Stable Paretian Distribution (Fat Tailed Distributions)

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Stable Paretian Distributions</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="98%" align="left" valign="top"><p class="word_list"> <a href="#reduced">reduced stable distribution</a><p class="word_list"> <a href="#stable">stable distribution</a><p class="word_list"> <a href="#stable Paretian distributions">stable Paretian distribution</a><p class="word_list"> <a href="#standardized">standardized stable distribution</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="left" id="table5"> <tr> <td width="300" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">Sums of random variables occur frequently in financial applications. Let's start with an example.</p> <p class="body">Assume that monthly <a href="../articles/return.htm">log returns</a> of some asset are independent and <a href="../articles/normal_distribution.htm">normally</a> distributed. The asset's one-year return will be the sum of 12 consecutive monthly returns. Will that one-year return also be normally distributed? The answer is "yes." This is because the 12 monthly returns are independent and normal, so they are <a href="../articles/joint_normal_distribution.htm">joint-normal</a>. A sum of joint-normal random variables must be normal.</p> <p class="body">What if the monthly returns are not normal? Assume that they are independent and identically distributed. Their common distribution is not normal, but its <a href="../articles/mean.htm">mean</a> and <a href="../articles/standard_deviation.htm">standard deviation</a> exist. Will the annual return also have that same distribution? Now the answer is "no." This follows from the <a href="../articles/central_limit_theorem.htm">central limit theorem</a>, which tell us that the distribution for the annual return will be approximately normal.</p> <p class="body">This poses a problem for finance. We don't want to have to change our financial models each time we change our unit of time. If we assume that returns have a certain distribution over a day, we would like them to have the same distribution (perhaps with a different mean and standard deviation) over a month. If we assume the returns are normally distributed, there is no problem, but what if we want to assume some other distribution?</p> <p class="body">This is the problem Benoit Mandelbrot contemplated in the early 1960s. It would lead to his groundbreaking (<a href="#Mandelbrot, Benoit B. (1963)">1963</a>) paper suggesting that asset returns be modeled with stable Paretian distributions. This article describes that work and introduces stable Paretian distributions.</p> <p class="body">Mandelbrot was trying to model cotton prices. His analysis of historical data indicated that returns had sample distributions that were highly <a href="../articles/kurtosis.htm">leptokurtic</a>. They had "fat tails" that made extreme market moves more likely than would be predicted by the normal distribution. This phenomena has been observed before, and today we know it is typical of most asset returns—<a href="../articles/common_stock.htm">stock</a>, <a href="../articles/coupon_bond.htm">bond</a>, commodity and energy returns routinely exhibit leptokurtosis. This is particularly extreme for energy returns.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="375" cellpadding="0" id="table6"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Fat-Tailed Return Distributions</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_stable_paretian_distribution.gif" width="350" height="125"></td> </tr> <tr> <td> <p class="exhibit_legend">Probability density functions (PDFs) are illustrated for a normal and a leptokurtic distribution. Sample asset return distributions tend to be leptokurtic. This means that markets tend to experience extreme moves more frequently than would be predicted based on an assumption that returns are normally distributed.</td> </tr> </table> </center> </div> <p class="body">Mandelbrot didn't want to have to assume one distribution for daily returns, another for monthly returns, and still another for annual returns. This would reduce any model to mere empirical distribution fitting. He wanted a consistent, flexible model that could be fit to different asset returns irrespective of the unit of time over which returns were calculated. This was possible with a model that assumed normally distributed returns, but normal distributions didn't fit historical data well. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table7"> <tr> <td align="right" width="9"> </td> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">The problem was the central limit theorem, which tells us that sums of random variables will converge to a normal random variable. The only way to get around the central limit theorem was to depart from one of its two main assumptions:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">that the random variables are independent, and</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">that they have finite standard deviations.</p> <p class="body">Departing from the first assumption would have meant abandoning the random walk hypothesis, which fits historical data well for many traded assets. Mandelbrot chose to depart from the second assumption and consider distributions whose standard deviations don't exist.</p> <p class="body">A probability distribution with distribution function <img border="0" src="../symbols/fi_cap.gif" align="absbottom" width="11" height="15"> is said to be <b><a name="stable">stable</a></b> if for any independent random variables <img border="0" src="../images/stable_paritian_distribution_X_X_X.gif" align="absbottom" width="94" height="15"> all having that distribution function <img border="0" src="../symbols/fi_cap.gif" align="absbottom" width="11" height="15">, there exists constants <i>a</i> and <i>b</i> such that the random variable</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/stable_paretian_distributions_[1].gif" width="193" height="15"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">also has that same distribution function <img border="0" src="../symbols/fi_cap.gif" align="absbottom" width="11" height="15">. Stated informally, such a distribution is "stable" under addition.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="right" id="table9"> <tr> <td width="11"> </td> <td width="160"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> </td> </tr> </table> <p class="body">As we have already noted, the central limit theorem suggests that the normal distribution is the only stable distribution whose standard deviation is defined. There are others whose standard deviations are not defined (or can be thought of as infinite). Paul Levy (<a href="#Levy, Paul (1925)">1925</a>, <a href="#Levy, Paul (1937)">1937</a>) identified the general class of stable distributions. Just as a normal distribution can be specified with <span style="font-size: 12.0pt; font-family: Times New Roman">a mean μ and standard deviation σ, stable distributions can be specified with four parameters:</span></p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> determines tail thickness. It satisfies 0 < <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> <img border="0" src="../symbols/less_than_or_equal.gif" align="absbottom" width="9" height="14"> 2. Generally, as <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> decreases, tail thickness increases.</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> determines asymmetry. It satisfies –1 <img border="0" src="../symbols/less_than_or_equal.gif" align="absbottom" width="9" height="14"> <img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> <img border="0" src="../symbols/less_than_or_equal.gif" align="absbottom" width="9" height="14"> 1, with <img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> = 0 corresponding to a symmetric distribution. A positive <img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> indicates that the right tail is fatter than the left tail. If <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> = 1, <img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> must equal 0.</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../symbols/gamma.gif" align="absbottom" width="7" height="12"> is a scale parameter satisfying <img border="0" src="../symbols/gamma.gif" align="absbottom" width="7" height="12"> <img border="0" src="../symbols/greater_than_or_equal.gif" align="absbottom" width="9" height="14"> 0, although the case <img border="0" src="../symbols/gamma.gif" align="absbottom" width="7" height="12"> = 0 is degenerate (similar to having a 0 standard deviation).</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><img border="0" src="../symbols/delta_lower_case.gif" align="absbottom" width="8" height="15"> is a location parameter and can take on any <a href="../articles/set.htm">real</a> value. If <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> > 1, <img border="0" src="../symbols/delta_lower_case.gif" align="absbottom" width="8" height="15"> equals the mean of the distribution.</p> <p class="body">This is one standard parameterization that is commonly used. Other very similar parameterizations are also used, so be careful to check any author's definitions carefully.</p> <p class="body">The distribution's mean exists so long as <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> > 1. Its standard deviation exists only in the case <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> = 2. Just as we define a <a href="../articles/normal_distribution.htm">standard normal</a> distribution as a normal distribution whose mean and standard deviation are 0 and 1, we say a stable distribution is <b><a name="standardized"> standardized</a></b> or <b><a name="reduced">reduced</a> </b>if <img border="0" src="../symbols/gamma.gif" align="absbottom" width="7" height="12"> = 1 and <img border="0" src="../symbols/delta_lower_case.gif" align="absbottom" width="8" height="15"> = 0.</p> <p class="body"><font face="Times New Roman">There are only three cases in which a <a href="../articles/closed_form_solution.htm">closed form expression</a> is known for a stable distribution's probability density function. These are the</font></p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">normal distribution: <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> = 2 (the value of <img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> becomes irrelevant in this case),</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/cauchy_distribution.htm">Cauchy distribution</a>: <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> = 1, <img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> = 0,</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">Levy distribution: <img border="0" src="../symbols/alpha.gif" align="absbottom" width="10" height="13"> = 0.5, <img border="0" src="../symbols/beta.gif" align="absbottom" width="10" height="15"> = 1.</p> <p class="body"><font face="Times New Roman">For this reason, theoretical work with stable distributions tends to be presented in terms of characteristic functions instead of probability density functions or distribution functions. However, density functions or distribution functions can always be valued using <a href="../articles/closed_form_solution.htm">numerical techniques</a>. The general formula for the characteristic function of a stable distribution is </font></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <img border="0" src="../symbols/stable_paretian_distributions_[2].gif" width="392" height="125"></td> <td width="5%" align="right">[2]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table11"> <tr> <td align="right" width="9" height="10"></td> <td width="336" height="10"> <font size="2"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="336">  <p style="margin-top: 0; margin-bottom: 2" class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I1"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82402"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82402;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> </tr> </table> <p class="body"><font face="Times New Roman">where <i>log</i> denotes a natural logarithm and <i>x</i>/|<i>x</i>| is understood to equal 0 when <i> x</i> = 0. </font></p> <p class="body"><font face="Times New Roman">Non-normal stable distributions have "fat tails" that generally satisfy a convergence property defined by Vilfredo Pareto. For this reason, non-normal stable distributions are often called <b><a name="stable Paretian distributions"> stable Paretian distributions</a></b>. </font></p> <p class="body"><font face="Times New Roman">Despite their merits for modeling asset returns, stable Paretian distributions have remained a fringe topic in finance, occasionally mentioned by researchers, but rarely implemented by practitioners. This may be because the mathematics of these distributions is more technical than that of more familiar distributions. Perhaps distributions with undefined standard deviations are simply too counter-intuitive for most people. Modest research is, however, ongoing.</font></p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table12"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Internal Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table13"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/cauchy_distribution.htm">Cauchy distribution</a> A bell-shaped distribution that is more peaked and has fatter tails than the normal distribution.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/central_limit_theorem.htm">central limit theorem</a> A theorem that explains why the normal distribution plays such an important role in probability theory.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/joint_normal_distribution.htm">joint normal distribution</a> A multivariate distribution with normal marginal distributions.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/kurtosis.htm">kurtosis</a> A parameter describing the peakedness and tails of a distribution.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/linear_polynomial_of_a_random_vecrtor.htm">linear polynomial of a random vector</a> A random variable or random vector that is defined as a linear polynomial of a random vector.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/normal_distribution.htm"><strong style="font-weight: 400">normal distribution</strong></a><strong style="font-weight: 400"> Perhaps the most important probability distribution for probability and statistics.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/standard_deviation.htm">standard deviation</a> A parameter describing the dispersion of a distribution.</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Books</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table14"> <tr> <td width="100%"> <p class="explanatory_text">Rachev et al (<a target="_blank" href="http://www.riskbook.com/link/rachev_menn_fabozzi_(2005).htm">2005</a>) Is an accessible introduction to stable Paretian distributions in finance.</p> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table15"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap422"> <area target="_blank" href="http://www.riskbook.com/link/rachev_menn_fabozzi_(2005).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_rachev_menn_and_fabozzi_(2005)_125.gif" usemap="#FPMap422" width="83" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/rachev_menn_fabozzi_(2005).htm"> Fat-Tailed<br> <font size="2">and Skewed Asset Return Distributions</font></a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>S. Rachev, C. Menn and F. Fabozzi</b></td> </tr> <tr> <td width="10%"> <map name="FPMap420"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="1, 2, 89, 15"> </map> <img border="0" src="../images/star_4.gif" usemap="#FPMap420" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <map name="FPMap421"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_3.gif" usemap="#FPMap421" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2005</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap366"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/rachev_menn_fabozzi_(2005).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap366" width="146" height="20"></td> </tr> </table> </td> </tr> </table> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_3.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%">  <p class="ad_legend" style="margin-left: 0; margin-bottom:2"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I2"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82436"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82436;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Cited Papers</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table17"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Levy, Paul (1925)">Levy, Paul (1925)</a> <i>Calcul des Probabilities</i>, Paris: Gauthier-Villars.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Levy, Paul (1937)">Levy, Paul (1937)</a> <i>Theorie de L'addition des Variables Aleatoires</i>, Paris: Gauthier-Villars.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Mandelbrot, Benoit B. (1963)">Mandelbrot, Benoit B. (1963)</a>. The variation of certain speculative prices, <i> Journal of Business</i>, 36, 394-419. Also appearing in Cootner, Paul H. (1964). <i>The Random Character of Stock Market Prices</i>, Cambridge: MIT Press.</td> </tr> </table> </td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table18"> <tr> <td width="100%" bgcolor="#FFFFFF"> <p align="center" style="margin-top: 18"><map name="FPMap365"> <area target="_top" href="http://www.riskglossary.com" shape="rect" coords="0, 0, 164, 102"> </map><img border="0" src="../images/logo.jpg" usemap="#FPMap365" width="165" height="103"></p> <p align="center"><span class="verdana10"> <a href="http://www.riskglossary.com/main/disclaimer.htm" target="_self"> Disclaimer</a></span></p> <p align="center">website: <span class="verdana10"> <a target="_top" href="http://www.contingencyanalysis.com"> http://www.contingencyanalysis.com</a></span><br> glossary direct link: <span class="verdana10"> <a target="_top" href="http://www.riskglossary.com"> http://www.riskglossary.com</a></span><br> copyright © Contingency Analysis, 2005<p align="center"><map name="FPMap1"> <area target="_top" href="http://www.contingencyanalysis.com" shape="rect" coords="0, 0, 240, 88"> </map><img border="0" src="../images/logo_glossary_white.gif" usemap="#FPMap1" width="241" height="89"></td> </tr> </table> </td> </tr> </table> </td> <td>  </td> </tr> </table> </div> </body>