Random Walk

<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">Random Walk</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="#arithmetic random walk">arithmetic random walk</a><p class="word_list"> <a href="#arithmetic random walk with drift">arithmetic random walk with drift</a><p class="word_list"> <a href="#geometric random walk">geometric random walk</a><p class="word_list"> <a href="#random walk">random walk</a><p class="word_list"> <a href="#simple random walk">simple random walk</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">A <b><a name="random walk">random walk</a></b> is a simple type of <a href="../articles/time_series_stochastic_process.htm">discrete stochastic process</a> whose increments form a <a href="../articles/white_noise.htm">white noise</a>. Since a white noise has zero mean, a random walk is a <a href="../articles/martingale.htm">martingale</a>. Let's formalize this.</p> <p class="body">If you have not already done so, see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation. A discrete univariate stochastic process <i><b>R</b></i> is called a random walk if its increments </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <i><sup>t</sup>W = <sup>t</sup>R – <sup>t–</sup></i><sup>1</sup><i>R</i></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">form a white noise. Because there are different types of white noises, there are different types of random walks. A <b> <a name="simple random walk">simple random walk</a></b>—what probability theorists generally call a random walk—is one whose increments form a <a href="../articles/white_noise.htm">strong white noise</a> whose terms only take on the values 1 or –1, each with probability 0.5. A realization of a simple white noise is indicated in Exhibit 1 along with the corresponding realization of the white noise of its increments.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table7"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Simple Random Walk</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_random_walk_a.gif" width="325" height="193"><br> <img border="0" src="../images/ex1_random_walk_b.gif" width="325" height="222"></td> </tr> <tr> <td> <p class="exhibit_legend">The top graph indicates a realization of a simple white noise. The bottom graph indicates the corresponding realization of the white noise of its increments.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table8"> <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">The top graph of Exhibit 1 illustrates how a simple random walk takes random "steps" up or down, which is what motivated the name "random walk."</p> <p class="body">In finance, an <b><a name="arithmetic random walk"> arithmetic random walk</a></b> is a random walk with increments that are a <a href="../articles/white_noise.htm">Gaussian white noise</a>. This can be represented as</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <i><sup>t</sup>R – <sup>t–</sup></i><sup>1</sup><i>R = </i> <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"><i><sup> t</sup>N</i></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">where the <i><sup>t</sup>N</i> are independent and identically distributed <a href="../articles/normal_distribution.htm"> standard normal</a> random variables, and <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"> is constant. If a constant drift term <img border="0" src="../symbols/mu.gif" align="absbottom" width="8" height="12"> is added, this becomes an <a name="arithmetic random walk with drift"><b>arithmetic random walk with drift</b></a>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <i><sup>t</sup>R – <sup>t–</sup></i><sup>1</sup><i>R = </i> <img border="0" src="../symbols/mu.gif" align="absbottom" width="8" height="12"><i> + </i> <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"><i><sup> t</sup>N</i></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">For modeling the behavior of certain asset prices, such as stock prices, arithmetic random walks have a number of limitations. They can take on negative values, which is an impossibility for many asset's prices. Also, asset price fluctuations tend to be proportional to those prices. For example, a 50 dollar stock might experience daily price fluctuations on the order of one dollar while a 200 dollar stock might experience daily price fluctuations on the order of four dollars. This is not reflected by arithmetic random walks, whose standard deviations don't increase with the value of the process. For these reasons, geometric random walks often provide superior modeling of asset prices over time.</p> <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"> </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">A <b><a name="geometric random walk">geometric random walk</a></b> is technically not a random walk, at least according to the general definition given above. It is a strictly positive stochastic process whose <a href="../articles/return.htm">log returns</a> follow a Gaussian white noise. This can be expressed as</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center"> <i>log</i>(<i><sup> t</sup>R <font size="4">/</font><sup> t–</sup></i><sup>1</sup><i>R </i>)<i> = </i> <img border="0" src="../symbols/mu.gif" align="absbottom" width="8" height="12"><i> + </i> <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"><i><sup> t</sup>N</i></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">where, again, the <i><sup>t</sup>N</i> are independent and identically distributed standard normal random variables, and <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"> is constant.</p> <p class="body" style="margin-bottom: 32">All the above concepts generalize naturally to multivariate random walks.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" 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