Arbitrage-Free Pricing

<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"> <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%"> <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"> <p class="header_gold">Arbitrage-Free Pricing</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="table5"> <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="#absolute pricing models">absolute pricing models</a><p class="word_list"> <a href="#arbitrage condition">arbitrage condition</a><p class="word_list"> <a href="#arbitrage-free pricing">arbitrage-free pricing</a><p class="word_list"> <a href="#equilibrium pricing models">equilibrium pricing model</a><p class="word_list"> <a href="#relative pricing models">relative pricing model</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="right" id="table6"> <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"><strong style="font-weight: 400">Most <a href="../articles/option_pricing_theory.htm">financial engineering</a> models are what are known as </strong><strong> <a name="relative pricing models">relative pricing models</a></strong><strong style="font-weight: 400">. They price instruments based on prices of other instruments quoted in the market—an instrument's price is determined <i>relative</i> to other prices quoted in the market. </strong></p> <p class="body"><strong style="font-weight: 400">Financial theorists and economists often use what are known as </strong><strong> <a name="absolute pricing models">absolute pricing models</a></strong><strong style="font-weight: 400">. These price instruments without regard for other prices quoted in the market. Instead, they base prices on economic theory, assumptions about the preferences of economic agents, supply and demand, etc.</strong></p> <p class="body"><strong style="font-weight: 400">Most absolute pricing models are what are known as </strong><strong> <a name="equilibrium pricing models">equilibrium pricing models</a></strong><strong style="font-weight: 400">. They calculate at what prices a market will reach equilibrium—where supply and demand balance and the market clears. A shortcoming of many equilibrium models is the fact that they cannot be calibrated to current market prices. They calculate hypothetical equilibrium prices that generally don't match actual prices currently observed in the market. Because this violates the <a href="../articles/law_of_one_price.htm">law of one price</a>, such models are useless in a trading context. Equilibrium models are largely theoretical tools. Sharpe's <a href="../articles/capital_asset_pricing_model.htm">capital asset pricing model</a> is an equilibrium pricing model.  </strong></p> <p class="body">Most relative pricing models employed by financial engineers are based on the theory of <b><a name="arbitrage-free pricing">arbitrage-free pricing</a></b>. Prices are determined relative to other prices quoted in the market in such a manner as to preclude any <a href="../articles/arbitrage.htm">arbitrage</a> opportunities. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table7"> <tr> <td width="336" height="10"><font size="2"> </font></td> <td width="9" height="10"></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">An <b><a name="arbitrage condition">arbitrage condition</a></b> is a relationship that must prevail between certain prices if they are to be <a href="../articles/arbitrage.htm">arbitrage-free</a>. Examples of arbitrage conditions are:</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/interest_rate_parity.htm">interest rate parity</a> for <a href="../articles/forward.htm">forward</a> exchange rates;</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/put_call_parity.htm">put-call parity</a> for European options;</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">cash-and-carry arbitrage conditions for forward commodity prices.</p> <p class="body">With arbitrage-free pricing, financial engineers apply arbitrage conditions to prices that are observable in the market in order to determine other prices that are not. Standard formulas for pricing forwards, <a href="../articles/swap.htm">swap</a>s and debt instruments are all derived using such arbitrage arguments. In complete markets, arbitrage-free pricing can be used to uniquely determine a price for any instrument. In incomplete markets, it may only place bounds on certain prices.</p> <p class="body">The groundbreaking Black-Scholes (<a href="#Black">1973</a>) approach to <a href="../articles/option_pricing_theory.htm">pricing options</a> is based on arbitrage-free pricing. Black and Scholes identified an arbitrage condition that, given certain simplifying assumptions, must hold between the price of an <a href="../articles/option.htm">option</a> and the value of a corresponding replicating portfolio. Based upon this, they were able to price options. That same approach, modified in many different ways, underlies essentially all models used today for pricing options and other <a href="../articles/derivative_instrument.htm">derivative instruments</a> in complete markets. In essence, much of financial engineering is little more than aggressive and creative use of arbitrage-free pricing. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table8"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Books</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="0" id="table9"> <tr> <td width="25%" align="center" valign="top"> <iframe src="http://rcm.amazon.com/e/cm?t=contingencyanaly&o=1&p=8&l=as1&asins=0786310251&fc1=000000&IS2=1&lt1=_blank&lc1=669933&bc1=FFFFCC&bg1=FFFFCC&f=ifr" style="width:120px;height:240px;" scrolling="no" marginwidth="0" marginheight="0" frameborder="0" name="I5"></iframe> </td> <td 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</td> </tr> <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="table10"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/arbitrage.htm">arbitrage</a> <strong style="font-weight: 400"> A transaction which generates a risk-free profit</strong>.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/arbitrage_free_model.htm">arbitrage-free model</a> A model for pricing instruments that produces arbitrage-free prices.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/EMM.htm">fundamental theorem of asset pricing</a> A theorem that relates the existence of an equivalent martingale measure to the no-arbitrage condition and completeness of markets.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/interest_rate_parity.htm">interest rate parity</a> An arbitrage condition that must hold between the spot interest rates of different currencies.</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/law_of_one_price.htm">law of one price</a> The notion that, if two assets have identical cash flows, they should have the same market value.</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/option_pricing_theory.htm">option pricing theory</a> The body of financial theory used by financial engineers to value options and other derivative instruments.</strong></p> <p class="explanatory_text" align="left"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/put_call_parity.htm" target="main"><strong style="FONT-WEIGHT: 400">put-call parity</strong></a><strong style="FONT-WEIGHT: 400"> A formula that relates the price of a put to the price of a corresponding call.</strong></p> </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 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</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">Cited 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="table11"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Black">Black</a>, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654.</p> </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>  </td> </tr> </table> </div> </body>