Black-Scholes (1973) Option Pricing Formula

<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">Black-Scholes (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="#Black-Scholes (1973) option pricing formula">Black-Scholes (1973) 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">In 1973, Fischer Black and Myron Scholes published their groundbreaking paper </strong> <a href="#Black (1973)">the pricing of options and corporate liabilities</a>. Not only did this specify the first successful options pricing formula, but it also described a general framework for pricing other <a href="../articles/derivative_instrument.htm">derivative instruments</a>. That paper launched the field of <a href="../articles/option_pricing_theory.htm">financial engineering</a>. Black and Scholes had a very hard time getting that paper published. Eventually, it took the intersession of Eugene Fama and Merton Miller to get it accepted by the <i>Journal of Political Economy</i>. In the mean time, Black and Scholes had published in the <i>Journal of Finance</i> a more accessible (<a href="#Black (1972)">1972</a>) paper that cited the as-yet unpublished (1973) option pricing formula in an empirical analysis of current options trading. </p> <p class="body"><strong style="font-weight: 400">The </strong><strong> <a name="Black-Scholes (1973) option pricing formula">Black-Scholes (1973) option pricing formula</a></strong><strong style="font-weight: 400"> prices <a href="../articles/option.htm">European</a> <a href="../articles/option.htm">put</a> or <a href="../articles/option.htm">call</a> <a href="../articles/option.htm">options</a> on a <a href="../articles/common_stock.htm">stock</a> that does not pay a <a href="../articles/common_stock.htm">dividend</a> or make other distributions. The formula assumes the underlying stock price follows a geometric <a href="../articles/brownian_motion.htm">Brownian motion</a> with constant <a href="../articles/volatility.htm">volatility</a>. It is historically significant as </strong>the original option pricing formula published by Black and Scholes in their landmark (<a href="#Black (1973)">1973</a>) paper.<strong style="font-weight: 400"> </strong></p> <p class="body"><strong style="font-weight: 400">Values for a call price <i> c</i> or put 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/black_1.gif" width="157" 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/black_2.gif" width="180" 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/black_3.gif" width="182" height="48"></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/black_4.gif" width="93" height="27"></td> <td width="5%" align="right">[4]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table10"> <tr> <td align="right" width="9" height="14"></td> <td width="336" height="14"> <font size="2"> </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">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</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 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 European call option on 100 <a href="../articles/common_stock.htm">shares</a> of non-dividend-paying stock ABC. The option is struck at <a href="../articles/currency_codes.htm">USD</a> 55 and expires in .34 years. ABC is trading at USD 56.25 and has 28% (that is .28) implied volatility. The continuously compounded risk free interest rate is .0285. Applying formula [1], the option's <a href="../articles/valuation.htm">market value</a> per share of ABC is USD 4.56. Since the call is for 100 shares, its total value is USD 456. Of this, USD 125 is <a href="../articles/time_value_and_intrinsic_value.htm">intrinsic value</a>, and USD 331 is <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>—<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="table11"> <tr> <td width="35%" align="center"> </td> <td width="60%" align="left">delta = <font face="Times New Roman"> Φ(<i>d</i><sub>1</sub>)</font></td> <td width="5%" align="right">[5]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12">gamma = <img border="0" src="../formulas/black_5.gif" align="absmiddle" width="41" height="44"></td> <td width="5%" align="right">[6]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12">vega = <img border="0" src="../formulas/black_6.gif" align="absmiddle" width="64" height="27"></td> <td width="5%" align="right">[7]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"> </td> <td width="60%" align="left" height="12">theta = <img border="0" src="../formulas/black_10.gif" align="absmiddle" width="165" height="44"></td> <td width="5%" align="right">[8]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" 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="table12"> <tr> <td width="35%" align="center"> </td> <td width="60%" align="left">delta = <font face="Times New Roman"> Φ(<i>d</i><sub>1</sub>) – 1</font></td> <td width="5%" align="right">[10]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12">gamma = <img border="0" src="../formulas/black_5.gif" align="absmiddle" width="41" height="44"></td> <td width="5%" align="right">[11]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12">vega = <img border="0" src="../formulas/black_6.gif" align="absmiddle" width="64" height="27"></td> <td width="5%" align="right">[12]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"> </td> <td width="60%" align="left" height="12">theta = <img border="0" src="../formulas/black_11.gif" align="absmiddle" width="170" height="44"></td> <td width="5%" align="right">[13]</td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"></td> <td width="5%" align="right"> </td> </tr> <tr> <td width="35%" align="center" height="12"></td> <td width="60%" align="left" height="12"> <img border="0" src="../formulas/black_8.gif" width="136" height="28"></td> <td width="5%" align="right">[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="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/merton_1973.htm">Merton (1973) option pricing formula</a> Used to price European options on dividend paying stocks or stock indexes.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a 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collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table15"> <tr> <td width="100%"> <p class="explanatory_text">To learn about the historical origins of the Black-Scholes formula, see Mehrling (<a target="_blank" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm">2005</a>) or Bernstein (<a target="_blank" href="http://www.riskbook.com/link/bernstein_p_(1993).htm">1993</a>). 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Hull (<a target="_blank" href="http://www.riskbook.com/link/hull_j_(2002).htm">2005</a>) is a standard introduction to financial engineering that covers the derivation of Black-Scholes in detail. </p> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap739"> <area target="_blank" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_mehrling_perry_(2005)_125.jpg" usemap="#FPMap739" 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="table17"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm"> Fischer Black<br> <font size="2">And the Revolutionary Idea of Finance</font></a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Perry Mehrling</b></td> </tr> <tr> <td width="10%"><map name="FPMap564"> <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_4.gif" usemap="#FPMap564" 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="FPMap738"> <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_1.gif" usemap="#FPMap738" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2005</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap563"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap563" 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="table18"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap559"> <area target="_blank" href="http://www.riskbook.com/link/bernstein_p_(1993).htm" shape="rect" coords="0, 0, 81, 124"> </map> <img border="0" src="../images/book_bernstein_peter_(1993)_125.jpg" usemap="#FPMap559" width="82" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table19"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/bernstein_p_(1993).htm"> Capital Ideas</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Peter L. 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Haug </span></b></td> </tr> <tr> <td width="10%"><map name="FPMap740"> <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="#FPMap740" 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="FPMap741"> <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="#FPMap741" 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="table22"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap501"> <area target="_blank" href="http://www.riskbook.com/link/natenberg_s_(1994).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_natenberg_s_(1994)_125.gif" usemap="#FPMap501" 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="table23"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/natenberg_s_(1994).htm"> Option Volatility & Pricing</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Sheldon Natenberg</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="#FPMap740" 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="#FPMap741" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">1994</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap497"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/natenberg_s_(1994).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap497" 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="table24"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap500"> <area target="_blank" href="http://www.riskbook.com/link/cox_and_rubinstein_(1985).htm" shape="rect" coords="0, 0, 80, 124"> </map> <img border="0" src="../images/book_cox_and_rubinstein_(1985)_125.gif" usemap="#FPMap500" width="81" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table25"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/cox_and_rubinstein_(1985).htm"> Options Markets</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>John C. 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">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="table26"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Black (1972)">Black</a>, Fischer and Myron S. Scholes (1972) The valuation of option contracts and a test of market efficiency, <i>Journal of Finance</i>, 27 (2), 399–418.</p> <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 (3), 637-654.</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_3.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">Related Forum Discussions</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="table27"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_bullhorn_narrow.gif" align="left" width="27" height="22"><b><a target="_top" href="http://www.contingencyanalysis.com/archive/link_thread/98-4/0000017e.htm">Black-Scholes</a></b> <i>07 Oct 1998</i><br> Intuitive understanding of the Black-Scholes formula.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_bullhorn_narrow.gif" align="left" width="27" height="22"><b><a target="_top" href="http://www.contingencyanalysis.com/archive/link_thread/98-4/00000119.htm">Options probability distributions</a></b> <i>02 Sep 1998</i><br> Risk neutrality and the probability of an option expiring in-the-money.</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>