Volatility Skew

<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">Volatility Skew</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="90%" align="left" valign="top"><p class="word_list"> <a href="#deterministic volatility function models">deterministic volatility function model</a><p class="word_list"> <a href="#implied tree models">implied tree model</a><p class="word_list"> <a href="#jump-diffusion model">jump-diffusion model</a><p class="word_list"> <a href="#Local volatility models">local volatility model</a><p class="word_list"> <a href="#Mixed distribution models">mixed distribution model</a><p class="word_list"> <a href="#skew">skew</a><p class="word_list"> <a href="#sticky delta">sticky delta</a><p class="word_list"> <a href="#sticky strike">sticky strike</a><p class="word_list"> <a href="#universal volatility models">universal volatility model</a><p class="word_list"> <a href="#volatility skew">volatility skew</a><p class="word_list"> <a href="#volatility smile">volatility smile</a><p class="word_list"> <a href="#volatility surface">volatility surface</a><p class="word_list"> <a href="#volatility term structure">volatility term structure</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body">Consider <a href="../articles/option.htm">put</a> or <a href="../articles/option.htm">call</a> <a href="../articles/option.htm">options</a> on a given <a href="../articles/derivative_instrument.htm">underlier</a>. They have different <a href="../articles/option.htm">strikes</a> but the same <a href="../articles/option.htm">expiration</a>. If we obtain market prices for those options, we can apply the <a href="../articles/black_scholes_1973.htm">Black-Scholes</a> (<a href="#Black (1973)">1973</a>) model to back-out <a href="../articles/volatility.htm">implied volatilities</a>. Intuitively, we might expect the implied volatilities to be identical. In practice, it is likely that they will not be.</p> <p class="body">Coffee options trade on New York's Coffee, Sugar and Cocoa Exchange (CSCE). Exhibit 1 indicates implied volatilities at various strikes for the May 2001 calls based upon their March 12, 2001 settlement prices. </p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="400" cellpadding="0" id="table5"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: Volatility Smile</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_volatility_skew.gif" width="325" height="212"></td> </tr> <tr> <td> <p class="exhibit_legend" style="margin-bottom: 18">CSCE May 2001 coffee call option implied volatilities as of March 12, 2001. Implied volatilities were calculated from settlement prices using <a href="../articles/black_1976.htm">Black's</a> (<a href="#Black, Fischer (1976)">1976</a>) option pricing model. The pattern of implied volatilities form a "smile" shape, which is called a volatility smile.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table6"> <tr> <td width="50%" align="left" valign="top"> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="left" id="table7"> <tr> <td width="300"> <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">The pattern of implied volatilities forms a "smile" shape, which is called a <b><a name="volatility smile">volatility smile</a></b>. Such a smile persists over time in the coffee options markets with <a href="../articles/time_value_and_intrinsic_value.htm">in-the-money</a> and <a href="../articles/time_value_and_intrinsic_value.htm"> out-of-the-money</a> volatilities generally higher than <a href="../articles/time_value_and_intrinsic_value.htm">at-the-money</a> volatilities.</p> <p class="body">Most derivatives markets exhibit persistent patterns of volatilities varying by strike. In some markets, those patterns form a smile. In others, such as equity index options markets, it is more of a skewed curve. This has motivated the name <b><a name="volatility skew"> volatility skew</a></b>. In practice, either the term "volatility smile" or "volatility skew" (or simply <b><a name="skew">skew</a></b>) may be used to refer to the general phenomena of volatilities varying by strike. Indeed, you may even hear of "volatility smirks" or "volatility sneers", but such names are often as much whimsical as they are descriptive of any particular volatility pattern.</p> </td> </tr> </table> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table8"> <tr> <td> <p class="exhibit_header" style="margin-top: 24"><span class="exhibit_header">Smile vs. Skew</span><br> <span class="exhibit_subheader">Exhibit 2</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex2_volatility_smile.gif" width="400" height="135"></td> </tr> <tr> <td> <p class="exhibit_legend">Volatility smile (left) is compared to volatility skew (right).</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table9"> <tr> <td align="right" width="9" height="10"></td> <td width="336" height="10"> <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">There are various explanations for why volatilities exhibit skew. Different explanations may apply in different markets. In most cases, multiple explanations may play a role. Some explanations relate to the idealized assumptions of the Black-Scholes approach to valuing options. Almost every one of those assumptions—<a href="../articles/lognormal_distribution.htm">lognormally</a> distributed <a href="../articles/return.htm">returns</a>, return <a href="../articles/heteroskedasticity.htm">homoskedasticity</a>, etc.—could play a role. For example, in most markets, returns appear more <a href="../articles/kurtosis.htm"> leptokurtic</a> than is assumed by a lognormal distribution. Market leptokurtosis would make way out-of-the-money or way in-the-money options more expensive than would be assumed by the Black-Scholes formulation. By increasing prices for such options, volatility smile could be the markets' indirect way of achieving such higher prices within the imperfect framework of the Black-Scholes model. Other explanations relate to relative supply and demand for options. In <a href="../articles/common_stock.htm">equity</a> markets, volatility skew could reflect investors' fear of market crashes—which would cause them to bid up the prices of options at strikes below current market levels. In electricity markets, utilities and other purchasers are concerned about price spikes. Not surprisingly, electricity volatilities exhibit the opposite skew—with volatilities elevated for higher strikes.</p> <p class="body">Another dimension to the problem of volatility skew is that of volatilities varying by expiration. This is illustrated for CSCE coffee options in Exhibit 3. It indicates what is known as a <b> <a name="volatility surface">volatility surface</a></b>—a three-dimensional graph indicating implied volatilities by both strike and expiration.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table10"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: Volatility Surface</span><br> <span class="exhibit_subheader">Exhibit 3</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex3_volatility_smile.gif" width="350" height="350"></td> </tr> <tr> <td> <p class="exhibit_legend">Volatility surface for CSCE cofee options on March 12, 2001.</td> </tr> </table> </center> </div> <p class="body">For early expirations, the graph exhibits a volatility smile. Volatilities dip and then rise, taking on a skew for the September contract before falling and flattening out. This is a pattern that recurs every year for coffee volatilities. September is the harvest month for Brazilian coffees. Brazil is a major exporter, and every year, the harvest is threatened by frost. Market participants watch the harvest, knowing that a frost will drive world coffee prices sharply higher. This explains both the rise in volatilities as well as the skew for September. For that month, traders are concerned about a spike in prices, and the volatility surface reflects this.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table11"> <tr> <td width="160"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> </td> <td width="11"> </td> </tr> </table> <p class="body">If we take a cross section of a volatility surface at a particular strike, we obtain a curve that describes implied volatilities as a function of expiration for that strike. This is called a <b> <a name="volatility term structure">volatility term structure</a></b>.</p> <p class="body">Traders aren't interested only in static volatility surfaces. They also want to know how skew will respond to the passage of time and changes in the underlier's value. While the dynamics of a volatility surface are complicated, there are two simple models that are useful for describing an aspect of those dynamics. Introduced by Derman (<a href="#Derman, Emanuel (1999)">1999</a>), these are called the <b><a name="sticky strike">sticky strike</a></b> and <b><a name="sticky delta">sticky delta</a></b> models. </p> <p class="body">If a skew is behaving according to the sticky strike model, its implied volatilities are associated with specific strikes. The curve of the skew will not shift with the value of the underlier.</p> <p class="body">With the sticky delta model, implied volatilities are associated with specific deltas. The entire curve of volatilities moves with the underlier. For example, suppose a strike 75 call has a delta of .65 and an implied volatility of 14%. If the value of the underlier rises so that the strike 80 option becomes the delta .65 option, then the 14% implied volatility will migrate to that option.</p> <p class="body">Obviously, both models are simplifications. No market exhibits one behavior or the other at all times, and actual volatility dynamics often blend the two.</p> <p class="body">Volatility skew complicates the tasks of pricing and hedging options. Consider the task of calculating an option's <a href="../articles/delta_and_gamma.htm">delta</a>. If we assume the sticky delta model, this will affect how we calculate the option's delta. Changes in implied volatilities that are expected to accompany changes in the value of the underlier will impact the option's value. Deltas need to be adjusted to reflect this. This is more than a theoretical consideration. If a trader is <a href="../articles/dynamic_hedging.htm">dynamically hedging</a> an options position and fails to incorporate skew into her delta calculations, her hedge ratio will be off.</p> <p class="body">Skew poses even greater challenges for <a href="../articles/option_pricing_theory.htm">financial engineers</a>. They need to adopt dynamic models for entire volatility surfaces. These are far more sophisticated than the simple sticky strike and sticky delta models. Standard models include</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">Merton's (<a href="#Merton, R. (1976)">1976</a>) jump-diffusion model;</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10"><a href="../articles/stochastic_volatility_model.htm">stochastic volatility models</a> of Hull and White (<a href="#Hull, John C. and Allen White (1988)">1988</a>) and Heston (<a href="#Heston, Steven L. (1993)">1993</a>);</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">local volatility models of Dupire (<a href="#Dupire, Bruno (1994)">1994</a>), Derman and Kani (<a href="#Derman, Emanuel and Iraj Kani (1994)">1994</a>), Rubinstein (<a href="#Rubinstein, Mark (1994)">1994</a>) and Andersen and Brotherton-Ratcliffe (<a href="#Andersen, L. and Brotherton-Ratcliffe, R. (1997)">1997</a>);</p> <p class="bullet_line_space"><img border="0" src="../images/bullet.gif" width="20" height="10">mixed distribution models such as Brigo and Mercurio (<a href="#Brigo, Damiano and Fabio Mercurio (2000)">2000</a>).</p> <p class="body">A <b><a name="jump-diffusion model">jump-diffusion model</a></b> adds random jumps to the geometric <a href="../articles/brownian_motion.htm">Brownian motion</a> that Black-Scholes (<a href="#Black (1973)">1973</a>) assumes for the underlier. Among other things, this has the effect of giving the underlier's value a leptokurtic distribution.</p> <p class="body">A <a href="../articles/stochastic_volatility_model.htm">stochastic volatility model</a> models both the underlier's value and its volatility as stochastic processes. As with a jump-diffusion model, this has the effect of giving the underlier's value a leptokurtic distribution. Heston's (<a href="#Heston, Steven L. (1993)">1993</a>) is a popular stochastic volatility model.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table12"> <tr> <td align="right" width="9"> <p style="margin-top: 0; margin-bottom: 0"> </td> <td width="336">  <p style="margin-top: 8; margin-bottom: 2" class="ad_legend"> <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=82402;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=82402;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82402"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82402;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> </tr> </table> <p class="body"><b><a name="Local volatility models">Local volatility models </a></b>have various names. They have been called <b> <a name="deterministic volatility function models">deterministic volatility function models</a></b> or <b><a name="implied tree models"> implied tree models</a></b> (implied binomial tree or implied trinomial tree models, depending on the type of tree). They assume that future volatilities will be a deterministic function of the underlier value and time. This function is implied by the current volatility skew and can be fully reflected in a suitably calibrated binomial or trinomial tree.</p> <p class="body"><b><a name="Mixed distribution models">Mixed distribution models</a></b> model the underlier's value with a mixture of distributions. This has the effect of giving the underlier a leptokurtic distribution. Brigo and Mercurio (<a href="#Brigo, Damiano and Fabio Mercurio (2000)">2000</a>) use a mixture of lognormals.</p> <p class="body">Other models are hybrids, combining aspects of those described above. Bates' (<a href="#Bates, David S. (1996)">1996</a>) stochastic volatility jump-diffusion model is an example. The <b> <a name="universal volatility models">universal volatility models</a></b> of Dupire (<a href="#Dupire, Bruno (1996)">1996</a>) and Britten-Jones and Neuberger (<a href="#Britten-Jones, M. and A. Neuberger (2000)">2000</a>), among others, combine elements of local volatility, jump-diffusion and stochastic volatility models. These are rich models, but they can be difficult to implement.</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%"> <span style="font-family: Times New Roman; letter-spacing: -0.2pt"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/ARCH_GARCH.htm">ARCH</a> A category of conditionally heteroskedastic stochastic processes.</p> </span> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/derivative_instrument.htm"><strong style="font-weight: 400">derivative instrument</strong></a><strong style="font-weight: 400"> An instrument which derives its value from the value of other financial instruments. 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