Random Walk Theory

<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="table8"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table9"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table10"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Random Walk Hypothesis</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="table11"> <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="0%" align="left" valign="top"><p class="word_list"> <a href="#Louis_Bachelier">Bachelier, Louis</a><p class="word_list"> <a href="#Alfred_Cowles_3rd">Cowles, Alfred</a><p class="word_list"> <a href="#Maurice_Kendall">Kendall, Maurice</a><p class="word_list"> <a href="#random_walk_hypothesis">random walk hypothesis</a><p class="word_list"> <a href="#Holbrook_Working">Working, Holbrook</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table12"> <tr> <td width="160">  </td> <td width="11"> </td> </tr> <tr> <td width="160"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> </td> <td width="11"> </td> </tr> </table> <p class="body">The <b><a name="random_walk_hypothesis">random walk hypothesis</a></b> is not so much a hypothesis as it is a model that has been found to be surprisingly useful for describing the behavior of prices in various markets, including major <a href="../articles/common_stock.htm">equity</a>, fixed income and commodities markets. It states that price series do not exhibit predictive patterns over time but can best be described with a <a href="../articles/random_walk.htm">random walk</a>. Accordingly, the random walk hypothesis is a rejection of technical analysis.</p> <p class="body"> <img border="0" src="../images/portrait_louis_b.gif" align="right" width="143" height="220">An early version of the random walk hypothesis was proposed by <b> <a name="Louis_Bachelier">Louis Bachelier</a></b> (<a href="#Bachelier">1900</a>) in his famous doctoral thesis <i>Th<font face="Times New Roman">é</font>orie de la Sp<font face="Times New Roman">é</font>culation</i>. Bachelier studied the market for forwards and options on French government <a href="../articles/coupon_bond.htm">bonds</a>, discovering a number of important results. As part of that work, he discovered the mathematics of <a href="../articles/brownian_motion.htm">Brownian motion</a>—five years before Albert Einstein independently did so. With regard to the markets, Bachelier <a name="[1]">noted</a><sup><font size="2"> <a href="#foot_[1]">[1]</a></font></sup></p> <p class="quote"> The influences that determine the movements of the exchange are innumerable; past, current and even anticipated events that often have no obvious connection with its changes ... it is thus impossible to hope for mathematical predictability. </p> <p class="body"> He went on to conclude that, if the market's movements cannot be predicted,</p> <p class="quote"> The mathematical expectation of the speculator is zero.</p> <p class="body"> Bachelier's thesis was decades ahead of its time. It was ignored for over a half century, but other researchers, acting independently, started to draw similar conclusions.</p> <p class="body"><b><a name="Holbrook_Working">Holbrook Working</a></b> was an agricultural economist with Stanford University’s Food Research Institute. In a <a href="#Working">1934</a> study, he compared <a href="../articles/time_series_stochastic_process.htm">time series</a> of historical commodity price changes to time series of random numbers. He wanted to determine what non-random price patterns might be exploited by traders to realize speculative profits. Using statistical techniques, he was unable to distinguish the series of price changes from the series of random numbers. He concluded that there were no predictive patterns in the price changes—that the prices were entirely random. He showed graphs of the time series to professional commodity traders. They too were unable to distinguish the series of price changes from the series of random numbers.</p> <p class="body"> <img border="0" src="../images/portrait_maurice_K.gif" align="right" width="165" height="220"><b><a name="Maurice_Kendall">Maurice Kendall</a></b> was one of the great statisticians of the 20th century. In <a href="#Kendall">1953</a>, he published a ground-breaking empirical study of weekly changes in nineteen indices of British industrial <a href="../articles/common_stock.htm">share</a> prices and in <a href="../articles/settlement.htm">spot</a> prices for New York cotton and Chicago wheat. His goal was to advance the field of technical analysis by introducing statistical rigor. He was startled to find that the random component of prices swamped any <a href="../articles/time_series_stochastic_process.htm">autocorrelations</a>. Frustrated, he <a name="[2]">concluded</a> [2]</p> <p class="quote"> The series looks like a wandering one, almost as if once a week the Demon of Chance drew a random number from a symmetrical population of fixed dispersion and added it to the current price to determine the next week's price.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table13"> <tr> <td width="336" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="336"> <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">Working's and Kendall's conclusion that past price data could not be used to predict future prices effectively rejecting the practice of technical analysis. But traders can base decisions on information other than past price data. An <a href="../articles/common_stock.htm">equities</a> trader might, for example, base trades on a firm's past earnings growth, its price/earnings ratio, how experienced its management team is, prospects for the firm's industry, and more. This is the realm of fundamental analysis. The studies by Working and Kendall did not address fundamental analysis, but earlier work by Alfred Cowles had.</p> <p class="body"><b><a name="Alfred_Cowles_3rd">Alfred Cowles 3<sup>rd</sup></a></b> was the son of a wealthy businessman. He subscribed to a number of investment newsletters during the 1920’s and 1930’s to help manage his family's fortune. Following the <a href="../articles/united_states_financial_regulation.htm">crash of 1929</a>, he looked back and realized that none of the newsletters had reasonably predicted the crash or subsequent events. Wondering if the poor performance was coincidence or typical, Cowles enlisted the aid of economists to conduct a massive study. Using an IBM punch-card machine—a precursor of the computer—they analyzed the historical investment decisions of 25 investment newsletters, 16 stock tip services and 12 fire insurance companies. </p> <p class="body"> <img border="0" src="../images/portrait_alfred_c.gif" align="left" width="147" height="220">Cowles published the results in a <a href="#Cowles">1933</a> paper entitled "Can Stock Market Forecasters Forecast?" He summarized his conclusions with a 3-word abstract: "It is doubtful." Most of the practitioners had underperformed the broad market. Their combined results had underperformed as well. The few practitioners who had done well did so modestly, so there was no indication that their performance was due to skill instead of luck. Cowles also experimented with series of forecasts constructed by randomly drawing cards from a deck. He found that the best random series of forecasts outperformed the best practitioners—and the worst series of random forecasts outperformed the worst practitioners.</p> <p class="body">Cowles did much to further research into the performance of stock pickers. In 1932, he launched the Cowles Commission, which has sponsored considerable research. In 1933, he helped found the prestigious journal <i>Econometrica</i>. He launched an equity index that became today's S&P 500. In <a href="#Cowles0">1944</a>, he published a large follow-up study to his 1933 study of stock pickers' performance. It drew a similarly dreary conclusion. </p> <p class="body">Cowles' efforts went unnoticed by brokers and investment managers, who made their living off investors believing it was possible to outperform the market. But academics took note. By the 1960s, there was an active literature emerging on what was then called the random walk hypothesis. In <a href="#Cootner">1965</a>, Paul Cootner published an influential book called <i> <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">The Random Character of Stock Market Prices</a></i>. This pulled together reprints of important articles up to 1963, including Cootner's own work and the early works of Bachelier and Kendall.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table14"> <tr> <td align="right" width="9"> <p style="margin-top: 0; margin-bottom: 0"> </td> <td width="336">  <p style="margin-top: 8; 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="I4"> <![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">Bachelier's original version of the random walk hypothesis was crude by today's standards. It had prices follow an <a href="../articles/random_walk.htm">arithmetic random walk</a> with zero drift. The modern version developed out of the work of multiple researchers, including those already mentioned, as well as Osborne (<a href="#Osborne">1959</a>), Moore (<a href="#Moore">1960</a>), Alexander (<a href="#Alexander">1961</a>) and Granger and Morgenstern (<a href="#Granger">1963</a>). It states that the log returns follow an arithmetic random walk with a drift reflecting the long-term return from equity investment. Stated another way, prices follow a <a href="../articles/random_walk.htm">geometric random walk</a> with drift.</p> <p class="body">The random walk hypothesis is more an empirical observation than a theoretical result. Fundamentally, it is an empirical observation that price series are well modeled with a random walk. However, researchers did offer a theoretical explanation for why prices should follow a random walk. They noted that prices change in response to news items—earnings reports, releases of economic indicators, merger announcements, etc. If news items are assumed to arise independently (the relative probabilities of upcoming news being good or bad is unaffected by whether recent news has been good or bad) then price changes should be independent. Also, the volume of news items affecting a price is sufficiently large that the <a href="../articles/central_limit_theorem.htm">central limit theorem</a> applies, and price changes over any discernible period should be approximately <a href="../articles/normal_distribution.htm">normal</a>. Somewhat after the fact, Samuelson (<a href="#Samuelson">1965</a>) and Mandelbrot (<a href="#Mandelbrot">1966</a>) rigorously formalized this theoretical justification of the random walk hypothesis.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table15"> <tr> <td width="336"> <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">In the early 1960s, the literature of the random walk hypothesis took two new directions. The first would extend results as an efficient market hypothesis. The other looked for flaws in the random walk model. Two obvious flaws, which were noted early on, were the fact that log price changes were</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/kurtosis.htm">leptokurtic</a>, rendering the normal distribution an imperfect representation, and</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/heteroskedasticity.htm">heteroskedastic</a>.</p> <p class="body">Neither flaw affords technicians trading opportunities, but researchers sifted through massive volumes of historical price data, looking for flaws that might. Their published results formed the literature on market anomalies which ultimately contributed to the dubious field of behavioral finance.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <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="table19"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><span class="verdana10"><strong style="font-weight: 400"><a href="../articles/capital_asset_pricing_model.htm">c</a></strong><a href="../articles/capital_asset_pricing_model.htm"><strong style="font-weight: 400">apital asset pricing model</strong></a></span><strong style="font-weight: 400"> A model for valuing financial assets based upon their systematic risk.</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/portfolio_theory.htm">portfolio theory</a> A body of theory relating to how investors optimize portfolio selections.</strong></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_1.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related 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="table20"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Alexander">Alexander</a>, Sidney (1961). Price movements in speculative markets: trends or random walks, <i>Industrial Management Review</i>, 2 (2), 7-26. [Reprinted in <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Cootner</a>.]<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Bachelier">Bachelier</a>, Louis (1900). <i>Théorie de la Spéculation</i>, Annales Scientifique de l'<font face="Arial">É</font>cole Normale Sup<font face="Arial">é</font>rieure, 3<sup>e</sup> s<font face="Arial">é</font>rie, tome 17, 21-86. [English translation in <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Cootner</a>; original French with a more recent English translation in <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http://www.amazon.com/History-Theory-Investments-Annotated-Bibliography/dp/0471770566?ie=UTF8&qid=1186264292&sr=1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325"> Davis and Etheridge</a>.]<p class="explanatory_text"> <img border="0" src="../images/bullet_book.gif" width="21" height="15"><a name="Cootner">Cootner</a>, Paul (1965). <i> <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">The Random Character of Stock Market Prices</a><img src="http://www.assoc-amazon.com/e/ir?t=contingencyanaly&l=ur2&o=1" width="1" height="1" border="0" alt="" style="border:none !important; margin:0px !important;" /></i>, Cambridge: MIT Press.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Cowles">Cowles</a>, Alfred 3rd (1933). Can stock market forecasters forecast?, <i> Econometrica</i>, 1 (3), 309-324.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Cowles0">Cowles</a>, Alfred 3rd (1944). Stock Market Forecasting, <i> Econometrica</i>, 12 (3 & 4), 206-214.<p class="explanatory_text"> <img border="0" src="../images/bullet_book.gif" width="21" height="15"><a name="Davis">Davis</a>, Mark and Alison Etheridge (2006). <i> <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FHistory-Theory-Investments-Annotated-Bibliography%2Fdp%2F0471770566%3Fie%3DUTF8%26qid%3D1186264292%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Louis Bachelier's Theory of Speculation</a></i><img src="http://www.assoc-amazon.com/e/ir?t=contingencyanaly&l=ur2&o=1" width="1" height="1" border="0" alt="" style="border:none !important; margin:0px !important;" />, Princeton: Princeton University Press.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Granger">Granger</a>, Clive and Oscar Morgenstern (1963) Spectral analysis of New York stock market prices, <i>Kyklos</i>, 16 (1), 1-27. [Reprinted in <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Cootner</a>.]<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Kendall">Kendall</a>, Maurice (1953). The analysis of time series, Part 1: Prices, <i> Journal of the Royal Statistical Society</i>, Series A (General), 116 (1), 11-34. [Reprinted in <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Cootner</a>.]<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Mandelbrot">Mandelbrot</a>, Benoit  (1966). Forecasts of future prices, unbiased markets, and martingale models, <i>Journal of Business</i>, 39 (Special Supplement, January), 242-255.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Moore">Moore</a>, Arnold (1960). Some characteristics of changes in common stock prices, doctoral thesis, University of Chicago. [Reprinted in <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Cootner</a>.]<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Osborne">Osborne</a>, M. F. M. (1959). Brownian motion in the stock market, <i> Operations Research</i>, 7, 145-173. [Reprinted in <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FRandom-Character-Stock-Market-Prices%2Fdp%2F1899332847%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1186256147%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Cootner</a>.]<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Samuelson">Samuelson</a>, Paul A. (1965). Proof that properly anticipated prices fluctuate randomly. <i>Industrial Management Review</i>, 6 (2), 41-49.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Working">Working</a>, Holbrook (1934). A random-difference series for use in the analysis of time series, <i>Journal of the American Statistical Association</i>, 29 (185), 11-24.</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_2.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="I11"> <![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">Footnotes</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="table21"> <tr> <td width="100%"> <p class="explanatory_text"> <a name="foot_[1]">[1]</a> Translations of Bachelier's thesis are from Mark Davis and Alison Etheridge (2006). <i> <a href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FLouis-Bacheliers-Theory-Speculation-Origins%2Fdp%2F0691117527%3Fie%3DUTF8%26s%3Dbooks%26qid%3D1185570380%26sr%3D1-1&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Louis Bachelier's Theory of Speculation</a><img src="http://www.assoc-amazon.com/e/ir?t=contingencyanaly&l=ur2&o=1" width="1" height="1" border="0" alt="" style="border:none !important; margin:0px !important;" /></i>, Princeton: Princeton University Press. [<a href="#[1]">return</a>]<p class="explanatory_text"> <a name="foot_[2]">[2]</a> Kendall (<a href="#Kendall">1953</a>), p. 13. [<a href="#[2]">return</a>]</td> </tr> <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, 2007<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>