Volatility

<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</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="#annualized volatilities">annualized volatility</a><p class="word_list"> <a href="#historical volatilities">historical volatility</a><p class="word_list"> <a href="#implied volatility">implied volatility</a><p class="word_list"> <a href="#square root of time rule">square root of time rule</a><p class="word_list"> <a href="#volatility">volatility</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> <p class="body">Consider the following two graphs.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table5"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: Two Time Series of Prices</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.gif" width="375" height="135"></td> </tr> <tr> <td> <p class="exhibit_legend" style="margin-bottom: 24">Time series are indicated for the prices of two hypothetical assets.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table6"> <tr> <td align="left" valign="top"> <p class="body">They depict time series of prices for two hypothetical assets. We may think of the price series on the left as more risky. We say it is the more "volatile" of the two. We formalize this by defining <b><a name="volatility">volatility</a></b> as follows. Let  <img border="0" src="../formulas/volatility_process.gif" align="absbottom" width="158" height="20">  be a <a href="../articles/time_series_stochastic_process.htm">stochastic process</a> (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation). Its terms <img border="0" src="../formulas/volatility_tq.gif" align="absbottom" width="16" height="19"> may represent prices, accumulated values, exchange rates, interest rates, etc. The volatility of the process at time <i>t</i>–1 is defined as the <a href="../articles/standard_deviation.htm">standard deviation</a> of the time <i>t</i> <a href="../articles/return.htm" name="[1]">return</a>. Typically, <a href="../articles/return.htm">log returns</a> are used, so the definition becomes</p> <div align="left"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table7"> <tr> <td align="center"> <img border="0" src="../formulas/volatility_eq1.gif" width="187" height="59"></td> <td width="30" align="left">[1]</td> </tr> </table> </div> </td> <td width="345" align="left" valign="top"> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table8"> <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></td> </tr> </table> <p class="body" style="margin-top: 6">where log denotes a natural logarithm. However, <a href="../articles/return.htm">simple returns</a> are sometimes used. This is especially true in the context of <a href="../articles/portfolio_theory.htm">portfolio theory</a>. </p> <p class="body">If we assume that returns are <a href="../articles/heteroskedasticity.htm">conditionally homoskedastic</a>, definition [<a href="#[1]">1</a>] is precise. However, if they are <a href="../articles/heteroskedasticity.htm">conditionally heteroskedastic</a>, we need to clarify the definition. Does volatility at time<i> t</i>–1 represent the unconditional standard deviation of the time <i>t</i> log return? Or does it represent the standard deviation of the time <i>t</i> log return conditional on information available at time <i>t</i>–1? The answer is the latter. To emphasize this, we might express definition [<a href="#[1]">1</a>] <a name="[2]">as</a></p> <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/volatility_eq2.gif" width="200" height="59"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">where the preceding superscript <i>t</i>–1 indicates that the standard deviation is conditional on information available at time <i> t</i>–1.</p> <p class="body">Another issue in defining volatility is that of the unit of time on which it is based. The standard deviation of a <a href="../articles/common_stock.htm">stock's</a> price return over a day might be .01. Over a year, it might be .16. Accordingly, for any quantity, we might speak of its daily volatility, weekly volatility, annual volatility, etc. All would be distinct notions.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table10"> <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">This leads to the question of whether, given a volatility based upon one time unit, there is a way to convert it to an equivalent volatility based upon another time unit. As a general rule, the answer is no. To understand why, consider Exhibit 2.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table11"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: Three Price Time Series</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.gif" width="250" height="405"></td> </tr> <tr> <td> <p class="exhibit_legend">Time series of three hypothetical prices are indicated. The first process is mean reverting. The second follows a random walk. The third trends.</td> </tr> </table> </center> </div> <p class="body">The first time series is a realization of a <a href="../articles/mean_reversion.htm">mean reverting</a> process. There is about as much uncertainty in the price a day in the future as there is a month in the future. If the one-day volatility is .02, then the monthly volatility might also be .02.</p> <p class="body">The second time series is a realization of a <a href="../articles/random_walk.htm">random walk</a>. The price is more uncertain a month in the future than it is a day in the future. If the daily volatility is .02, then the monthly volatility would be .05.</p> <p>The third time series is a realization of a process that tends to follow long-term trends. Because of those trends, there is far more uncertainty in prices a month in the future than a day in the future. If the daily volatility is .02, the monthly volatility might be .08.</p> <p>As this example illustrates, volatilities for different units of time are fundamentally different notions. There is no direct relationship between, say a weekly volatility and an annual volatility. However, there is an exception to this observation. The exception is called the <b> <a name="square root of time rule">square root of time rule</a></b>. If fluctuations in a stochastic process from one period to the next are independent (i.e., there are no serial <a href="../articles/correlation.htm">correlations</a> or other dependencies) volatility increases with the square root of the unit of time. Any price that follows a random walk, <a href="../articles/brownian_motion.htm">Brownian motion</a> or geometric Brownian motion satisfies this independence condition. The square root of time rule is exact if volatilities are based upon log returns. It is approximately correct if volatilities are based upon simple returns.</p> <p>Let's consider an example. Suppose a price has .04 monthly volatility. If follows a random walk, so, by the square root of time rule, its annual volatility is</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/volatility_eq3.gif" width="88" height="23"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p>Consider another example. A price has .24 annual volatility. What is its daily volatility, assuming it follows a geometric Brownian motion? We can apply the square root of time rule, but this raises an question. In converting from a year to a day, should we count actual days (including weekends and holidays) or should we count only trading days? The latter approach seems more plausible because prices cannot change on days that no trading takes place. In fact, empirical research supports this approach. In most markets, price fluctuations from one trading day to the next appear typically not to be much larger if there is an intervening weekend or holiday as when there is not. Returning to our example, in the United States, there are about 252 trading days in a year. Accordingly, our price will have daily volatility of </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/volatility_eq4.gif" width="88" height="44"></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="table14"> <tr> <td align="right" width="9" height="10"></td> <td width="336" height="10"> <font size="4">  </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="336">  <p style="margin-top: 0; 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>Volatilities play an important role in <a href="../articles/option_pricing_theory.htm">financial engineering</a> and especially in the valuation of various forms of <a href="../articles/option.htm">options</a>. In their landmark (<a href="#Black (1973)">1973</a>) paper, Black and Scholes derived a formula for pricing a vanilla put or call option on a <a href="../articles/common_stock.htm">non-dividend paying</a> stock. <a href="../articles/black_scholes_1973.htm">That formula</a> requires as inputs</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the <a href="../articles/derivative_instrument.htm"> underlier's</a> current price,</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the option's <a href="../articles/option.htm">strike</a> price</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the option's time to <a href="../articles/option.htm"> expiration</a>,</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">a risk free interest rate, and</p> <p class="bullet"><img border="0" src="../images/bullet.gif" width="20" height="10">the underlier's annual volatility (based on log returns).</p> <p>All of these quantities are (typically) observable in the marketplace—except the volatility. Accordingly, financial engineering has spawned a tremendous need to estimate volatilities for underliers. Those underliers were first prices—say prices of stocks or commodities—but they have come to include quantities such as exchange rates, interest rates or even weather conditions.</p> <p>A standard way to estimate a volatility for a given underlier is to use the price of an option on that underlier. Suppose a call option on the underlier is actively trade, so the option's price is readily obtainable. Then, by applying a suitable option pricing formula—in a sense backwards—we calculate the annual volatility that would have to be input into the option pricing formula to obtain that price for the option. In this manner, we obtain the volatility implied by the option price—what is called the <b><a name="implied volatility">implied volatility</a></b> for the underlier.</p> <p>Such an implied volatility can then be used to price other options on that same underlier—perhaps options that are not actively traded or for which prices are otherwise not readily available. In practice, this is what financial engineers do to price a variety of derivative instruments—obtain implied volatilities from quoted prices for certain derivatives on an underlier so that they can price other derivatives on that same underlier. However, the process tends to be more involved than this. In practice, different implied volatilities may be obtained for a given underlier depending upon the strike or expiration of the option from which they are obtained. See the article <a href="../articles/volatility_skew.htm"> volatility skew</a>.</p> <p>Another approach to estimating volatilities is to apply techniques of <a href="../articles/time_series_stochastic_process.htm">time series analysis</a> to historical data for the variable whose volatility is to be estimated.  Volatilities calculated in this manner are called <b> <a name="historical volatilities">historical volatilities</a></b>. Historical volatilities are routinely used in applications other than financial engineering—such as <a href="../articles/value_at_risk.htm"> value-at-risk</a> or <a href="../articles/portfolio_theory.htm">portfolio theory</a>—where volatilities are required for quantities on which options are not traded. They might also be used by financial engineers for underliers for which implied volatilities are unavailable—perhaps because options are not actively traded on those underliers. Financial engineers also might use historical volatilities as a "reality check" to supplement implied volatilities.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table15"> <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>Historical volatilities are usually calculated from daily data. This means that they are daily volatilities. Because volatilities are usually quoted on an annual basis (especially for option pricing) such daily historical volatilities are routinely converted to an annual basis by applying the square root of time rule. This is done even if conditions for applying that rule are not satisfied. The resulting volatilities are referred to as <b><a name="annualized volatilities">annualized volatilities</a></b>—as opposed to annual volatilities—to alert people to the fact that this is just a quoting convention.</p> <p>A question that frequently arises is whether implied or historical volatilities offer a better indication of <a href="../articles/market_risk.htm">market risk</a>. The answer is that each has its strengths as well as limitations. Implied volatilities are often referred to as a "market consensus" of volatility—an indication of risk that combines the insights of many market participants. For the most part, this is a reasonable interpretation. However, implied volatilities are essentially prices. They can be biased by such things as <a href="../articles/spreads.htm">bid-ask spreads</a> as well as supply and demand for options. For example, at the height of his speculative activity in 1995, <a href="../articles/barings_debacle.htm">Nick Leeson</a> was selling so many Nikkei options that he drove that implied volatility far below its historical levels. Historical volatility, on the other hand, reflects actual market fluctuations. However, the data upon which an historical volatility is based may be stale—perhaps encompassing a period not reflective of current market conditions. For this reason, implied volatilities tend to be more responsive to current market conditions.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <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="table17"> <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> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/brownian_motion.htm">Brownian motion</a> A simple continuous stochastic process that is widely used in physics and finance for modeling random behavior that evolves over time.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/heteroskedasticity.htm">heteroskedasticity</a> A condition where a stochastic process has non-constant second moments.</p> </span> <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"><strong style="font-weight: 400"><a href="../articles/portfolio_theory.htm">p</a></strong><a href="../articles/portfolio_theory.htm"><strong style="font-weight: 400">ortfolio theory</strong></a><strong style="font-weight: 400"> A body of theory relating to how investors optimize portfolio selections.</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/standard_deviation.htm">standard deviation</a> </strong>A parameter describing the dispersion of a probability distribution.<span style="font-family: Times New Roman; 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