Lognormal Distribution

<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"> <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">Lognormal Distribution</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="0%" align="left" valign="top"><p class="word_list"> <a href="#lognormal distribution">lognormal distribution</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" 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">In probability theory, the lognormal distribution is defined with reference to the <a href="../articles/normal_distribution.htm">normal distribution</a>. A random variable <i>X</i> is <b> <a name="lognormal distribution">lognormal</a></b> if its natural logarithm, <img border="0" src="../formulas/lognormal_y=log(x).gif" align="absbottom" width="71" height="15">, is normal.</p> <p class="body">Consider a <a href="../articles/time_series_stochastic_process.htm">stochastic process</a>  <img border="0" src="../formulas/lognormal_stochastic_process.gif" align="absbottom" width="135" height="20">  representing accumulated values over time for some asset. It might represent daily accumulated values of a <a href="../articles/common_stock.htm">common stock</a> or a physical commodity position. At some time <i>t</i>, the realization <img border="0" src="../formulas/lognormal_tq.gif" align="absbottom" width="12" height="19"> of <img border="0" src="../formulas/lognormal__tQ.gif" align="absbottom" width="16" height="19"> is known, but the realization of the subsequent value <img border="0" src="../formulas/lognormal__t+1Q.gif" align="absbottom" width="30" height="20"> is unknown. Accordingly, the single-period <a href="../articles/return.htm">log return</a> <img border="0" src="../formulas/lognormal_log_return.gif" align="absbottom" width="88" height="21"> is random. Assume that its conditional <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15"> as of time <i>t</i> is normal. Then, by the above definition, <img border="0" src="../formulas/lognormal_total_return.gif" align="absbottom" width="54" height="21"> must be lognormal, and it is easy to demonstrate (try it!) that <img border="0" src="../formulas/lognormal__t+1Q.gif" align="absbottom" width="30" height="20"> is also lognormal. This model—of log returns being normal and corresponding prices being lognormal—is one of the most ubiquitous models in finance. It is the reason that the lognormal <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15"> plays such an important role in finance.</p> <p class="body">The lognormal <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15"> for a random variable <i>X</i> may be specified with its <a href="../articles/mean.htm">mean</a> <font face="Times New Roman">μ</font> and <a href="../articles/standard_deviation.htm">variance</a> <img border="0" src="../formulas/lognormal_sigma_squared.gif" align="absbottom" width="14" height="20">. Alternatively, it may be specified with the mean <i>m</i> and variance <i> s</i><sup>2</sup> of the normally distributed <i>log</i>(<i>X</i>). We denote a lognormal <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15"> as <img border="0" src="../formulas/lognormal_lambda_notation.gif" align="absbottom" width="52" height="20">, but its probability density function (PDF) is most easily expressed in terms of <i>m</i> and <i>s</i>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/lognormal_pdf.gif" width="286" height="120"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">This is graphed in Exhibit 1:</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"> <font face="Verdana">λ(μ,σ<sup>2</sup>) Probability Density Function</font></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_lognormal_distribution.gif" width="300" height="149"></td> </tr> <tr> <td> <p class="exhibit_legend">The λ(μ,σ<sup>2</sup>) distribution is positively skewed.</td> </tr> </table> </center> </div> <p class="body"> <span style="font-size: 12.0pt; font-family: Times New Roman">The expectation, standard deviation, <a href="../articles/skewness.htm">skewness</a> and <a href="../articles/kurtosis.htm">kurtosis</a> of</span> a lognormal <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15"> are, in terms of <i>m</i> and <i>s</i>:</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/lognormal_mean.gif" width="141" height="27"></td> <td width="5%" align="right"><a name="[2]">[2]</a></td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/lognormal_standard_deviation.gif" width="229" height="29"></td> <td width="5%" align="right"><a name="[3]">[3]</a></td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/lognornormal_skewness.gif" width="197" height="31"></td> <td width="5%" align="right">[4]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/lognormal_kurtosis.gif" width="275" height="28"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">If we know <font face="Times New Roman">μ</font> and <font face="Times New Roman">σ</font> instead of <i>m</i> and <i>s</i>, we can convert between these with:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/lognormal_convert_to_m.gif" width="137" height="59"></td> <td width="5%" align="right">[6]</td> </tr> <tr> <td width="100%" align="center">  </td> <td width="5%" align="right"> </td> </tr> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/lognormal_convert_to_s.gif" width="139" height="31"></td> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">The reverse conversion is provided by [<a href="#[2]">2</a>] and [<a href="#[3]">3</a>].</p> <p class="body">As with the normal <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15">, the cumulative <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15"> function (CDF) of a lognormal <img border="0" src="../formulas/lognormal_distribution_word_distr.gif" align="absbottom" width="65" height="15"> exists but cannot be expressed in terms of standard functions. 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