Asian Option

<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">Asian Option</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="#asian option">Asian option</a><p class="word_list"> <a href="#average option">average option</a><p class="word_list"> <a href="#average price option">average price option</a><p class="word_list"><a href="#average rate option">average rate option</a><p class="word_list"><a href="#average strike option"> average strike option</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"><strong style="font-weight: 400">An </strong><strong> <a name="asian option">Asian option</a></strong><strong style="font-weight: 400"> (also called an </strong><strong><a name="average option">average option</a></strong><strong style="font-weight: 400">) is an <a href="../articles/option.htm">option</a> whose <a href="../articles/options_spread.htm">payoff</a> is linked to the average value of the <a href="../articles/derivative_instrument.htm">underlier</a> on a specific <a href="../articles/set.htm">set</a> of dates during the life of the option. There are two basic forms:</strong></p> <p class="bullet_line_space"><strong style="font-weight: 400"> <img border="0" src="../images/bullet.gif" width="20" height="10">An </strong><strong> <a name="average rate option"> <em style="font-style: normal">average rate</em> option</a></strong><strong style="font-weight: 400"> (or </strong><strong><a name="average price option">average price option</a></strong><strong style="font-weight: 400">) is a <a href="../articles/physical%20settlement.htm">cash-settled</a> option whose payoff is based on the difference between a the average value of the underlier during the life of the option and a fixed <a href="../articles/option.htm">strike</a>.</strong></p> <p class="bullet_line_space"><strong style="font-weight: 400"> <img border="0" src="../images/bullet.gif" width="20" height="10">An </strong><strong> <a name="average strike option"> <em style="font-style: normal">average strike</em> option</a></strong><strong style="font-weight: 400"> is a cash settled or <a href="../articles/physical%20settlement.htm">physically settled</a> option. It is structured like a <a href="../articles/option.htm">vanilla</a> option except that its strike is set equal to the average value of the underlier over the life of the option.</strong></p> <p align="left" class="body">Both forms can be structured as <a href="../articles/option.htm">puts</a> or <a href="../articles/option.htm">calls</a>. Exercise is generally <a href="../articles/option.htm">European</a>, but it is possible to specify early exercise provisions based upon an average-to-date. Averages can be calculated arithmetically:</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="../images/asian_1.gif" width="239" height="41"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p align="left" class="body">or geometrically:</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="../images/asian_2.gif" align="middle" width="204" height="28"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p align="left" class="body">They can also be weighted with some weights <i>w<sub>i</sub></i>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="12" id="table8"> <tr> <td width="100%" align="center"> <img border="0" src="../images/asian_3.gif" width="352" height="45"></td> <td width="5%" align="right">[3]</td> </tr> <tr> <td width="100%" align="center"> <img border="0" src="../images/blank_10x10.gif" width="10" height="10"></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="../images/asian_4.gif" width="343" height="36"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p align="left" class="body">The name "Asian" option has no particular significance. David Spaughton tells the story of how both he and Mark Standish were both working for Bankers Trust in 1987. They were in Tokyo on business when they developed the first commercially used pricing formula for options linked to the average price of crude oil. Because they were in Asia, they called the options "Asian options." (see Falloon and Turner (<a href="#Falloon">1999</a>))</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table10"> <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 align="left" class="body"><strong style="font-weight: 400">End-users of commodities or energies tend to be exposed to average prices over time, so Asian options are attractive for them. Asian options are also popular with <a href="../articles/corporation.htm">corporations</a>, such as exporters, who have ongoing currency exposures. Asian options are also attractive because they tend to be less expensive—sell at lower <a href="../articles/time_value_and_intrinsic_value.htm"> premiums</a>—than comparable <a href="../articles/option.htm">vanilla</a> puts or calls. This is because the <a href="../articles/volatility.htm">volatility</a> in the average value of an underlier tends to be lower than the volatility of the value of the underlier. Also, in situations where the underlier is thinly traded or there is the potential for its price to be manipulated, an Asian option offers some protection. It is more difficult to manipulate the average value of an underlier over an extended period of time than it is to manipulate it just at the expiration of an option.</strong></p> <p align="left" class="body">Exact analytic formulas for average rate options don't exist. This is primarily due to the fact that the arithmetic average of a set of <a href="../articles/lognormal_distribution.htm">lognormal</a> random variables has a distribution that is largely intractable. Analytic approximations have been proposed by: Turnbull and Wakeman (<a href="#Turnbull">1991</a>), Levy (<a href="#Levy">1992</a>) and Curran (<a href="#Curran">1992</a>). If the underlier is assumed lognormal, then its geometric average is lognormal. In a classic paper, Kemna and Vorst (<a href="#Kemna">1990</a>) use this fact to derive an analytic solution for the price of a geometric average rate option. They use that solution as a <a href="../articles/monte_carlo_method.htm">control variate</a> for a <a href="../articles/monte_carlo_method.htm">Monte Carlo</a> solution for the price of an arithmetic average rate option. See also Wilmott, Dewynne and Howison<i> </i>(<a target="_blank" href="http://www.riskbook.com/link/wilmott_et_al_(1997).htm">2000</a>) and Seydel (<a target="_blank" href="http://www.riskbook.com/link/seydel_r_(2002).htm">2002</a>) for <a href="../articles/closed_form_solution.htm">numerical solutions</a> based on exact differential equations. Tavella and Randell (<a target="_blank" href="http://www.riskbook.com/link/tavella_and_randall_(2000).htm">2000</a>) focus specifically on finite differences. Klassen (<a href="#Klassen">2001</a>) discusses the use of binomial methods. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table11"> <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="table12"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/barrier_option.htm">barrier option</a> A path-dependent option that terminates or is activated by the underlier reaching some "barrier" level.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/binary_option.htm"><strong style="font-weight: 400">binary option</strong></a><strong style="font-weight: 400"> A type of option which features a discontinuous expiration value.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/derivative_instrument.htm">d</a></strong><a href="../articles/derivative_instrument.htm"><strong style="font-weight: 400">erivative instrument</strong></a><strong style="font-weight: 400"> An instrument which derives its value from the value of other financial instruments. Article includes a list of vanilla and exotic derivatives.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/lookback_option.htm">lookback option</a> </strong> <strong style="FONT-WEIGHT: 400">A path dependent option whose payout depends upon the maximum or minimum underlier value achieved during the entire life of the option.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/option_pricing_theory.htm">option pricing theory</a> The body of financial theory used by financial engineers to value options and other derivative instruments.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/path_dependence.htm">path dependence</a> A property of certain exotic options whose terminal value depends upon the path taken by the underlier during the life of the option.</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_1.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> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Books</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">Bryis et al (<a target="_blank" href="http://www.riskbook.com/link/briys_et_al_(1998).htm">1998</a>) describes classic pricing methodologies. 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