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% November 7, 2000
% robert chen
% burton rosenberg
%
% Estimating volatility in markets with price collars
%


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\title{Inferring model parameters in markets with collars (full paper)}
\author{Robert W. Chen$^1$, Burton Rosenberg$^1$, \& Yi-Tsung Lee$^2$}
\address{${}^1$University of Miami\\
${}^2$National Chung Cheng University}


\begin{document}
\maketitle

%\hrule
%\begin{center}
%{\Large\sc Inferring model parameters in markets with collars }
%{\em ~
%\\ ~
%\\ Robert W. Chen
%\\ Burton Rosenberg
%\\ University of Miami
%\\~
%\\ Yi-Tsung Lee
%\\ National Chung Cheng University}
%\end{center}
%\hrule

\begin{abstract}

Security prices are set by a continuous auction, the rules of
which are set by the exchange or by the government. For many
exchanges, there is a general free-flow of price information
resulting in stock prices which can be modelled by a random walk
following a Weiner-Levy process. However, many markets have
collars, under which the rules of the auction will not let prices
move too rapidly. In this paper we present methods for estimating
the volatility of the underlying price data when the true price
information is obscured by such collars. Numerical simulations are
presented which demonstrate and contrast the methods.
~\\
%{\em AMS 2000 subject classifications.}
%Primary --- 65C20, 62P05. 
% Probabilistic methods, simulation and stochastic differential equations
% 65C20 Models, numerical methods
% 62P05 Applications to actuarial sciences and financial mathematics
% 62F35 Robustness and adaptive procedures
% 60G40 Stopping times; optimal stopping problems; gambling theory
%Secondary --- 60G40.
%~\\
%{\em Key words and phrases.}
%Estimation of volatility, market models, market collars.

%~\\
%~\\
%{\em Running title.} Volatility estimation with collars

\end{abstract}

%\newpage

%\addtolength{\baselineskip}{.5\baselineskip}
%\begin{abstract}
%We develop methods to estimate the volatility of a random
%walk under the regime that observed values are collared by a fixed percentage
%per observation. Such a regime models securities whose daily prices changes
%are collared within limits set by the previous day's close.
%
%{~}
%
%\noindent{\sc Keywords:}
%
%\end{abstract}

\section{Introduction}

Security prices are set by a continuous auction. The rules of the
auction are set by the exchange or by the government. For many
exchanges, there is a general free-flow of price information
resulting in stock prices which can be modelled by a random walk
following a Weiner-Levy process, perhaps with an added component
of drift depending on the season, investor psychology, or other
poorly understood factors. This model of stock prices has been
very important both theoretically and practically to the
development of option pricing.

However, many markets have collars, under which the rules of the
auction will not let prices move too rapidly. We consider the case
where the previous day's closing price is set as the center of a
percentage band outside of which the stock will not trade for that
day. Since this affects the volatility of stock prices, and
options are priced as a function of volatility, the question
arises as to the proper pricing of options in markets with
collars.

%We propose a simply model in which there is an ideal stock price
%and an exchange price. The ideal stock price is the market
%clearing price which may be outside of a legally allowed price
%range. The exchange price is the price quoted by the exchange and
%is subject to the exchange or government restrictions on price
%ranges. We assume that the ideal or market-clearing price is not
%affected by the exchange price, that the exchange price is derived
%by applying a deterministic algorithm to the ideal price. We also
%assume that the exchange price will always return to the ideal
%price, so that investors will always have the opportunity to trade
%at the ideal price. Conditions on collar size can insure that this
%is so.

In this paper we present three methods by which the volatility of
the ideal price can be estimated by looking only at the exchange
price:
%\begin{enumerate}
%\item 
a method based on the measure of waiting times;
%\item 
a method based on the estimate of the likelihood of an at-market
day being followed by an at-limit day (definitions follow);
%\item 
the renewal time method introduced by Chiang and Wei. \cite{ChiangWei}.
%\end{enumerate}
%The renewal time method is due to Chiang and Wei. \cite{ChiangWei}.
In the full paper we provide a rigorous proof of strong consistency
for their method.

In addition, we compare the results of these methods to the
volatility of the actual stock price and perform simulations on
which of the two values are more appropriate for option pricing.

\section{Methods}

\subsection{Definitions}

Our ideal stock trace is a sequence of closing prices $S_0, S_1,
\ldots$. The inter-day increments are independent, identically
distributed random variables with log-normal distribution of
common mean $\mu$ and variance $\sigma^2$,
\[
 Z_i = \ln ( S_i/S_{i-1} ) \sim {\cal N} ( \mu, \sigma^2).
\]
If $r$ is the risk-free rate, then $\mu = r - \sigma^2/2$.

Collars are applied to the ideal price trace $S_i$ to create an
observable sequence $S^*_i$ of exchange prices. Given lower and
upper collars,
\[
    \kappa_l < 1 < \kappa_u,
\]
we define
$S^*_i, \,i=0, 1, 2, \ldots $ according to,
\begin{eqnarray*}
    S^*_i &=& \left \{
               \begin{array}{ll}
                  S_0 & \mbox{if $i=0$,}\\
                  \kappa_l\, S^*_{i-1} & \mbox{if $S_i <\kappa_l\, S^*_{i-1}$,} \\
                  \kappa_u\, S^*_{i-1} & \mbox{if $S_i > \kappa_u\, S^*_{i-1},$} \\
                  S_i & \mbox{otherwise.} \\
               \end{array}
            \right.
\end{eqnarray*}
We classify each observation $S^*_i$ as either {\em at-market} or
{\em at-limit}. Since we cannot always observe $S_i$, an
observation $S^*_i$ will be considered at-market only if it is
strictly within that observation's collar. Else we must conclude
that the observation is at-limit. Formally, the set of market
observations is,
\[
           T = \{ i>0 \,|\,(\kappa_l\, S^*_{i-1}) < S^*_i
                  < (\kappa_u\, S^*_{i-1}) \} \cup \{0\}
\]
and the set of limit observations is the complement,
\[
           L = T^c = \{ i>0 \,|\, S^*_i  = (\kappa_l \,S^*_{i-1})
                     \mbox{ or } S^*_i = (\kappa_u\, S^*_{i-1}) \}.
\]
We enumerate the indices of $T$ and $L$ in ascending order as $T_0, T_1,\ldots$
and $L_0, L_1, \ldots$. Note that by construction $T_0=0$, and that $L$ might be
empty.

An example trace of 90 day data is given in figure 1. The ideal
price, exchange price and their difference are plot, with the
difference trace shifted and scaled for display. The simulated
stock has zero mean,  $\sigma=47\%$, and the collars were set to
$(5/4 )\,\sigma$.

\begin{figure}
\epsfig{file=pricechart_5_4.eps} \caption{Ideal, exchange prices
and delta}
\end{figure}

\subsection{Estimation of the mean}

We estimate the mean $\mu$ of $S_i$ and prove some additional
lemmas.

If we consider any two consecutive at-market observations, we have
a random variable,
\[
   X_i = \ln \left( S_{T_i}/S_{T_{i-1}} \right)
       = \sum_{j=T_{i-1}+1}^{T_i} Z_j.
\]
Since $Z_i$ and $Z_j$ are independent for $i\not = j$, the $X_i$
are independent identically distributed random variables. This is
also true that the random variable $T'_i = T_i - T_{i-1}$ are
independent and identically distributed.

% Definition of B
Define $B_k(a)$ to be the event that the ideal price is above the
upper limit of the exchange price for at least $k$ consecutive
days given that at day zero the ideal price is $a$ above the
zero-th day upper limit,
\begin{eqnarray*}
    B_k(a) &=& \{\, (a + Z_1+ \ldots + Z_j) > j \ln \kappa_u , \mbox{ for }
    j=1,\ldots, k \,\},
\end{eqnarray*}

\begin{lemma} Let
$\lambda = (\ln \kappa_u - \mu)/\sigma$. If $\lambda>0$ then for
large enough $k$ and $a\ge 0$,
\[\Prob( B_k(a) ) < { e^{-k\lambda^2/4} \over
       \lambda\sqrt{\pi k}}.
\]
\end{lemma}
\begin{proof}
\includeproof{proof-of-mean-1.tex}
\end{proof}

\begin{lemma} If $\ln \kappa_l < \mu < \ln \kappa_u$ then $E(T_1)$ and $E(T_1^2)$ are
bound.
\end{lemma}
\begin{proof}
\includeproof{proof-of-mean-2.tex}
\end{proof}


\begin{lemma}
$E(X_{T_1}) = \mu E(T_1)$.
\end{lemma}
\begin{proof}
\includeproof{proof-of-mean-3.tex}
\end{proof}

Since $E(X_{T_1})$ and $E(T_1)$ are finite, the Strong Law of
Large Numbers gives us strong consistency for the sample
expectations of of $X_{T_1}$ and $T_1$,
\begin{eqnarray*}
   \widehat{E(X_{T_1})} = (1/n) \sum_{i=1}^n X_i \\
   \widehat{E(T_1)} = (1/n) \sum_{i=1}^n (T_i - T_{i-1}) = T_n/n
\end{eqnarray*}
and a strongly consistent approximation $\hat{\mu}$ to $\mu$,
\[
   \hat{\mu} = E(X_{T_1}) / E(T_1)
   = (\ln S_{T_n} - \ln S_{0}) / T_n
\]
where $n=|T|$ is the number of days observed at-market in our
sample.

\begin{theorem}
Under the given assumptions $\hat{\mu}$ is strongly consistent.
\end{theorem}

\subsection{Estimation of $\sigma$: the waiting time method}

To estimate the variance, we look at the waiting time for the next
at-limit observation beginning from an at-market observation.
Define $T'$ to be the first at-market day in a sequence of at
market days, and $L'$ to be the first at-limit day in a sequence
of at-limit days,
\begin{eqnarray*}
         T' &=& \{ i\in T \,|\, i>1, \, i-1 \not\in T \} \cup \{0\} \\
         L' &=& \{ i\in L \,|\, i-1\in T \}
\end{eqnarray*}
Enumerate these indices as $T'_1, T'_2, \ldots, T'_m$ and $L'_1,
L'_2, \dots, L'_m$. Pairing them up we calculate the $i$-th
waiting time
$
          W_i = L'_i - T'_i.
$ The $W_i$ are independent identically distributed random
variables with geometric distribution $\{pq^{k-1}\}$ where $q$ is
the probability that $Z_i$ falls within the collar $
\ln\kappa_l<Z_i<\ln\kappa_u$.
Hence,
\[
  E(W_1) = 1/(1-q)  < \infty.
\]
This gives us the implicit relation between $E(W_1)$ and $\sigma$,
\[
      1 - { 1 \over E(W_1) } =
             {1\over\sqrt{2\pi}} \int_{(\ln\kappa_l-\mu)/\sigma}^{(\ln\kappa_u-\mu)/\sigma}
            e^{-x^2/2}\, dx.
\]
Since $E(W_1)<\infty$, the Strong Law of Large Numbers allows that
the sample mean $\widehat{E(W_1)}$ is a strongly consistent
estimate for $E(W_1)$,
\[
    \widehat{E(W_1)} = (1/m) \sum_{i=1}^m W_i.
\]
We use numerical methods to solve for $\hat{\sigma}$ given
$\widehat{E(W_1)}$. Due to the continuity of this relation, that
$\widehat{E(W_1)}$ is a strongly consistent estimate for $E(W_1)$
implies that $\hat{\sigma}$ is a strongly consistent estimate for
$\sigma$.

\begin{theorem}
Under the given assumptions $\hat{\sigma}$ is strongly consistent.
\end{theorem}

\subsection{Estimation of $\sigma$: the transition counting method}

As referred to in the previous section, we can view the outcome of
what follows an at-market day as the flip of a biased coin. The
bias on this coin relates to the area under the gaussian
distribution which might put the next market step into the limit
region --- and this is related monotonically to the variance. By
observing the ratio of at-market days followed by at-market days
versus by at-limit days, this bias can be estimated.

Define $R$ to be the
set of indices such that two consecutive days are at market and $S$
to be the set of indices such that the first day is at market and the
next is at limit,
\begin{eqnarray*}
     T_{MM} &=& \{ i \,|\, (i \in T) \land (i+1\in T )\},\\
     T_{ML} &=& \{ i \,|\, (i \in T) \land (i+1\in L) \}.
\end{eqnarray*}
We are flipping a coin which comes up ``limit'' with probability
$|T_{ML}|/(|T_{ML}|+|T_{MM}|)$.
Therefore,
\[
     E(W) = 1 + ( |T_{MM}|/|T_{ML}|).
\]

This formula agrees with the formula for $E(W)$ of the waiting
time method if the sequence of observations ends on an at-limit
day. Else the estimate of $E(W)$ will be slightly lower, as the
sequence of ending observations will bias downward the estimation
of the probability of hitting the limit.

Because this method agrees so closely with the method of waiting
times, we will not consider it further.

\subsection{Estimation of $\sigma$: the renewal time method}

Recall that our at-market days have indices $T_0, T_1, \ldots$.
The renewal time from one at-market day to another is
$T'_i=T_{i-1}-T_i$, for $i=1, 2, \ldots$. The $T'_i$ are i.i.d.
random variables. The change in log market price during interval
$T'_i$ is,
\[
   X_{T'_i}= \sum_{j=T_{i-1}-1}^{T_i} ( \mu + \sigma Z_j)
\]
where the $Z_i$ are i.i.d. standard gaussian normal random
variables.

\begin{lemma}
$E\bigl((X_{T_1}-\mu\,T_1)^2\bigr) = \sigma^2 E(T_1)$.
\end{lemma}
\begin{proof}
\includeproofB{proof-of-renewal-1.tex}
\end{proof}

\begin{theorem}
The estimate,
\[
   \hat{\sigma}^2 = {\sum_{i=1}^n (X_{T'_i}-\hat{\mu}T'_i)^2
           \over \sum_{i=1}^n T'_i}
\]
is strongly consistent.
\end{theorem}
\begin{proof}
\includeproofB{proof-of-renewal-2.tex}
\end{proof}

\section{Results}


In graphs in figures 2 through 6 plot results of the estimation of
$\sigma$ for the waiting time method, the renewal time method, and
the direct calculation on the observed price stream, with this
data displayed as box charts, left to right as listed.

For each trial, a random walk of 250 days was generated, with
$\mu=0$ and $\sigma=1$. The upper and lower collars equal a common
$\kappa$, where these collars are applied to the log data,
\[
   \ln \kappa_l = \ln \kappa_u = \kappa .
\]
Various ratios $\kappa/\sigma$ are tried. A plot of 100 trials
with a bisected box indicating the median and one quartile above
and below the median is given.

For lower $\kappa/\sigma$, the waiting time method shows greater
accuracy (median is closer to 1) and less dispersion (quartiles
are closer to median). By $\kappa/\sigma=2$ the two methods
perform about equally, and above that the renewal time method
tends to excel. In this region, hitting a collar is rare, so we do
not get enough waiting time events to accurately measure $\sigma$.



\begin{thebibliography}{99}
\bibitem{Kast} Robert Cast, Andr\'e Lapied,
   {\em Fondements Micro\'economique de la Th\'eorie des March\'es
Financiers,} Economica, Paris, 1992
\bibitem{ChiangWei} R.~Chiang, K.~Wei,
{\em GMM and MLE approaches for estimation of
volatility and regression models when daily prices are subject to price
limits}, manuscript.
\bibitem{Feller}William Feller, {\em An Introduction to Probability
Theory and its Applications}, Volume 2, John Wiley and Sons, 1996.
\bibitem{Hall} John C. Hall,
   {\em Options, Futures, and Other Derivatives,} Prentice-Hall, 1997.
\end{thebibliography}

%\begin{thebibliography}{99}
%\end{thebibliography}

%\noindent 
%{\sc 
%Chiang, R. and K.~Wei}. 
%GMM and MLE approaches for
%estimation of volatility and regression models when daily prices
%are subject to price limits. {\em manuscript.}
%\\
%{\sc 
%Feller, William (1966)}. 
%An Introduction to Probability Theory
%and its Applications. John Wiley and Sons.
%\\
%{\sc 
%Hall, John C. (1997)}.  
%Options, Futures, and Other Derivatives. Prentice-Hall.


\newpage
\includeappendix{markets_programs.tex}


\begin{figure}[b]
\epsfig{file=compete_1_2.eps} \caption{$\kappa/\sigma=1/2$}
\end{figure}

\begin{figure}
\epsfig{file=compete_2_3.eps} \caption{$\kappa/\sigma=2/3$}
\end{figure}

\begin{figure}
\epsfig{file=compete_1.eps} \caption{$\kappa/\sigma=1$}
\end{figure}

\begin{figure}
\epsfig{file=compete_3_2.eps} \caption{$\kappa/\sigma=3/2$}
\end{figure}

\begin{figure}
\epsfig{file=compete_2.eps} \caption{$\kappa/\sigma=2$}
\end{figure}

\end{document}
