Black's Model: Black '76

<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" align="left" valign="top"> <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">Black (1976) 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="#Blacks (1976) option pricing formula">Black (1976) option pricing formula</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"><strong style="font-weight: 400">A challenge in pricing <a href="../articles/option.htm">options</a> on commodities is non-randomness in the evolution of many commodity prices. For example, the <a href="../articles/settlement.htm">spot price</a> of an agricultural product will generally rise prior to a harvest and fall following the harvest. Natural gas tends to be more expensive during Winter months than Summer months. Because of such non-randomness, many spot commodity prices cannot be modeled with a geometric <a href="../articles/brownian_motion.htm">Brownian motion</a>, and the Black-Scholes (<a href="#Black (1973)">1973</a>) or Merton (<a href="#Merton">1973</a>) models for options on <a href="../articles/common_stock.htm">stocks</a> do not apply. In 1976, Fischer Black published a paper addressing this problem. His solution was to model <a href="../articles/forward.htm">forward prices</a> as opposed to spot prices. Forward prices do not exhibit the same non-randomness of spot prices. Consider a forward price for delivery shortly after a harvest of an agricultural product. Prior to the harvest, the spot price may be high, reflecting depleted supplies of the product, but the forward price will not be high. Because it is for delivery after the harvest, it will be low in anticipation of a drop in prices following the harvest. While it is not reasonable to model the spot price with a Brownian motion, it may be reasonable to model the forward price with one. </strong> <strong><a name="Blacks (1976) option pricing formula">Black's (1976) option pricing formula</a></strong><strong style="font-weight: 400"> reflects this solution, modeling a forward price as an underlier in place of a spot price. The model is widely used for modeling <a href="../articles/option.htm">European</a> options on physical commodities, <a href="../articles/forward.htm">forwards</a> or <a href="../articles/future.htm">futures</a>. It is also used for pricing interest rate <a href="../articles/cap.htm">caps</a> and <a href="../articles/floor.htm">floors</a>. The model is popularly known as </strong><strong><a name="Black 76">Black '76</a></strong><strong style="font-weight: 400"> or simply </strong><strong><a name="Blacks model">Black's model</a></strong><strong style="font-weight: 400">.</strong><p> </p> <p class="body"><strong style="font-weight: 400">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> are:</strong></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"> <img border="0" src="../formulas/black76_1.gif" width="177" height="28"></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="table7"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/black76_2.gif" width="199" height="28"></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="table8"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/black76_3.gif" width="177" 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="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/black76_4.gif" width="97" height="25"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body" style="margin-top: 20; margin-bottom:24">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>f</i> = the current <a href="../articles/derivative_instrument.htm">underlying</a> forward price</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</a> price</p> <p class="bullet_line_space"> <img border="0" src="../images/bullet.gif" width="20" height="10"><i>r</i> = the <a href="../articles/compounding.htm">continuously compounded</a> <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>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 forward price</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" style="margin-bottom: 16; margin-top:24"><strong style="font-weight: 400">The <a href="../articles/greeks.htm">Greeks</a>—<a href="../articles/delta_and_gamma.htm">delta</a>, <a href="../articles/delta_and_gamma.htm">gamma</a>, <a href="../articles/vega.htm">vega</a>, <a href="../articles/theta.htm">theta</a> and <a href="../articles/rho.htm">rho</a>—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="20%" align="center"> </td> <td align="left" width="60%"> <img border="0" src="../formulas/black76_5.gif" width="120" height="28"></td> <td align="right" width="5%">[5]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_6.gif" width="145" height="48"></td> <td align="right" width="5%">[6]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_7.gif" width="148" height="28"></td> <td align="right" width="5%">[7]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"> </td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_8.gif" width="339" height="49"></td> <td align="right" width="5%">[8]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_9.gif" width="124" height="28"></td> <td align="right" width="5%">[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="20%" align="center"> </td> <td align="left" width="60%"> <img border="0" src="../formulas/black76_10.gif" width="140" height="28"></td> <td align="right" width="5%">[10]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_6.gif" width="145" height="48"></td> <td align="right" width="5%">[11]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_7.gif" width="148" height="28"></td> <td align="right" width="5%">[12]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"> </td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_13.gif" width="357" height="49"></td> <td align="right" width="5%">[13]</td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"></td> <td align="right" width="5%"> </td> </tr> <tr> <td width="20%" align="center" height="12"></td> <td align="left" height="12" width="60%"> <img border="0" src="../formulas/black76_14.gif" width="143" height="28"></td> <td align="right" width="5%">[14]</td> </tr> </table> <p class="body">Note that gamma formulas [6] and [11] are identical for puts and calls, as are vega formulas [7] and [12].</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table12"> <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="table13"> <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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James and Webber (<a target="_blank" href="http://www.riskbook.com/link/james_and_webber_(2000).htm">2000</a>) discuss the use of Black '76 to price caps, floors and other fixed income derivatives.</p> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table15"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap496"> <area target="_blank" href="http://www.riskbook.com/link/haug_e_(1997).htm" shape="rect" coords="0, 0, 100, 124"> </map> <img border="0" src="../images/book_haug_espen_(1997)_125.gif" usemap="#FPMap496" width="101" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/haug_e_(1997).htm"> <font size="2">Complete Guide to </font><br> Option Pricing Formulas </a></td> </tr> <tr> <td colspan="3"> <p class="author"><b><span class="verdana10bold">Espen G. Haug </span></b></td> </tr> <tr> <td width="10%"><map name="FPMap481"> <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_5.gif" usemap="#FPMap481" 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="FPMap482"> <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="#FPMap482" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">1997</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap366"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/haug_e_(1997).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap366" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table17"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap476"> <area target="_blank" href="http://www.riskbook.com/link/james_and_webber_(2000).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_james_125.gif" usemap="#FPMap476" width="83" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table18"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/james_and_webber_(2000).htm"> Interest Rate Modelling</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Jessica James and Nick Webber</b></td> </tr> <tr> <td width="10%"> <map name="FPMap477"> <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_6.gif" usemap="#FPMap477" 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="FPMap478"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 1, 89, 15"> </map><img border="0" src="../images/tech_4.gif" usemap="#FPMap478" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2000</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap479"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/james_and_webber_(2000).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap479" 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">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 class="ad_legend" style="margin-left: 0; margin-bottom:2"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 0; 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="I1"> <![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"> <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="table19"> <tr> <td width="100%"> <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="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> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15">Black, Fischer (1976). The pricing of commodity contracts, <i>Journal of Financial Economics</i>, 3, 167-179.</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>