\label{remark2}
This result 
%was also derived in 
corrects that of Sutrick et al (1997) where there appears to be a mistake
in the computations.
%but we notice a couple of
%differences in the result. Specifically, while the expression for $I(k)$ is
%the same, the \underline{second last term } of $J(k)$ is different, and we cannot reconcile
%the difference. We believe that there is a computational error in the expression
%that Sutrick et al (1997) obtain for $J(k)$, especially because of the lack of 
%symmetry.
\end{Remar}

\begin{Remar}\label{equiv}
The probabilistic starting point for both Kon\'{e} (1996) and Sutrick et al 
(1997) is 
$p_t(x;a,b)$ $dx:=P(a<m_t<M_t<b,x\leq X_t<x+dx|X_0=0)$. The former uses a result 
of Feller (1951) that can be traced to L\'{e}vy (1948):
\begin{equation}
%P(a<m_t<M_t<b,x)
p_t(x;a,b)=\frac{1}{\sigma\sqrt{t}}\sum_{k=-\infty}^\infty
\Bigg{[}\phi\Big{(}\frac{2k(b-a)-x}{\sigma\sqrt{t}}\Big{)}
-\phi\Big{(}\frac{2k(b-a)-2b+x}{\sigma\sqrt{t}}\Big{)}\Bigg{]},
\label{Feller}
\end{equation}
 while the latter uses a result of Billingsley (1968):
\begin{equation}
%P(a<m_t<M_t<b,x)
p_t(x;a,b)=\frac{1}{\sigma\sqrt{t}}\sum_{k=-\infty}^\infty
\Bigg{[}\phi\Big{(}\frac{x+2k(b-a)}{\sigma\sqrt{t}}\Big{)}
-\phi\Big{(}\frac{2b-x+2k(b-a)}{\sigma\sqrt{t}}\Big{)}\Bigg{]}.
\label{Billingsley}
\end{equation}
The probabilistic results 
%used as starting points by Kon\'{e}
%(1996) and Sutrick et al (1997) are equivalent, %two arms of the same body.
%very close
%%not only related, but are in fact equivalent (see Remark \ref{equiv}). 
%Specifically, a result presented by 
%Karatzas and Shreve (1988) and traced to Feller (1951) is equivalent to that
%of Billingsley (1968).
%That is, equations 
(\ref{Feller}) and (\ref{Billingsley}) are in fact equivalent, as one can be obtained from the other by 
appropriately replacing the summation index $k$ %=-\infty\dots\infty$ 
with 
$-k$ %=-\infty\dots\infty$ 
and $\phi(x)$ with $\phi(-x)$.
\end{Remar}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The method of moments applied to volatility estimation
using  daily high, low, opening and closing prices}
\recount
\recoup
In this section we apply Theorem \ref{theor1} to the estimation of the drift 
and  volatility parameters of the stock price from 
%a system of two implicit equations obtained from 
market data on high, low, opening and closing prices.



\begin{Defin}\label{days}
%\begin{enumerate}
%\item 
i) A {\em trading day} is the period elapsed between the opening and
the closing bells of a calendar day.

%\item 
%\noindent 
ii) A {\em virtual trading day} is the after-hours period 
beginning from the closing of one trading day and ending at the opening 
of the next trading day.
%\end{enumerate}

iii) A {\em one-day period} consists of one trading day followed by one
virtual trading day.
\end{Defin}

We assume that the stock price $S_t$ %during a one-day period $t\in[0,1]$ 
has
%Let $S_t$ be the stock price at time $t\in[0,1]$ during one day.  It follows 
the usual geometric Brownian motion dynamics:
\begin{equation}
\frac{dS_t}{S_t}=\mu_s\; dt+\sigma\; dW_t, \;\;t\geq 0.
%\leq 1. 
\label{equation6} 
\end{equation}
Then the log-stock price $\log S_t$ is the arithmetic Brownian motion $X_t$ 
defined in (\ref{equation1})
%:
%\begin{equation}
%$\log S_t=X_t$, %                   \label{equation7}
%\end{equation}
with drift coefficient 
\begin{equation}
\mu=\mu_s-\frac{\sigma^2}{2}.   \label{equation7}
\end{equation} 
Note that $\mu_s$ is the {\em one-day period} drift of the stock price $S_t$, while 
$\mu$ is the similar drift of the log-price $X_t=\log S_t$;  the volatility 
parameter $\sigma$ is the same for both $S_t$ and $X_t$.

%\begin{Remar}\label{insight}
%Recall that $S_i$ is the closing price at the end of day $i\in\{1,\dots n\}$
%The stock price was assumed to follow a geometric Brownian motion (see
%(\ref{equation6})). 
The market data used for parameter estimation is as follows: for each one-period
day $i\in\{1, 2, \dots, n\}$ we denote by $S_{i-1}$ the opening price and 
by 
%$S_i$ the closing price. Denote also 
$H_i$ and $L_i$ the intra-day high and low prices, respectively (i.e. the high and 
low are observed only during the trading day, and not the virtual trading day
- see Figure \ref{fig0}). 
%Thus:
%The opening price on day 1 is $S_{0+}$, and the closing 
%price is $S_1$. 
%, and this has an important impact on the annualization
%of the daily volatility obtained in the previous section.
%\begin{equation}
%\mbox{OPEN}(i)=S_{(i-1)+},\;
%\mbox{CLOSE}(i)=S_{i},\;
%\mbox{HIGH}(i)=H_i,\;
%\mbox{LOW}(i)=L_i.
%\label{notation}
%\end{equation}

The after-hours arrival of information in the market determines a jump between 
the closing price 
%$S_{(i-1)}$ 
of one trading day and the opening price 
%$S_{(i-1)+}$
of the next day. We model this jump by letting the same geometric
Brownian motion $S_t$ have an unobserved evolution during a virtual trading
day. The length of this virtual trading day is assumed to be, on average,
a fraction $f$
%$1-t^*$ 
of the unit length of the one-day period. 
\begin{Remar}\label{important}This assumption follows 
Garman and Klass (1980) and
Yang and Zhang (2000), except that they assume the after-hours trading day precedes
the actual trading day. They call it the opening jump (from $C_{i-1}$ to $O_i$),
and assume it is modeled
by a Poisson process.  % (which is thus needed in the estimation).
\end{Remar}
Thus for $i\in\{1, 2, \dots, n\}$ we have (see Figure \ref{fig0}):
\begin{equation}
\mbox{OPEN}(i)=O_i=S_{i-1},\;
\mbox{CLOSE}(i)=C_i=S_{i-f},\;
\mbox{HIGH}(i)=H_i,\;
\mbox{LOW}(i)=L_i,
\label{notation}
\end{equation}
where:
\begin{equation}
H_i=\sup_{t\in[i-1,i-f]}S_t,\;\;L_i=\inf_{t\in[i-1,i-f]}S_t.
\label{high_low}
\end{equation}
\begin{figure}[htb]
%\centerline{\includegraphics[scale=0.6]{trading_day.png}}
%\centerline{\epsfig{file=plot1t.eps,scale=1.0,angle=0}}
\includegraphics{plot1t}
\caption{A one-day period consisting of a trading day and an after-hours period}
\label{fig0}
\end{figure}
The evolution of the price during the trading period $i-1\leq t<i-f$ is given by:
%alignment of colums alternate right, left, right, left, so we need two &&
%\begin{alignat}{4}
\begin{equation}
log \;S_t=log\; O_i %OPEN(i)
%S_{(i-1)}
+\mu\;(t-i+1)+\sigma (W_t-W_{t-i+1}),
%\;\; i-1\leq t<i-1+t^*,
\label{trad}
\end{equation}
%log \;S_t&=log \;CLOSE(i),&&
%S_i,&&
%t=i-1+t^*,\nonumber\\
and during the after-hours virtual trading period $i-f\leq t<i$ by:
\begin{equation}
log \;S_t=log \;C_i
%CLOSE(i)
+\mu (t-i+f)+\sigma (W_t-W_{i-f}).
%i-1+t^*\leq t<i.
\label{virt}
\end{equation}
%log \;S_t&=log \;S_{i+}, &&t=i.\nonumber
%\end{alignat}

%The choice of $t^*$ is similar to Garman and Klass (1980), and to $f$ of 
%Yang and Zhang (2000). The latter paper, however, incorporates
%$t^*$ in the drift and volatility parameters of the equations 
%(\ref{trad})-(\ref{virt}) in a model with Poisson jumps, while we assume it only 
%intervenes in the time parameter in a model without jumps.

%The opening price at the beginning of day 1 is $S_0$.
%, and the
%opening price at the beginning of day $i$ is $S_{i+}$. Generally 
%$S_i\neq S_{i+}$, and we will account for this jump in (\ref{equation22}).
%Suppose the closing stock prices $S_0,S_1, S_2,\dots, S_n$ and the daily 
%high and low stock prices $H_1, H_2, \dots, H_n$ and $L_1, L_2, \dots, L_n$
%are available.
Taking expectation in (\ref{trad}) when $t\nearrow (i-f)$ 
and in (\ref{virt}) when $t\nearrow i$ gives:
\begin{eqnarray}
&&E\Big{[}\log \frac{C_i
%CLOSE(i)
}{O_i
%OPEN(i)
}\Big{]}
=E\Big{[}\log\frac{S_{i-f}}{S_{i-1}}\Big{]}=\mu\;(1-f),\label{drift1}\\
&&E\Big{[}\log \frac{
O_{i+1}
%OPEN(i+1)
}{C_i
%CLOSE(i)
}\Big{]}
=E\Big{[}\log\frac{S_{i}}{S_{i-f}}\Big{]}=\mu\;f.\label{drift2}
\end{eqnarray}
Using $W_t-W_s$ identically distributed to $W_{t-s}$, the trading day and 
virtual trading day variances are obtained, respectively, as:
\begin{alignat}{4}
\operatorname{VAR}&\Big{[}\log
\frac{C_i
%CLOSE(i)
}{O_i
%OPEN(i)
}\Big{]}=
\operatorname{VAR}\Big{[}\log
\frac{S_{i-f}}{S_{i-1}}\Big{]}
=\;\sigma^2\;(1-f),\nonumber\\
\operatorname{VAR}&\Big{[}\log
\frac{O_{i+1}
%OPEN(i+1)
}{C_i
%CLOSE(i)
}\Big{]}=
\operatorname{VAR}\Big{[}\log
\frac{S_{i}}{S_{i-f}}\Big{]}
%\operatorname{VAR}&\left\{\log
%\frac{S_{i+}}{S_i}\right\}
=\;\sigma^2\;f.\label{variances}
\end{alignat}
Thus, we can write heuristically:
\[
\sigma^2=\operatorname{VAR}\Big{[}\log
\frac{C_{i}}{O_i}\Big{]}+
\operatorname{VAR}\Big{[}\log
\frac{O_{i+1}}{C_i}\Big{]}
=\operatorname{VAR(trading\;day)}+
\operatorname{VAR(after\;hours)}.
\]
%\begin{Remar}\label{yz}
%Under the assumptions of the model this is equivalent to having after-hours 
%preceed the trading day (as in Garman and Klass (1980) and Yang and Zhang (2000)):
%\[
%\sigma^2=
%\operatorname{VAR}\Big{[}\log
%\frac{O_{i+1}}{C_i}\Big{]}
%+\operatorname{VAR}\Big{[}\log
%\frac{C_{i+1}}{O_{i+1}}\Big{]}.
%\]
%\end{Remar}
%Let $H_1, H_2, \dots, H_n$ and $L_1, L_2, \dots, L_n$ be the daily high and 
%low closing prices of the stock. 

To estimate the variance over the trading day we use the method of moments.
The range $R_{1-f}=\log H_1-\log L_1$ 
of the arithmetic Brownian motion $X_t=\log S_t$ over the trading day $[0,1-f]$
was obtained in equation (\ref{er}):
\begin{equation}
E(R_{1-f})=ER(\mu,\sigma,1-f)=ER(\mu(1-f),\sigma\sqrt{1-f},1).\label{new_er}
\end{equation} 
In (\ref{new_er}) we estimate $E(R_{1-f})$ using the daily range data:
\begin{equation}
%\widehat{E[R_{t^*}]}
k_1:=\frac 1n \sum_{i=1}^n \log \frac{H_i}{L_i}.%=:k_1.
\label{equation9}
\end{equation}
 and  $\mu(1-f)$  by (see (\ref{drift1})): 
%$\hat{\mu}$, $\hat{\mu}_s$ and
%$\hat{\sigma}$ be the unbiased estimators of $\mu$, $\mu_s$ and $\sigma$, 
%respectively. 
%Then:
%Since $\mu$ is the expected daily return of the log-price,we have:
%If $S_0,S_1, S_2,\dots, S_n$ are closing stock prices for $n+1$ consecutive 
%trading days, then 
%an unbiased estimator of $\mu$ is:
\begin{equation}
%\hat{\mu}=\hat{\mu_s}-\frac{\hat{\sigma}^2}{2}:=\frac{1}{t^*}\;
k_2:=\frac 1n \sum_{i=1}^n 
\log\frac{C_i
%CLOSE(i)
}{O_i
%OPEN(i)
}. %=:\frac{1}{t^*}\; k_2.
%=\frac 1n\log\frac{S_{n+1}}{S_1}.
\label{equation8}
\end{equation}
This leads to the following equation to be solved for $x$, the estimate of 
$\sigma\sqrt{1-f}$:
%=\widehat{\sigma\sqrt{1-f}}$: 
\begin{equation}
k_1=h\Big{(}\frac{k_2}{x},\frac{x^2}{k_2}\Big{)}.
%\Big{(}\hat\mu {t^*}+\frac{\hat\sigma^2}{\hat\mu}\Big{)}
%\Big{(}1-2\Phi\Big{(}-\sqrt{{t^*}}\frac{\hat\mu}{\hat\sigma}\Big{)}\Big{)}
%+2\frac{\hat\sigma\sqrt{{t^*}}}{\sqrt{2\pi}}
%\exp\Big{(}-\frac{{t^*}\hat\mu^2}{2\hat\sigma^2}\Big{)}
%=k_1.
%\frac 1n \sum_{i=1}^n \log \frac{H_i}{L_i}. 
\label{main_eq}
\end{equation}
The squared of this solution gives an estimate $V_i=x^2$ of
the variance (volatility squared) corresponding to the trading day part of 
a one-day period. 

For the after-hours part of the one-day period we have two choices: $V_0$
(centered approach) used in Yang and Zhang (2000), or $V_0'$ (non-centered)
used in Garman and Klass (1980). Using the former (i.e. the sample standard variance $V_0$),
we obtain the estimate for the variance of the entire one-day period as:
\begin{equation}
V_Z:=V_0+V_i,\label{est}
\end{equation}
%with $V_0$ being the sample standard variance. 
%given by:
%\begin{equation}
%V_0'=\frac 1n \sum_{i=1}^n\Big{(}\log\frac{O_{i}}{C_{i-1}}\Big{)}^2. \label{v_0}
%\end{equation}
or, in annualized form, as:
\begin{equation}
\sigma_a^2:=252 V_Z.\label{ann}
\end{equation}

%\vspace*{0.9cm}

\medskip

\noindent Denoting by $V_C$ the sample variance of $\log(C_i/O_i)$ used in their estimator 
by Yang and Zhang (2000) :
$$V_{YZ}=V_0+kV_C+(1-k)V_{RS},$$ 
where $V_{RS}$ is the estimator of Rogers and Satchell (1991) and 
Rogers, Satchell and Yoon (1994) and $k$ is a constant, we note the following.
\begin{Remar}\label{comparison}
i) The term $V_i$ replaces the linear combination of $V_C$ and
$V_{RS}$ used by Yang and Zhang (2000) for the intra-day trading period, and it does not need estimating the value
of $k$ that achieves minimum variance. 

ii) Our estimator is a true range-based estimator (log-range to be precise
since it uses $\log({H_i}/{L_i})$), 
unlike that of Yang and Zhang (2000). 
%The data used in $V_i$ is:
%$$\log\frac{C_i}{O_i}\;\; \mbox{and} \;\;\log\frac{H_i}{L_i},$$ 
%instead of:  
%$$\Big{(}\log\frac{C_i}{O_i}\Big{)} \;\mbox{for}\; V_C \;\mbox{and}\;
%\Big{(} \log\frac{H_i}{O_i}, \log\frac{H_i}{C_i},\log\frac{L_i}{O_i
%\log\frac{L_i}{C_i}\Big{)}\;\mbox{for}\;V_{RS}.$$


iii) Our estimator $V_Z$ is independent of both the drift and the weight $f$ of
the after-hours information.
\end{Remar}


%Replacing $\hat\mu$ from (\ref{equation8}) in (\ref{equation10}) gives an
%implicit equation in $\hat\sigma$ that depends on the estimates $k_1$,
%$k_2$ and $t^*$ obtained from the data.
%%\[
%%k_1:=\widehat{E[R_t]},\;\;k_2.
%%\]
%
%The equation to be solved numerically is: % ($t=1$ for daily data):
%\begin{equation}
%\Big{(}k_2+\frac{{t^*}\hat\sigma^2}{k_2}\Big{)}\Big{(}2\Phi\Big{(}\frac{k_2}
%{\hat{\sigma}\sqrt{{t^*}}}\Big{)}-1\Big{)}+2\hat\sigma\sqrt{{t^*}}\;
%\phi\Big{(}\frac{k_2}{\hat\sigma\sqrt{{t^*}}}
%\Big{)}-k_1=0.                      \label{main_eq}
%\end{equation}
%Once $\hat\sigma$ is known, equation (\ref{equation8}) gives $\hat\mu_s$ because of
%our knowledge of $k_1=\hat\mu$.

%We are left only to estimate $t^*$ from the data; one possibility is the 
%non-centered approach:
%\begin{equation}
%t^*
%%=\frac{\frac{1}{n}\Sigma_i\Big{(}\log \frac{S_{i+1}}{S_{i+}}\Big{)}^2}
%%{\frac{1}{n}\Sigma_i\Big{(}\log \frac{S_{i+}}{S_{i}}\Big{)}^2}
%=\frac{\sum_{i=1}^n\Big{(}\log \frac{{CLOSE(i)}}{{OPEN(i)}}\Big{)}^2}
%{\sum_{i=1}^n\Big{(}\log \frac{{OPEN(i+1)}}{{CLOSE(i)}}\Big{)}^2}.
%\label{equation12}
%\end{equation}
%
%
%As noted by Yang and Zhang (2000), the random variable:
%\begin{equation}
%\frac{1-t^*}{t^*}
%\frac{\sum_{i=1}^n\Big{(}\log \frac{\mbox{CLOSE(i)}}{\mbox{OPEN(i)}}\Big{)}^2}
%{\sum_{i=1}^n\Big{(}\log \frac{\mbox{OPEN(i+1)}}{\mbox{CLOSE(i)}}\Big{)}^2}
%\label{F}
%\end{equation}
%has an $F(n-1,n-1)$ distribution.
%\newpage


\begin{Examp}\label{examp30}
Consider the market data on the high, low, opening and closing prices for the IBM stock 
for the period from May 26, 2010 to June 18, 2010.
%March 15th, 2007 to May 25th, 2007.
%We use for interest rate the average value over the same period of the 3-month 
%US Treasury bill yield from 
%\href{http://www.treasury.gov/offices/domestic-finance/debt-management/interest-rate/yield_historical_2007.shtml}
%{Treasury data} $r=0.0509$. %$r=0.04955$. % \times 10^{-2}$.
For each of these days we consider the historical 3-month \footnote{For parameter estimation Hull 
(2006, p. 287) recommends using historical data of 90 to 180 days.} estimates of $k_1$ and $k_2$, 
and we solve the corresponding equation (\ref{main_eq}). 

The solution is our estimate of the volatility $\sigma\sqrt{1-f}$ corresponding to the trading day, 
and we present it in annualized form (i.e. multiplied by $\sqrt{252}$) in Figure \ref{fig_chart0}.
We compare our estimate of the volatility corresponding to
a one-day period with the one of Yang and Zhang (2000). On June 18, 2010 they are
%$V_i: 0.1543$, %$V_0:0.2313$, $V_C:0.1636$, $V_{RS}:0.1921$, 
$\sigma_a=0.2781$ (see (\ref{ann})) and $0.2982$ (annualized
volatility corresponding to $V_{YZ}$).

\begin{figure}[htb]
%\centerline{\includegraphics[scale=1.0]{ivol.png}}
\centerline{\epsfig{file=ivol.eps,scale=0.7,angle=0}}
\caption{Estimated intra-day IBM volatility - May 26 to June 18, 2010 }
\label{fig_chart0}
\end{figure}



%$V_P:0.1781$ $V_{CC}:0.1915$ $V_{Z}:0.2781$ $V_i=0.1543$.

%This means that we underestimated volatility. Applying the method of 
%Yang and Zhang (2000) gives $\sigma_a=0.2337$ in comparison to our
%$\sigma_a=0.150062$.



\end{Examp}

%\newpage

%\begin{Examp}\label{examp31}
%Consider the market data on the high, low, opening and closing prices for the IBM stock 
%for the period from October 31, 2006 to March 13, 2007 (90 trading days).
%%March 15th, 2007 to May 25th, 2007.
%%We use for interest rate the average value over the same period of the 3-month 
%%US Treasury bill yield from 
%%\href{http://www.treasury.gov/offices/domestic-finance/debt-management/interest-rate/yield_historical_2007.shtml}
%%{Treasury data} $r=0.0509$. %$r=0.04955$. % \times 10^{-2}$.
%For each of these days we consider the historical 3-month \footnote{For parameter estimation Hull 
%(2006, p. 287) recommends using historical data of 90 to 180 days.} estimates of $k_1$ and $k_2$, 
%and we solve the corresponding equation (\ref{main_eq}). 

%The solution is our estimate of the volatility $\sigma(1-f)$ corresponding to the trading day, 
%and we present it in annualized form (i.e. multiplied by $\sqrt{252}$) in Figure \ref{fig_chart}.

%\begin{figure}[htb]
%%\centerline{\includegraphics[scale=1.0]{chart_3m_vol.png}}
%\centerline{\epsfig{file=chart_3m_vol.eps,scale=0.7,angle=0}}
%\caption{Annualized volatility for the trading day based on a 3-month window}
%\label{fig_chart}
%\end{figure}
%\end{Examp}


%\begin{eqnarray*}
%ibm\_ data\_ mar\_ 2010.ods[k1-k2]
%&spreadsheet\\
%kone84.r& R\; solver\\
%k84musigma.ods& output\; vol
%\end{eqnarray*}


\section{European options: mispricing opportunities}

%Brenner and Subrahmanyam (1988) formula for at-the-money implied volatility:
%\[
%\sigma\sqrt{T}=\sqrt{2\pi}\;\frac CS,
%\]
%where S denotes a stock price, C a call option price, %X a discounted strike price, 
%$\sigma$ a standard deviation, and T the time until option maturity.

We use the resulting annualized volatility to compute the Black-Scholes prices of European options on the
stock. 
%using the parameter values estimated previously, 
We then seek those instances when the computed prices differ the most from the 
market prices, and devise trading strategies to take advantage of the price
difference.
% (in effect identifying the most profitable arbitrage opportunities).

\medskip

%The opening price is generally different
%from the previous closing price due to information that accumulates
%between trading days. 
%We want t

%First we clarify the terminology.

% of the 
%next trading day introduces a volatility that is due to the after-hours 
%arrival of information in the market. 







%If $t$ is measured in days, then the length of the trading day is $t=1$. 
%Based 
%on the variation of the stock price during a trading day, and that during a 
%virtual trading day, we will deduce the length of the latter.
% with respect to the regular trading day.



%The intra-day variance of the stock price $S_t$ 
%during one trading day of unit length ($t=1$) is:
%%varying from the opening price $S_{i+}$ to the closing price $S_{i+1}$:
%\[
%\operatorname{VAR}\;\ln\frac{S_{i+1}}{S_{i+}}=\sigma^2 \cdot 1.
%\]
%Let us denote by $t^*$ the fraction of the volatility contribution of the 
%unobserved {virtual trading day} relative to the volatility of a regular 
%{trading day}:
%\begin{equation}
%t^*=
%\frac{\operatorname{VAR}\ln\frac{S_{i+}}{S_{i}}}
%{\operatorname{VAR}\ln\frac{S_{i+1}}{S_{i+}}}.
%\label{fraction}
%\end{equation}
%Then $t^*$ is the length of a virtual trading day, and we have:
%\[
%\operatorname{VAR}\;\ln\frac{S_{i+}}{S_{i}}=\sigma^2 \cdot t^*.
%\]


%We estimate the fraction
%of the volatility contribution of the unobserved { virtual trading day} 
%relative to the volatility of a regular { trading day}.
%The opening price of the
%virtual trading day is the closing price of the current trading day, and the
%closing price of the virtual trading day is the opening price of the next
%trading day.
%To account for this jump 
%when the daily volatility is annualized, so 
%we add to each trading day a ``virtual" trading 
%day corresponding to the unobserved after-hours trading. 
%We regard that fraction as being the length of the { virtual trading day}
%(assuming the { trading day} has unit length).
% as having a length equal to that
%fraction of the length of a trading day.
%The annual volatility is given by the intra-day volatility of 
%252 trading days of unit length and by the volatility of the 252 after-hours 
%virtual trading days of length $t^*$ (on average).
% a fraction $t^*$ of the intra-day volatility.


We now use the volatility parameter estimated above to price European call 
options
%turn now to pricing European call options 
using the Black-Scholes formula:
\begin{equation}
C_t=S_t\;\Phi(d_1)-Ke^{-r(T-t)}\;\Phi(d_2),                      \label{equation14}
\end{equation}
with
\begin{eqnarray*}
d_1&=&\frac{\log (S_t/K)+(r+\sigma_a^2/2)(T-t)}{\sigma_a\sqrt{T-t}},\\
d_2&=&d_1-\sigma_a\sqrt{T-t}.
\end{eqnarray*}
%where the parameter $\sigma$ in (\ref{equation14}) is actually $\sigma_a$
%obtained in (\ref{equation13}).
%The price of the European put option can be easily derived from the call-put parity:
%\begin{equation}
%P_t=Ke^{-r(T-t)}\Phi(-d_2)-S_t\Phi(-d_1).\label{equation15}
%\end{equation}

We devise a trading strategy to take advantage of the information differential
between our estimated prices and market prices. For simplicity we trade only in
European call options, and assume that at expiry there is a payment equal to
the payoff so that no actual trading occurs in the underlying stock (naked
trading).

Having assumed a constant volatility there is no volatility smile and no 
stochastic volatility\footnote{Alternative approaches like
stochastic volatility or econometric models (ARMA, GARCH etc) are not discussed
here.}, so we restrict our analysis to 
European call options 
%that are close to at the 
%money and have a chance of being in the money at least once during the 
%observation period. Because of the volatility smile we choose those options 
whose strike prices are relatively close to the stock price at the beginning 
of the period (preferably in the money), and whose expiry dates are up to 
three months 
(the parameters can be re-estimated later in view of new data).
\begin{Examp}\label{examp2-0}
Consider the market prices for the European call options on IBM for the period
May 26, 2010 to June 18, 2010 with expiry dates June 18 and July 16,
and strike prices $K\in\{115, 120, 125, 130\}$ %presented in Table \ref{tab_data} 
(the stock price on May 26 was 125.91).
%$\checkmark$
%\begin{table}[htb]
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|c|c|}
%\hline
%expiry/K&$<$115&115&120&125&130&$>$130\\
%\hline
%June 18&-&\checkmark&\checkmark&\checkmark&\checkmark&-\\
%July 16&-&\checkmark&\checkmark&\checkmark&\checkmark&-\\
%\hline
%\end{tabular}
%\caption{Strike prices $K$ and expiration dates $T$ selected for option data}
%\label{tab_data}
%\end{center}
%\end{table}
We compare these market prices with the Black-Scholes prices calculated
using (\ref{equation14}). Here the inputs are the stock price, the volatility estimated in 
Example \ref{examp30}, and $r$ the value of the 1-month US Treasury 
bill yield for the previous day (online
\href{
http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/default.aspx
%http://www.treasury.gov/offices/domestic-finance/debt-management/interest-rate/yield_historical.shtml}
{Treasury data}\footnote
%{http://www.treasury.gov/offices/domestic-finance/debt-management/interest-rate/yield\_historical.\\shtml}). 
{http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/default.aspx}
%$r=0.04955$. % \times 10^{-2}$.
\end{Examp}

Since  %leave aside for now incorporating after-hours volatility and use straight
%compute the prices of European call options using the estimated intra-day volatility $\sqrt{252}\; x$, with $x$ solution of
the intra-day volatility of Example \ref{examp30} that we use in Black-Scholes formula
 % (\ref{main_eq}). %  to compute European call option prices. 
does not include the effect of the after-hours evolution,
we compensate by allowing our prices to differ by up to 10\% from the bid-ask spread. Thus, 
we trade when our estimated call price falls outside the interval (0.9$\times$bid-price, 1.1$\times$ask-price). 
%The flag returns -1 if our price is lower, and 1 if it is higher.
%Our trading strategy deals only with call options, unless they reach the
%maturity date in which case stock might be traded as well. More specifically we have 

There are two cases.
%{\bf The case flag=-1:} 
If our price is lower, then we short-sell the option at the bid price and wait for the first day when the 
estimated price is no longer lower to buy back the option at the then ask price. If it expires and the call 
option is exercised then we buy the stock in the market and deliver it.

%{\bf The case flag=1:} 
If our price is higher, then we buy the option at ask price and wait for the first day when the price is no
longer higher to sell it at the then bid price. If it reaches expiry date, then we exercise it.

This trading strategy is summarized in Algorithm \ref{algo0} for $t$ between May 26, 2010 and
June 18, 2010 for European call options expiring at close June 18, 2010. The data is retrieved once a day,
except on expiration date when it is retrieved several times a day (this can be implemented as an algorithmic
trading strategy and deployed continuously without much effort, especially by those interested in technical
trading).



%
%
%\addtocounter{footnote}{1}
%\footnotetext[\value{footnote}]{at 12:27pm}
%
%\addtocounter{footnote}{1}
%\footnotetext[\value{footnote}]{at 13:26pm}
%
%\addtocounter{footnote}{1}
%\footnotetext[\value{footnote}]{at 15:58pm}
%
%\addtocounter{footnote}{1}
%\footnotetext[\value{footnote}]{at 16:00pm}
%
%\addtocounter{footnote}{-1}
%
%We notice that flag=-1 on May 26 for $K=130$, on May 28 for $K=130$, on June 7 for $K=125$ and $K=130$, 
%on June 8 for 
%$K=120$, on June 9 for $K=125$, and on June 18 for $K=130$.
\begin{table}[htb]
\begin{center}
\begin{threeparttable}
\begin{tabular}{|c|c|c|c|c||c|c|c|c|}
\hline
 t&K&$\hat{C}(t)$ &(bid,ask)& trade&  t& (bid,ask) &trade& profit
%&\%\tnote{f}
\\
\hline
May 26&130&0.90&(\textcolor{blue}{1.16},{1.17})&sell&May 27&(0.96,\textcolor{red}{0.99})&buy&0.17
%&15
\\
May 28&130&0.57&(\textcolor{blue}{0.75},0.78)&sell&Jun 2&(0.67,\textcolor{red}{0.70})&buy&0.05
%&7
\\
Jun 7&125&1.92&(\textcolor{blue}{2.20},2.23)&sell&Jun 8&(1.21,\textcolor{red}{1.23})&buy&0.97
%&44
\\
Jun 7&130&0.29&(\textcolor{blue}{0.42},0.44)&sell&Jun 8&(0.15,\textcolor{red}{0.17})&buy&0.25
%&60
\\
Jun 8&120&3.69&(\textcolor{blue}{4.15},4.20)&sell&Jun 9&(4.60,\textcolor{red}{4.75})&buy&(0.6)
%&(14)
\\
Jun 9&125&1.09&(\textcolor{blue}{1.30},1.38)&sell&Jun 10&(3.05,\textcolor{red}{3.15})&buy&(1.77)
%&(128)
\\
Jun 17&130&1.09&(0.90,\textcolor{red}{0.94})&buy&Jun 19&(\textcolor{blue}{1.00},1.05)&sell&0.06
%&6
\\
Jun 18\tnote{a} &130&0.60&(0.48,\textcolor{red}{0.51})&buy&Jun 18\tnote{b} &(\textcolor{blue}{0.63},0.69)&sell&0.12
%&24
\\
%(12:27)&&&&&(13.26)&&&&\\
Jun 18\tnote{c} &130&0.18&(\textcolor{blue}{0.21},0.25)&sell&Jun 18\tnote{d} &$S_t$=130.14&ex\tnote{e}&0.07
%&33
\\
%(15:58)&&&&&(16:00)&&&&\\
\hline
\end{tabular}
\begin{tablenotes}[para]
\item [a] at 12:27pm 
\item  [b] at 13:36pm
\item [c] at 15:58pm
\item [d] at 16:00pm
\item [e] if exercised
%\item [f] profit rate
\end{tablenotes}
\end{threeparttable}
\caption{Trading in the call option expiring June 18, 2010 \newline
\centerline{(left: open a position, right: close position)}}
\label{table10}
\end{center}
\end{table}
\begin{Remar}\label{algo}
This strategy results in an overall loss of 0.68 (see Table \ref{table10}). This is due mostly 
to one large loss induced by a large sudden move in the stock price
on June 10 (127.3 versus 123.9 the day before). That is because we use yesterday's
intra-day volatility to trade in today's world. 

Over a time horizon longer than a month the strategy can absorb
such shocks in the stock prices, provided they are sparse. Alternatively, one can
implement an additional stopping rule when the change in the stock price exceeds a pre-determined margin.

A similar behaviour is exhibited when applying the same trading strategy to
the European call option expiring at close July 16, 2010, but as the expiry date
is longer than a couple of months the limitations of the assumptions of the model become apparent.
\end{Remar}
%\newpage

%For the call option expiring in April we notice in Figure \ref{fig1} 
%that for the strike price 
%closest to the stock price ($K=95$) the our estimated prices are below market 
%prices.
%charges a premium in comparison 
%to our prices. Furthermore, this premium seems to be constant in time, at 
%about \$ 0.3. Shorting this option earns a profit.


%\vspace*{2cm}
%\begin{figure}[htb!]
%%\centerline{\includegraphics[scale=0.82]{testK95apr07.png}}
%\centerline{\epsfig{file=testK95apr07.eps,scale=1.0,angle=0}}
%\caption{Call option expiring in April 2007 (blue=market price, red=estimated price)}
%\label{fig1}
%\end{figure}

%For strike prices further into the money the difference between the market
%prices and our estimates seem smaller, with occasional overlapping areas
%(see Figures \ref{fig2} and \ref{fig3}).


%\begin{figure}[h!tb]
%\centerline{\includegraphics[width=3in]{testK90apr07.png}}
%\caption{Call option expiring in April 2007 (blue=market price, red=estimated price)}
%\label{fig2}
%\end{figure}


%\begin{figure}[h!tb]
%\centerline{\includegraphics[width=3in]{testK85apr07.png}}
%\caption{Call option expiring in April 2007 (blue=market price, red=estimated price)}
%\label{fig3}
%\end{figure}

%We compare the market option price data with the estimates obtained using
%our model of 
%volatility. There are occasionally big differences in the prices, 
%which
%can be grouped in two categories. 
%The first one is when our estimated price, although
%different from the reported market price, is within the bid-ask spread of the
%market.

%The second group of differences occurs for European put options with expiration
%times longer than a couple of months. We find invariably for this type of 
%options that the market charges a premium to what our estimates would suggest,
%even when comparing with the midpoint of the bid-ask spread (see plots attached
%as appendices).

%straddle view: highlighted options are in-the-money
%\end{Examp}

%\newpage

%There are instances where the difference between the market price and our
%estimated prices are significant. We plan to take advantage of this situation
%by trading in the options (``naked trading", i.e. no trading in the stock).

%\medskip

%The risks of this strategy are two-fold. On one hand, as time approaches 
%expiration long positions in out-of-the-money calls become worthless. On the 
%other hand, changes in the stock price could render a position unprofitable.

%To manage the former risks we restrict ourselves to short positions,
%while for the latter risks we implement a strategy of  
%cutting the losses short when the stock price moves too much. This is done 
%based on the greek $\Delta=\Phi(d_1)$, which intervenes in the linear 
%approximation:
%\begin{equation}
%C_{t_1}-C_{t_0}\approx \Phi(d_1^0) (S_{t_1}-S_{t_0}).\label{delta}
%\end{equation}
%Namely, we define the exit time to be the first time the price of the call option
%moves by an amount larger than some constant $\delta$, which is equivalent
%by (\ref{delta}) to the stock price moving by an amount 
%$\displaystyle \delta/\Phi(d_1^0)$: 
%\begin{equation}
%\tau:=\inf\{t\in[t_0,T]: |C_t-C_{t_0}|>\delta\}=
%\inf\{t\in[t_0,T]: |S_t-S_{t_0}|>\delta/\Phi(d_1^0)\}. \label{tau}
%\end{equation}
%One possible choice of $\delta$ is an {\em a priori} ten percent of the call 
%price. Another possibility is to choose $\delta$ such that based on some 
%historic data sample we have:
%\begin{equation}
%P(|S_t-S_{t_0}>\delta/\Phi(d_1^0)|)<0.1, \label{prob}
%\end{equation}
%and then use this value of $\delta$ for the current data set.


%\begin{Examp}\label{examp2}
%Consider the market prices for the European call options on IBM for the period
%March 15th, 2007 to May 25th, 2007 with expiry dates and strike prices presented
%in Table \ref{tab_data}.

%$\checkmark$



%We compare them with the Black-Scholes prices (\ref{equation14}) calculated
%using the volatility estimated in Example \ref{examp31} and
%$r=0.0509$, obtained as the average value of the  3-month US Treasury 
%bill yield for the same period from online
%\href{http://www.treasury.gov/offices/domestic-finance/debt-management/interest-rate/yield_historical_2007.shtml}
%{Treasury data}\footnote{http://www.treasury.gov/offices/domestic-finance/debt-management/interest-rate/\\yield\_historical\_2007.shtml}. %$r=0.04955$. % \times 10^{-2}$.

%For the call option expiring in April we notice in Figure \ref{fig1} 
%that for the strike price 
%closest to the stock price ($K=95$) the our estimated prices are below market 
%prices.
%charges a premium in comparison 
%to our prices. Furthermore, this premium seems to be constant in time, at 
%about \$ 0.3. Shorting this option earns a profit.
%This means that we understimated volatility. Applying the method of 
%Yang and Zhang (2000) gives $\sigma_a=0.2337$ in comparison to our
%$\sigma_a=0.150062$.

%\vspace*{2cm}
%\begin{figure}[htb!]
%%\centerline{\includegraphics[scale=0.82]{testK95apr07.png}}
%\centerline{\epsfig{file=testK95apr07.eps,scale=1.0,angle=0}}
%\caption{Call option expiring in April 2007 (blue=market price, red=estimated price)}
%\label{fig1}
%\end{figure}

%For strike prices further into the money the difference between the market
%prices and our estimates seem smaller, with occasional overlapping areas
%(see Figures \ref{fig2} and \ref{fig3}).


%\begin{figure}[h!tb]
%\centerline{\includegraphics[width=3in]{testK90apr07.png}}
%\caption{Call option expiring in April 2007 (blue=market price, red=estimated price)}
%\label{fig2}
%\end{figure}


%\begin{figure}[h!tb]
%\centerline{\includegraphics[width=3in]{testK85apr07.png}}
%\caption{Call option expiring in April 2007 (blue=market price, red=estimated price)}
%\label{fig3}
%\end{figure}

%We compare the market option price data with the estimates obtained using
%our model of 
%volatility. There are occasionally big differences in the prices, 
%which
%can be grouped in two categories. 
%The first one is when our estimated price, although
%different from the reported market price, is within the bid-ask spread of the
%market.

%The second group of differences occurs for European put options with expiration
%times longer than a couple of months. We find invariably for this type of 
%options that the market charges a premium to what our estimates would suggest,
%even when comparing with the midpoint of the bid-ask spread (see plots attached
%as appendices).

%straddle view: highlighted options are in-the-money
%\end{Examp}


\section{Conclusions}
We have used the method of moments to estimate the volatility of the stock price
and used this to identify arbitrage opportunities in the market of European
options. As a by-product we have derived the density and expectation of the
range of an arithmetic Brownian motion.

In comparison to the estimate of Yang and Zhang (2000), our volatility estimate takes
advantage of the actual range of the Brownian motion and perhaps does not
overestimate as much. It is most useful for short expiration dates and for strike prices
that are not far out. We believe it is an efficient alternative that
can be easily computed and has a practical implementation. These traits recommend it
to the attention of practioners in the field.
%\newpage
%\begin{thebibliography}{99}
%\begin{center}

\newpage
%\medskip 

\hspace*{0.6cm}{\bf REFERENCES}
%\end{center}


\bigskip

\noindent
\textsc{ Billingsley, P.} (1968):
{\em Convergence of probability measures.} John Wiley, New York.

\noindent
\textsc{ Chan, L. and D. Lien} (2003):
Using high, low, open, and closing prices to estimate the effects of cash
settlements on futures prices.
{\em International Review of Financial Analysis\/} 12, 35--47.

%\noindent
%\textsc{ Domin\'{e}, M.} (1996):
%First passage time distribution of a Wiener process with drift concerning two
%elastic barriers.
%{\em Journal of Applied Probability\/} 33, 164--175.

%\noindent
%\textsc{ Douady, R., A. Shiryaev and M. Yor } (2000):
%On probability characteristics of downfalls in a standard Brownian motion.
%{\em SIAM Theory Probability Appl.\/} 44(1), 29--38.

\noindent
\textsc{ Fama, E.F.} (1965):
The behaviour of stock market prices.
{\em Journal of Business\/} 38, 34--105.

\noindent
\textsc{ Feller, W.} (1951):
The asymptotic distribution of the range of sums of independent random 
variables.
{\em Annals of Mathematical Statistics\/} 22, 427--432.

%\noindent
%\textsc{ French, K.R.} (1980):
%Stock returns and the weekend effect.
%{\em Journal of Financial Economics\/} 8, 55--69.

%\noindent
%\textsc{ French, K.R. and Roll, R.} (1986):
%Stock return variances: the arrival of information and the reaction of traders.
%{\em Journal of Financial Economics\/} 17, 5--26.

\noindent
\textsc{ Garman, M. and M. Klass } (1980):
On the estimation of security price volatilities from historical data.
{\em Journal of Business\/} 53(1), 67--78.

%\noindent
%\textsc{ Girsanov, I.} (1960):
%On transforming a certain class of stochastic processes by absolutely
%continuous substitution of measures.
%{\em Theory of Probability and its Applications}, V(3), 285--301.

\noindent
\textsc{ Harrison, J.M.} (1985):
{\em Brownian motion and stochastic flow systems.}
Wiley, New York.

\noindent
\textsc{ Hull, J.C.} (2006):
{\em Options, futures and other derivatives.} 6th ed,
Prentice Hall, New Jersey.

\noindent
\textsc{ Karatzas, I. and S. E. Shreve} (1998):
{\em Brownian motion and stochastic calculus.}
Springer, New York.


%\noindent
%%\bibitem[\protect\citeauthoryear{Atkinson, Pliska, and Wilmott}{Atkinson
%%  et~al.}{1997}]{[APW]}
%\textsc{ Atkinson, C., S.~R. Pliska, and P.~Wilmott} (1997):
%Portfolio management with transaction costs,
%{\em Proceedings of the Royal Society of London Series A\/} 453 (1958), 551--568.

%\noindent
%%\bibitem[\protect\citeauthoryear{Atkinson and Wilmott}{Atkinson and
%%  Wilmott}{1995}]{[AW]}
%\textsc{ Atkinson, C. and P.~Wilmott} (1995):
%Portfolio management with transaction costs: An asymptotic analysis
%  of the {M}orton and {P}liska model,
%{\em Mathematical Finance\/} 5(4), 357--367.

%\noindent%
%\textsc{ Barber, B., and Odean, T.} (2003): Are individual investors tax savvy? 
%Evidence from retail and discount brokerage accounts, {\em Journal of Public 
%Economics} 88, 419--442.

%\noindent
%%\bibitem[\protect\citeauthoryear{Bielecki and Pliska}{Bielecki and
%%  Pliska}{2000}]{[BP2]}
%\textsc{ Bielecki, T. and S.~Pliska} (2004):
%Risk-sensitive ICAPM with application to fixed-income management,
%{\em IEEE Transactions on automatic control} 49(3), 420--432.

\noindent
\textsc{ Kon\'{e}, F. J.} (1996):
{\em Estimation of the volatility of stocks using the high, low and 
closing prices.}
Ph.D. thesis, State University of New York at Stony Brook.

\noindent
\textsc{ L\'{e}vy, P. } (1948):
{\em Processus stochastique et mouvement brownien.} Gauthier-Villars, Paris. 

\noindent
\textsc{ Magdon-Ismail, M. and A. Atiya } (2000):
Volatility estimation using high, low and close data - a maximum likelihood
approach.
{\em Computational Finance\/}, June CF2000 Proceedings.

\noindent
\textsc{ Magdon-Ismail, M., A. Atiya, A. Pratap and Y. Abu-Mustafa } (2004):
On the maximum drawdown of a Brownian motion.
{\em Journal of Applied Probability\/}, 41(1), 147--161.

%\noindent
%%\bibitem[\protect\citeauthoryear{Boyce and DiPrima}{Boyce and
%  DiPrima}{1992}]{[BD]}
%\textsc{ Boyce, W. and R.~DiPrima} (1992):
%{\em Elementary Differential Equations and Boundary Value Problems}.
%New York: Wiley \& Sons.

\noindent
\textsc{ Parkinson, M.} (1980):
The extreme value method for estimating the variance of the rate of return.
{\em Journal of Business} 53(1), 61--65.

\noindent
\textsc{ Rogers, L.C.G. and S. Satchell } (1991):
{ Estimating variance from high, low and closing prices}. 
{\em Annals of Applied Probability\/} 1(4), 504--512.

\noindent
\textsc{ Rogers, L.C.G., S. Satchell and Y. Yoon} (1994):
Estimating the volatility of stock prices: a comparison of methods that use
high and low prices.
{\em Applied Financial Economics\/} 4, 241--247.

\noindent
\textsc{ Rogers, L.C.G. and F. Zhou} (2008):
Estimating correlation from high, low, opening and closing prices.
{\em The Annals of Applied Probabilty\/} 18(2), 813--823.

\noindent
%\bibitem[\protect\citeauthoryear{Buescu}{Buescu}{2003}]{[cbuescu]}
\textsc{ Sutrick K., J. Teall, A. Tucker and J. Wei} (1997):
{ The range of  Brownian motion processes: density functions and derivative
pricing applications}.
{\em The Journal of Financial Engineering} 6(1), 31--46.


\noindent
%\bibitem[\protect\citeauthoryear{Buescu}{Buescu}{2003}]{[cbuescu]}
\textsc{ Yang D. and Q. Zhang} (2000):
{  Drift-independent volatility estimation based on high,
low, open, and close prices}.
{\em Journal of Business} 73, 477--491.

%\noindent
%%\bibitem[\protect\citeauthoryear{Taksar, Klass, and Assaf}{Taksar
%%  et~al.}{1988}]{[TKA]}
%\textsc{ Taksar, M.~I., M.~J. Klass, and D.~Assaf} (1988):
%A diffusion model for optimal portfolio selection in the presence of
%  brokerage fees,
%{\em Mathematics of Operations Research\/} 13(2), 277--294.


%\begin{figure}[h!t]
%\centerline{\epsfig{file=boxn.eps,scale=0.7,angle=0}}
%\caption{Boundaries of the approximating grid}%Lower boundary of the domain}
%\label{box}
%\end{figure}

%\hspace*{-2cm}\includegraphics[scale=0.85]{option_short.pdf}
%\hspace*{-2cm}\includegraphics[scale=0.95]{option_long.pdf}
%\hspace*{-2cm}\includegraphics[scale=0.95]{testK95apr07.pdf}
%\caption{European put option charge a premium for expiration longer than two 
%months\\
%Straddle view, highlighted options are in-the-money}

%\listofalgorithms

\IncMargin{1em}
\begin{algorithm}
% \SetKwData{Left}{left}
% \SetKwData{This}{this}
% \SetKwData{Up}{up}
 %\SetKwFunction{Union}{Union}
 %\SetKwFunction{FindCompress}{FindCompress}
 \SetKwFunction{BlackScholesCall}{BlackScholesCall}
 \SetKwFunction{profit}{profit}
 %\SetKwFunction{flag}{flag}
 \SetKwInOut{Input}{input}
 \SetKwInOut{Output}{output}
\BlankLine
\Input{Parameters $t$, $\sigma_a$, $r$, $S_t$, bid(t) and ask(t)  (European call prices)}
\Output{Profit of trading strategy}
\BlankLine
\profit $\leftarrow$ 0 \tcp*{initialize}
%$t_0 \leftarrow$ May 26, 2010\tcp*{starting date}
T $\leftarrow$ June 18, 2010\tcp*{expiry date}
%\For
%{
%  \emph{t} $\leftarrow t_0$ \KwTo \emph{T}
%}
%{ %begin for
  compute $\sigma_a$\tcp*{Example \ref{examp30}}
  \For
  {
    \emph{K} $\leftarrow$ \emph{115} \KwTo \emph{130}
  }
  { %begin 2nd for
     $\hat{C}(t)\leftarrow$ \BlackScholesCall{$t, T, K, \sigma_a, r, S_t$}\tcp*{(\ref{equation14})}
 %    flag(t,K)==-2\tcp*{initialize}
     \eIf
     {
       $\hat{C}<0.9\times$ \emph{bid(t)}
     }
     {
%        flag(t,K) $\leftarrow$ -1\; 
      \profit $\leftarrow$ \profit + bid(t)\tcp*{short-sell call}
      \While{$t<T$ \emph{and} $\hat{C}$\emph{(t)}$<$\emph{0.9}$\times$\emph{bid(t)}}{t $\leftarrow$ t+1
                                                   \tcp*{wait 1 day}
          compute $\sigma_a$\tcp*{Example \ref{examp30}}
        $\hat{C}(t)\leftarrow$ \BlackScholesCall{$t, T, K, \sigma_a, r, S_t$}\tcp{(\ref{equation14})}
         }
    \eIf{$t<T$}{\profit $\leftarrow$ \profit - ask(t) \tcp*{buy back call}}
      {\If{\emph{call is exercised}}{\profit $\leftarrow$ \profit - ($S_t$ -K)\;\tcp{buy stock and deliver for K}
         }}
      }
      {
           \If
          {  
             $\hat{C}>1.1\times$\emph{ask(t)}
           }
 %         {flag(t,K) $\leftarrow$ 1\;}
        {
          \profit $\leftarrow$ \profit - ask(t)\tcp*{buy call}
          \While{$t<T$ \emph{and} $\hat{C}$\emph{(t)}$>$\emph{1.1}$\times$\emph{ask(t)}}
         {
          t $\leftarrow$ t+1\tcp*{wait 1 day}
          compute $\sigma_a$\tcp*{Example \ref{examp30}}
           $\hat{C}(t)\leftarrow$ \BlackScholesCall{$t, T, K, \sigma_a, r, S_t$}\;%\tcp{(\ref{equation14})}
         }
    \eIf{$t<T$}{\profit $\leftarrow$ \profit + bid(t) \tcp*{sell call}}
         { \profit $\leftarrow$ \profit + ($S_t$ -K)\tcp*{exercise call}
         }}    
    } %end 2nd for
 %  \lForEach{element $e$ of the line $i$}{\FindCompress{p}}
} %end for
\Return{\profit}\;
\BlankLine
\caption{Trading strategy for a mispricing opportunity found at time $t$ 
%between May 26, 2010 and June 18, 2010
}\label{algo0}
\end{algorithm}\DecMargin{1em}
\end{document}

1112.4534.extracted_bib
\bibitem[\protect\citeauthoryear{Atkinson, Pliska, and Wilmott}{Atkinson
%%  et~al.}{1997}]{[APW]}
%\textsc{ Atkinson, C., S.~R. Pliska, and P.~Wilmott} (1997):
%Portfolio management with transaction costs,
%{\em Proceedings of the Royal Society of London Series A\/} 453 (1958), 551--568.

%\noindent
%%
\bibitem[\protect\citeauthoryear{Atkinson and Wilmott}{Atkinson and
%%  Wilmott}{1995}]{[AW]}
%\textsc{ Atkinson, C. and P.~Wilmott} (1995):
%Portfolio management with transaction costs: An asymptotic analysis
%  of the {M}orton and {P}liska model,
%{\em Mathematical Finance\/} 5(4), 357--367.

%\noindent%
%\textsc{ Barber, B., and Odean, T.} (2003): Are individual investors tax savvy? 
%Evidence from retail and discount brokerage accounts, {\em Journal of Public 
%Economics} 88, 419--442.

%\noindent
%%
\bibitem[\protect\citeauthoryear{Bielecki and Pliska}{Bielecki and
%%  Pliska}{2000}]{[BP2]}
%\textsc{ Bielecki, T. and S.~Pliska} (2004):
%Risk-sensitive ICAPM with application to fixed-income management,
%{\em IEEE Transactions on automatic control} 49(3), 420--432.

\noindent
\textsc{ Kon\'{e}, F. J.} (1996):
{\em Estimation of the volatility of stocks using the high, low and 
closing prices.}
Ph.D. thesis, State University of New York at Stony Brook.

\noindent
\textsc{ L\'{e}vy, P. } (1948):
{\em Processus stochastique et mouvement brownien.} Gauthier-Villars, Paris. 

\noindent
\textsc{ Magdon-Ismail, M. and A. Atiya } (2000):
Volatility estimation using high, low and close data - a maximum likelihood
approach.
{\em Computational Finance\/}, June CF2000 Proceedings.

\noindent
\textsc{ Magdon-Ismail, M., A. Atiya, A. Pratap and Y. Abu-Mustafa } (2004):
On the maximum drawdown of a Brownian motion.
{\em Journal of Applied Probability\/}, 41(1), 147--161.

%\noindent
%%
\bibitem[\protect\citeauthoryear{Boyce and DiPrima}{Boyce and
%  DiPrima}{1992}]{[BD]}
%\textsc{ Boyce, W. and R.~DiPrima} (1992):
%{\em Elementary Differential Equations and Boundary Value Problems}.
%New York: Wiley \& Sons.

\noindent
\textsc{ Parkinson, M.} (1980):
The extreme value method for estimating the variance of the rate of return.
{\em Journal of Business} 53(1), 61--65.

\noindent
\textsc{ Rogers, L.C.G. and S. Satchell } (1991):
{ Estimating variance from high, low and closing prices}. 
{\em Annals of Applied Probability\/} 1(4), 504--512.

\noindent
\textsc{ Rogers, L.C.G., S. Satchell and Y. Yoon} (1994):
Estimating the volatility of stock prices: a comparison of methods that use
high and low prices.
{\em Applied Financial Economics\/} 4, 241--247.

\noindent
\textsc{ Rogers, L.C.G. and F. Zhou} (2008):
Estimating correlation from high, low, opening and closing prices.
{\em The Annals of Applied Probabilty\/} 18(2), 813--823.

\noindent
%
\bibitem[\protect\citeauthoryear{Buescu}{Buescu}{2003}]{[cbuescu]}
\textsc{ Sutrick K., J. Teall, A. Tucker and J. Wei} (1997):
{ The range of  Brownian motion processes: density functions and derivative
pricing applications}.
{\em The Journal of Financial Engineering} 6(1), 31--46.


\noindent
%
\bibitem[\protect\citeauthoryear{Buescu}{Buescu}{2003}]{[cbuescu]}
\textsc{ Yang D. and Q. Zhang} (2000):
{  Drift-independent volatility estimation based on high,
low, open, and close prices}.
{\em Journal of Business} 73, 477--491.

%\noindent
%%
\bibitem[\protect\citeauthoryear{Taksar, Klass, and Assaf}{Taksar
%%  et~al.}{1988}]{[TKA]}
%\textsc{ Taksar, M.~I., M.~J. Klass, and D.~Assaf} (1988):
%A diffusion model for optimal portfolio selection in the presence of
%  brokerage fees,
%{\em Mathematics of Operations Research\/} 13(2), 277--294.


%\begin{figure}[h!t]
%\centerline{\epsfig{file=boxn.eps,scale=0.7,angle=0}}
%\caption{Boundaries of the approximating grid}%Lower boundary of the domain}
%\label{box}
%