Arbitrage-Free 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="table2"> <tr> <td width="740" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table3"> <tr> <td height="35" width="100%" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table4"> <tr> <td width="100%" align="left" valign="top" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <p class="header_gold">Arbitrage-Free Model</td> <td width="45" align="right" valign="bottom" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <map name="FPMap7"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap7" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table5"> <tr> <td width="0%" align="left" valign="top" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <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" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"><p class="word_list"> <a href="#arbitrage-free model">arbitrage-free model</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"><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="table6"> <tr> <td width="300" height="14" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"></td> </tr> <tr> <td width="300" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> </td> </tr> </table> <p class="body">A<strong style="font-weight: 400">n </strong><strong> <a name="arbitrage-free model">arbitrage-free model</a></strong><strong style="font-weight: 400"> is a <a href="../articles/option_pricing_theory.htm">financial engineering</a> model that assigns prices to <a href="../articles/derivative_instrument.htm">derivatives</a> or other instruments in such a way that it is impossible to construct <a href="../articles/arbitrage.htm">arbitrages</a> between two or more of those prices. </strong>For example, if an option pricing formula assigned prices to <a href="../articles/option.htm">put</a> and <a href="../articles/option.htm">call</a> <a href="../articles/option.htm">options</a> that violated <a href="../articles/put_call_parity.htm">put-call parity</a>, that model would not be arbitrage-free.<strong style="font-weight: 400"> While technically not required by the definition, arbitrage-free models used for trading are generally calibrated to one or more market prices to preclude arbitrages between prices assigned by the model and those quoted prices.</strong></p> <p class="body">The <a href="../articles/black_scholes_1973.htm">Black-Scholes</a> (1973) option pricing formula is, of course, arbitrage free. Problems arose more with <a href="../articles/term_structure.htm">term structure</a> models developed for pricing fixed income derivatives during the 1980s. Early term structure models—including Vasicek (<a href="#Vasicek">1977</a>), Rendleman and Bartter (<a href="#Rendleman">1980</a>), and Cox, Ingersoll and Ross (<a href="#Cox">1985</a>)—were <a href="../articles/arbitrage_free_pricing.htm">equilibrium models</a>. They had two shortcomings:</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" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> </td> <td width="336" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <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"><img border="0" src="../images/bullet.gif" width="20" height="10">They constructed a current equilibrium term structure that was generally different from the actual term structure observed in the market. For this reason, they ascribed prices to <a href="../articles/treasury_securities.htm">Treasury securities</a> different from those quoted in the market.</p> <p class="body"><img border="0" src="../images/bullet.gif" width="20" height="10">They were not arbitrage-free.</p> <p class="body">The Ho and Lee (<a href="#Ho">1986</a>) model was the first term structure model to solve these problems. It could be calibrated to the current term structure, so it ascribed prices to Treasury securities that were the same as those observed in the market. Also, it was arbitrage-free. The model is one example of a larger class of arbitrage-free models specified by Heath, Jarrow and Morton (<a href="#Heath">1992</a>).</p> <p class="body">Today, the theory of arbitrage-free term structure modeling is well developed. 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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="I7"> <![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" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <p class="header2_black">Cited Papers</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table13"> <tr> <td width="100%" style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Black (1973)">Black</a>, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities, <i>Journal of Political Economy</i>, 81, 637-654.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Cox">Cox</a>, John, Jonathan Ingersoll, and Stephen Ross (1985), A theory of the term structure of interest rates, <i>Econometrica</i>, 53 (2), 385–407.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Heath">Heath</a>, David, Robert Jarrow, and Andrew Morton (1992), Bond pricing and the term structure of interest rates, <i>Econometrica</i>, 60 (1), 77–105.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Ho">Ho</a>, Thomas and Sang-Bin Lee (1986), Term structure movements and pricing interest rate contingent claims, <i>Journal of Finance</i>, 41 (5), 1011–1029.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Rendleman">Rendleman</a>, Richard and Brit J Bartter (1980). The pricing of options on debt securities, <i>Journal of Financial and Quantitative Analysis</i>, 15 (1), 11-24</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Vasicek">Vasicek</a>, Oldrich (1977), An equilibrium characterization of the term structure, <i>Journal of Financial Economics</i>, 5 (8), 177–188.</td> </tr> </table> </td> </tr> </table>  <p align="center"><map name="FPMap0_I1"> <area target="_top" href="http://www.riskglossary.com" shape="rect" coords="0, 0, 164, 102"> </map><img border="0" src="../images/logo.jpg" usemap="#FPMap0_I1" width="165" height="103"></p> <p align="center"><span class="verdana10"> <a href="http://www.riskglossary.com/main/disclaimer.htm">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, 1996 - current</p> <p align="center"><map name="FPMap1_I1"> <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_I1" width="241" height="89"></p> </td> <td style="font-family: verdana,arial,helvetica,sans-serif; font-size: x-small">  </td> </tr> </table> </div> </body>