Merton (1973) Option Pricing Formula

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue"> <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">Merton (1973) Option Pricing Formula</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="100%" align="left" valign="top"><p class="word_list"> <a href="#Merton (1973) option pricing formula">Merton (1973) option pricing formula</a></td> </tr> </table> </td> </tr> </table> <p class="body"><strong style="font-weight: 400">The </strong><strong> <a name="Merton (1973) option pricing formula">Merton (1973) option pricing formula</a></strong><strong style="font-weight: 400"> generalization the <a href="../articles/black_scholes_1973.htm">Black-Scholes (1973) formula</a> so it can price <a href="../articles/option.htm">European options</a> on <a href="../articles/common_stock.htm">stocks</a> or stock indices paying a known <a href="../articles/yield.htm">dividend yield</a>. The yield is expressed as an annual <a href="../articles/compounding.htm">continuously compounded</a> rate <i>q</i>. Values for a <a href="../articles/option.htm">call</a> price <i> c</i> or <a href="../articles/option.htm">put</a> price <i>p</i> <a name="[1]">are:</a></strong></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table5"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/merton1.gif" width="190" height="29"></td> <td width="5%" align="right">[1]</td> </tr> <tr> <td width="100%" align="center" height="12"></td> <td width="5%" align="right"> </td> </tr> </table> <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/merton2.gif" width="213" height="29"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body"><strong style="font-weight: 400">where:</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"> <img border="0" src="../formulas/merton3.gif" width="206" height="50"></td> <td width="5%" align="right">[3]</td> </tr> <tr> <td width="100%" align="center" height="12"></td> <td width="5%" align="right"> </td> </tr> </table> <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/merton4.gif" width="94" height="27"></td> <td width="5%" align="right">[4]</td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table9"> <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">Here, log denotes the natural logarithm, and:</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>s</i> = the price of the <a href="../articles/derivative_instrument.htm">underlying</a> stock</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>x</i> = the <a href="../articles/option.htm">strike price</a></p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>r</i> = the continuously compounded <a href="../articles_old/glossaryriskfreerate.htm">risk free interest rate</a></p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>q</i> = the continuously compounded annual dividend yield</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>t</i> = the time in years until the <a href="../articles/option.htm">expiration</a> of the option</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><font face="Times New Roman">σ = the <a href="../articles/volatility.htm">implied volatility</a> for the underlying stock</font></p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><font face="Times New Roman">Φ = the <a href="../articles/normal_distribution.htm">standard normal</a> cumulative distribution function.</font></p> <p class="body">Consider a call option on a stock index. The option is struck at <a href="../articles/currency_codes.htm">EUR</a> 8000 and expires in .18 years. The index is trading at EUR 7986 and has 24% (that is .24) implied volatility. The continuously compounded risk free interest rate is .0293. Based upon recent dividends, assume an annual dividend yield of<i> q</i> =<i> </i>.0254. Applying formula [<a href="#[1]">1</a>], the option has <a href="../articles/valuation.htm">market value</a> EUR 319. Because the option is <a href="../articles/time_value_and_intrinsic_value.htm">out-of-the-money</a>, that value is entirely <a href="../articles/time_value_and_intrinsic_value.htm">time value</a>.</p> <p class="body"><strong style="font-weight: 400">The <a href="../articles/greeks.htm">Greeks</a>—delta, gamma, vega, theta and rho—for a call are:</strong></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table10"> <tr> <td width="25%" align="center"> </td> <td width="70%" align="left"> <img border="0" src="../formulas/merton5.gif" width="118" height="29"></td> <td width="5%" align="right">[5]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/merton6.gif" width="135" height="50"></td> <td width="5%" align="right">[6]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/merton7.gif" width="139" height="29"></td> <td width="5%" align="right">[7]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"> </td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/merton8.gif" width="336" height="50"></td> <td width="5%" align="right">[8]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/black_12.gif" width="121" height="27"></td> <td width="5%" align="right">[9]</td> </tr> </table> <p class="body"><strong style="font-weight: 400">where <img border="0" src="../formulas/black_9.gif" align="top" width="14" height="21"> denotes the standard normal probability density function. For a put, the Greeks are:</strong></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table11"> <tr> <td width="25%" align="center"> </td> <td width="70%" align="left"> <img border="0" src="../formulas/merton10.gif" width="145" height="29"></td> <td width="5%" align="right">[10]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/merton6.gif" width="135" height="50"></td> <td width="5%" align="right">[11]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/merton7.gif" width="139" height="29"></td> <td width="5%" align="right">[12]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"> </td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/merton13.gif" width="375" height="50"></td> <td width="5%" align="right">[13]</td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="25%" align="center" height="12"></td> <td width="70%" align="left" height="12"> <img border="0" src="../formulas/merton14.gif" width="135" height="27"></td> <td width="5%" align="right">[14]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table12"> <tr> <td width="336" height="14"><font size="4"> </font></td> <td width="9" height="14"></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">A shortcoming of the Merton formula is its assumption that dividends are paid out continuously. For a stock index, this is an imperfect but usually reasonable approximation. For individual stocks, which typically distribute dividends in two payments each year, it is more problematic. The stock's annual yield is immaterial. The quantity <i>q</i> needs to reflect the dividends that will be earned prior to the option's expiration. If the stock has no dividend <a href="../articles/record_date.htm">record date</a> prior to the option's expiration, set <i>q</i> = 0. Otherwise, calculate the stock's dividend yield through expiration and annualize. Another problem is the fact that the model assumes that the dividend yield is a known constant. Often a dividend payment will be scheduled during the life of an option, but the amount of the payment has not yet been announced. This is an additional source of uncertainty the Merton model can not reflect.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table13"> <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="table14"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/black_scholes_1973.htm">Black-Scholes (1973) option pricing formula</a> The original option pricing formula published by Black and Scholes in their landmark (1973) paper. 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Cox and Mark Rubinstein</b></td> </tr> <tr> <td width="10%"><map name="FPMap499"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/star_classic.gif" usemap="#FPMap499" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <map name="FPMap427"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map><img border="0" src="../images/tech_2.gif" usemap="#FPMap427" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">1985</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap498"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/cox_and_rubinstein_(1985).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap498" width="146" height="20"></td> </tr> </table> </td> </tr> </table> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related 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="table22"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15">Black, 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="Merton">Merton</a>, Robert C. (1973). Theory of rational option pricing, <i>Bell Journal of Economics and Management Science</i>, 4 (1), 141-183. Available in Merton (<a target="_blank" href="http://www.riskbook.com/link/merton_r_(1992).htm">1990</a>).</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%"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </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> <p>This page uses frames, but your browser doesn't support them.</p> </body>