Put-Call Parity

<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">Put-Call Parity</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="#Put-call parity">put-call parity</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="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" style="margin-top: 26"><strong><a name="Put-call parity">Put-call parity</a></strong><strong style="font-weight: 400"> is a relationship, first identified by Stoll (<a href="#Stoll">1969</a>), that must exist between the prices of European put and call <a href="../articles/option.htm">options</a> that both have the same <a href="../articles/derivative_instrument.htm">underlier</a>, <a href="../articles/option.htm">strike price</a> and <a href="../articles/option.htm">expiration date</a>. The relationship is derived using <a href="../articles/arbitrage_free_pricing.htm">arbitrage arguments</a>. Consider two portfolios consisting of: </strong></p> <p class="body"><img border="0" src="../images/bullet.gif" width="20" height="10"><strong style="font-weight: 400">The call option and an amount of cash equal to the present value of the strike price. </strong></p> <p class="body"><img border="0" src="../images/bullet.gif" width="20" height="10"><strong style="font-weight: 400">The put option and the underlier.</strong></p> <p class="body" style="margin-bottom: 30"><strong style="font-weight: 400">Exhibit 1 compares the expiration value for these two portfolios, with <i>x</i> representing the common strike price:</strong> </p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="450" cellpadding="0" id="table6"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Put-Call Parity:<br> Two Portfolios Have Identical Expiration Values</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_put_call_parity.gif" width="450" height="346"></td> </tr> <tr> <td> <p class="exhibit_legend">A portfolio comprising a call option and an amount <i>x</i> of cash equal to the present value of the option's strike price has the same expiration value as a portfolio comprising the corresponding put option and the underlier. For European options, early exercise is not possible. If the expiration values of the two portfolios are the same, then their present values must also be the same. This equivalence is put-call parity.</td> </tr> </table> </center> </div> <p class="body"><strong style="font-weight: 400">What is significant about Exhibit 1 is the fact that the two portfolios (call + cash and put + underlier) have identical expiration values. Irrespective of the value of the underlier at expiration, each portfolio will have the same value as the other.</strong></p> <p class="body"><strong style="font-weight: 400">If the two portfolios are going to have the same value at expiration, then they must have the same value today. Otherwise, an investor could make an arbitrage profit by purchasing the less expensive portfolio, selling the more expensive one and holding the long-short position to expiration. Accordingly, we have the price <a name="[1]">equality:</a></strong></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <strong style="font-weight: 400"><i>c</i> + <i>PV</i>(<i>x</i>) = <i>p</i> + <i>s </i></strong></td> <td width="5%" align="right">[1]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table8"> <tr> <td align="right" width="9" height="14"></td> <td width="336" height="14"> <font size="4"> </font></td> </tr> <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"><strong style="font-weight: 400">where: </strong></p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><strong style="font-weight: 400"><i>c</i> = the current <a href="../articles/valuation.htm">market value</a> of the call;</strong></p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><strong style="font-weight: 400"><i>PV</i>(<i>x</i>) = the present value of the strike price <i>x</i>, discounted from the expiration date at a suitable risk free rate;</strong></p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><strong style="font-weight: 400"><i>p</i> = the current market value of the put;</strong></p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10"><strong style="font-weight: 400"><i>s</i> = the current market value of the underlier.</strong></p> <p class="body"><strong style="font-weight: 400">Equation [<a href="#[1]">1</a>] is the put-call parity. Note that it is not based on any <a href="../articles/option_pricing_theory.htm">option pricing model</a>. It was derived purely using arbitrage arguments. It applies only to European options, since a possibility of early exercise could cause a divergence in the present values of the two portfolios.  </strong></p> <p class="body"><strong style="font-weight: 400">Put-call parity offers a simple test of option pricing models. Any option pricing model that produces put and call prices that do not satisfy put-call parity must be rejected as unsound. 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The relationship between put and call option prices, <i>Journal of Finance</i>, 23, 801-824. </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>