Stochastic Volatility Model

<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">Stochastic Volatility Model</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="2" 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="#Stochastic volatility models">stochastic volatility model</a></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table5"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300"> <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"><b><a name="Stochastic volatility models">Stochastic volatility models</a></b> are a category of <a href="../articles/time_series_stochastic_process.htm">stochastic processes</a> that have stochastic (random) second moments. Stated another way, they have random <a href="../articles/volatility.htm">volatility</a> or are <a href="../articles/heteroskedasticity.htm">conditionally heteroskedastic</a>. "Stochastic volatility model" is a technical term. While all stochastic volatility models have stochastic second moments, not all models that have stochastic second moments are called stochastic volatility models. In finance, two categories of stochastic processes are widely used to model stochastic second moments. One is stochastic volatility models. The other is <a href="../articles/ARCH_GARCH.htm">ARCH/GARCH</a> models. </p> <p class="body">Both ARCH/GARCH and stochastic volatility models derive their randomness from <a href="../articles/white_noise.htm">white noise</a> processes. The difference is that an ARCH/GARCH process depends on just one white noise <b><i>W</i></b>. That white noise directly determines innovations in the ARCH/GARCH process while also indirectly determining innovations in its second moments. Stochastic volatility models generally depend on two white noises, <i><b>V</b></i> and <i><b>W</b></i>. One directly determines innovations in the stochastic volatility process. The other directly determines innovations in its second moments.</p> <p class="body">Stochastic volatility models come in forms far more diverse than those of ARCH or GARCH models. An example of a simple univariate stochastic volatility model <i><b>X</b></i> is (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation)</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" rowspan="2"> <img border="0" src="../formulas/stochastic_volatility_model_[1]_[2].gif" width="257" height="68"></td> <td width="5%" align="right">[1]</td> </tr> <tr> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">where <i><b>V</b></i> and <i><b>W</b></i> are independent <a href="../articles/normal_distribution.htm">standard normal</a> <a href="../articles/white_noise.htm">Gaussian white noises</a>. Stochastic volatility models often employ logarithms as in [2] to ensure conditional <a href="../articles/standard_deviation.htm">variances</a> <img border="0" src="../formulas/ARCH_GARCH_stdev.gif" align="absbottom" width="37" height="20"> are nonnegative.</p> <p class="body">While our example is of a <a href="../articles/time_series_stochastic_process.htm">discrete-time</a> model, <a href="../articles/time_series_stochastic_process.htm"> continuous-time</a> stochastic volatility models are widely used in <a href="../articles/option_pricing_theory.htm">financial engineering</a>.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table7"> <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="table8"> <tr> <td width="100%"> <span style="font-family: Times New Roman; letter-spacing: -0.2pt"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/ARCH_GARCH.htm">ARCH</a> A category of conditionally heteroskedastic stochastic processes.</p> <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/ARMA.htm">autoregressive moving-average process</a> A type of stochastic process.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/autoregressive_process.htm">autoregressive process</a> <strong style="font-weight: 400">A type of stochastic process.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/heteroskedasticity.htm">heteroskedasticity</a> A condition where a stochastic process has non-constant second moments.</p> <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/moving_average_process.htm">moving-average process</a> A type of stochastic process.</strong></p> <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/random_walk.htm">random walk</a> A discrete stochastic process whose increments form a white noise.</strong></span><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/time_series_stochastic_process.htm">time series and stochastic processes</a> An introductory article.</strong></p> <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/volatility.htm">volatility</a> A metric of  variability in a stochastic process.</strong></p> <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/volatility_clustering.htm">volatility clustering</a> A property of some stochastic processes that they experience periods of high and low variance.</strong></p> </span> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/volatility_skew.htm">volatility skew</a> A condition where implied volatilities vary by strike.</p> <span style="font-family: Times New Roman; 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