Garman and Kohlhagen (1983) 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="780"> <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">Garman and Kohlhagen (1983) 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="#Garman and Kohlhagen (1983) option pricing formula">Garman and Kohlhagen (1983) option pricing formula</a></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="135" align="left" id="table5"> <tr> <td width="125" height="16"> <font size="4"> </font></td> <td width="10" height="16"></td> </tr> <tr> <td width="125"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="10"> </td> </tr> </table> <p class="body"><strong style="font-weight: 400">In the interbank foreign exchange market, <a href="../articles/option.htm">options</a> are not quoted with prices. They are quoted indirectly with <a href="../articles/volatility.htm">implied volatilities</a>. The convention for converting volatilities to prices is the </strong><b><a name="Garman and Kohlhagen (1983) option pricing formula">Garman and Kohlhagen (1983) option pricing formula</a></b><strong style="font-weight: 400">. Mathematically, the formula is identical to <a href="../articles/merton_1973.htm">Merton's (1973) formula</a> for options on <a href="../articles/common_stock.htm">dividend-paying</a> <a href="../articles/common_stock.htm">stocks</a>. Only the term<i> q</i>,<i> </i> which did represent a stock's <a href="../articles/yield.htm">dividend yield</a>, now represents the foreign currency's <a href="../articles/compounding.htm">continuously compounded</a> <a href="../articles_old/glossaryriskfreerate.htm">risk-free rate</a>. Like the Merton formula, the </strong>Garman and Kohlhagen formula applies only to <a href="../articles/option.htm">European</a> options. Generally, <a href="../articles/exchange_traded.htm">OTC</a> currency options are European.</p> <p class="body"><strong style="font-weight: 400">Values for a <a href="../articles/option.htm">call</a> 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/merton1.gif" width="190" height="29"></td> <td width="5%" align="right"><a name="[2]">[1]</a></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/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="table8"> <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="table9"> <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> <p class="body">Here, log denotes the natural logarithm, and:</p> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table10"> <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="bullet" align="left"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>s</i> = the current exchange rate (domestic currency per unit of foreign currency)</p> <p class="bullet" align="left"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>x</i> = the <a href="../articles/option.htm">strike</a> exchange rate</p> <p class="bullet" align="left"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>r</i> = the continuously compounded domestic risk free interest rate</p> <p class="bullet" align="left"><img border="0" src="../images/bullet.gif" width="20" height="10"><i>q</i> = the continuously compounded foreign risk free interest rate</p> <p class="bullet" align="left"><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" align="left"><img border="0" src="../images/bullet.gif" width="20" height="10"><font face="Times New Roman">σ = the implied volatility for the <a href="../articles/derivative_instrument.htm">underlying</a> exchange rate</font></p> <p class="bullet" align="left"><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 put <a href="../articles/currency_codes.htm">GBP</a> call USD option on GBP10<a href="../articles/mm.htm">MM</a> for which we want a USD value. The option is struck at 1.65 USD/GBP and expires in .09 years. The current exchange rate is 1.62 USD/GBP. Assume 18% (that is .18) implied volatility. USD and GBP continuously compounded risk free interest rates are .0294 and .0327. Applying formula [<a href="#[2]">2</a>], the option has <a href="../articles/valuation.htm">market value</a> USD .0524 per GBP. Based upon the <a href="../articles/notional_amount.htm">notional amount</a> of GBP 10MM, that becomes USD 0.524MM. 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">Because their prices are affected by two—one domestic and the other foreign—risk free rates, currency options have two rho sensitivities. All the <a href="../articles/greeks.htm">Greeks</a>—delta, gamma, vega, theta, domestic rho and foreign 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="table11"> <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/garman9.gif" width="180" height="27"></td> <td width="5%" align="right">[9]</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/garman20.gif" width="179" height="29"></td> <td width="5%" align="right">[10]</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 <a href="../articles/option.htm">put</a>, 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="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">[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/merton6.gif" width="135" height="50"></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/merton7.gif" width="139" height="29"></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/merton13.gif" width="375" height="50"></td> <td width="5%" align="right">[14]</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/garman15.gif" width="200" height="27"></td> <td width="5%" align="right">[15]</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/garman21.gif" width="179" height="29"></td> <td width="5%" align="right">[16]</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> </table> <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_1976.htm">Black (1976) option pricing formula</a> Used to price options on forwards.</p> <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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Walmsley (<a target="_blank" href="http://www.riskbook.com/link/walmsley_j_(2000).htm">2000</a>) is a sophisticated book on the foreign exchange markets.</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="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="table17"> <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="table18"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap382"> <area target="_blank" href="http://www.riskbook.com/link/walmsley_j_(2000).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_walmsley_125.gif" usemap="#FPMap382" 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="table19"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/walmsley_j_(2000).htm"> The Foreign Exchange and Money Markets Guide</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Julian Walmsley</b></td> </tr> <tr> <td width="10%"> <map name="FPMap362"> <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="#FPMap362" 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="FPMap39"> <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="#FPMap39" 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="FPMap37"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/walmsley_j_(2000).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap37" 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="table20"> <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">Garman, Mark B. and Steven W. Kohlhagen (1983). Foreign currency option values, <i>Journal of International Money and Finance</i>, 2, 231-237. </p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15">Merton, Robert C. (1973). Theory of rational option pricing, <i>Bell Journal of Economics and Management Science</i>, 4 (1), 141-183. 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