Delta and Gamma

<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">Delta and Gamma</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="#Delta">delta</a><p class="word_list"> <a href="#delta approximation">delta approximation</a><p class="word_list"> <a href="#delta-gamma approximation">delta-gamma approximation</a><p class="word_list"> <a href="#Gamma">gamma</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body"><strong style="font-weight: 400">Changes in the value of an <a href="../articles/derivative_instrument.htm">underlier</a> are often the primary source of <a href="../articles/risk.htm">risk</a> in a <a href="../articles/derivative_instrument.htm">derivatives</a> portfolio, so there are two <a href="../articles/greeks.htm">Greek</a> factor sensitivities for measuring such risk. Delta and gamma represent first- and second-order measures of sensitivity to an underlier.</strong></p> <p class="body"><strong style="font-weight: 400">Consider a hypothetical portfolio whose value depends upon some underlier whose current value is USD 101. Exhibit 1 illustrates how the current <a href="../_private/footer_glossary.htm">market value</a> of a hypothetical portfolio depends upon the current underlier value.  </strong></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: Portfolio Value as a Function of Underlier Value</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_delta_and_gamma.gif" width="350" height="175"></td> </tr> <tr> <td> <p class="exhibit_legend">The market value in USD <a href="../articles/mm.htm">MM</a> of a hypothetical portfolio as a function of underlier value in USD. The current underlier value is USD 101. The graph indicates what the portfolio value would be if the underlier had a current value anywhere in the interval USD 99 to USD 103.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table6"> <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">Exhibit 1 fully describes the relationship between the portfolio's current market value and the underlier, assuming other market variables are unchanged. With just two numbers—delta and gamma—we can summarize the information contained in Exhibit 1. Certainly, two numbers cannot describe the wealth of detail contained in a picture, but with delta and gamma we capture the two most important pieces of information in the picture. </strong></p> <p class="body"><strong style="font-weight: 400">Let's start with delta. The most significant information that Exhibit 1 provides about this particular portfolio is the fact that its value will increase if the underlier increases, and it will decrease if the underlier decreases. This is the information that delta conveys, along with the magnitude of such sensitivity. </strong></p> <p class="body"><strong style="font-weight: 400">If we fit a tangent line to the curve in Exhibit 1 at the underlier's current value of 101, the slope of that line will capture the direction and magnitude of the portfolio's sensitivity to the underlier. Delta is the slope of that tangent line.</strong></p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table7"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Delta is the Slope of the Tangent Line</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_delta_and_gamma.gif" width="350" height="175"></td> </tr> <tr> <td> <p class="exhibit_legend">Delta is the slope of the tangent line fit to the portfolio's value function at the current underlier value. In this example, the current underlier value is USD 101, and the slope of the tangent line 0.8MM.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table8"> <tr> <td width="160">  </td> <td width="11"> </td> </tr> <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"><strong style="font-weight: 400">For example, in Exhibit 2, the slope of the tangent line is 0.8<a href="../articles/mm.htm">MM</a> (for each unit increase in the underlier, the portfolio's price appreciates by 0.8MM). Accordingly, the portfolio's delta is 0.8MM. </strong></p> <p class="body"><strong style="font-weight: 400">Fitting tangent lines to functions is the province of calculus, so we turn to calculus for the formal definition of delta. Let 0 be the current time. Let <img border="0" src="../images/delta_0p.gif" align="absbottom" width="14" height="18"> and <img border="0" src="../images/delta_0s.gif" align="absbottom" width="12" height="18"> be current values for the portfolio and underlier</strong> (see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation)<strong style="font-weight: 400">. </strong><strong><a name="Delta">Delta</a></strong><strong style="font-weight: 400"> is the first partial derivative of a portfolio's value<i> </i>with respect to the value<i> </i>of the underlier:</strong></p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../images/delta_1.gif" width="81" height="50"></td> <td width="5%" align="right">[1]</td> </tr> </table> </center> </div> <p class="body"><strong style="font-weight: 400">This technical definition leads to an approximation for the behavior of a <a name="[2]">portfolio</a>.</strong></p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <img border="0" src="../images/delta_2.gif" width="110" height="28"></td> <td width="5%" align="right">[2]</td> </tr> </table> </center> </div> <p class="body"><strong style="font-weight: 400">Where <img border="0" src="../images/delta_delta_0s.gif" align="absbottom" width="22" height="18"> is a small change in the underlier's current value, and <img border="0" src="../images/delta_delta_0p.gif" align="absbottom" width="25" height="18"> is the corresponding change in the portfolio's current value. This is called the </strong><strong><a name="delta approximation">delta approximation</a></strong><strong style="font-weight: 400">.</strong></p> <p class="body"><strong style="font-weight: 400">Suppose a portfolio is exposed to IBM <a href="../articles/common_stock.htm">stock</a> and has a IBM delta of 1.5MM <a href="../articles/common_stock.htm">shares</a>. This means that, for small market moves, the portfolio behaves like a position comprising 1.5MM shares of IBM. It will gain about USD 1.5MM if IBM stock rises USD 1 or lose about USD 3.0MM if the stock falls USD 2. Plug these numbers into formula [<a href="#[2]">2</a>] and confirm that it is saying the same thing! Note that the portfolio's exposure could result from direct holdings in IBM stock, a derivatives position with IBM stock as an underlier, or some combination of the two. If it is caused entirely by an outright position in IBM stock, then that position must consist of exactly 1.5 million shares of IBM stock because the delta of one unit of the underlier always equals 1.0. </strong></p> <p class="body"><strong style="font-weight: 400">If the portfolio is exposed to several stocks, then it will have a delta for each. For example, it's Exxon delta might be –2.5MM shares. This would behave similarly to a short position in 2.5 million shares of American Airlines stock. If American Airlines stock rose USD 1, the portfolio would lose about USD 2.5MM. If the stock declined USD 0.5, the portfolio would gain about USD 1.25MM.</strong></p> <p class="body"><strong style="font-weight: 400">Now let's consider gamma. If delta summarizes the most significant piece of information about a portfolio's sensitivity to an underlier, gamma summarizes the second-most significant piece of information. Delta captured the fact that the graph in Exhibit 1 was upward sloping. It did not capture its downward curvature. Gamma describes curvature.</strong></p> <p class="body"><strong style="font-weight: 400">Exhibit 3 shows the best-fitting parabola for the graph of Exhibit 1: </strong></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">Best-Fitting Parabola</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_delta_and_gamma.gif" width="350" height="175"></td> </tr> <tr> <td> <p class="exhibit_legend">The parabola that best fits the portfolio's value function.</td> </tr> </table> </center> </div> <p class="body"><strong style="font-weight: 400">Note that the best-fitting parabola does not exactly overlay the curve in Exhibit 3 because the curve is not itself a parabola. In general, the best-fitting parabola will have the form:</strong></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/delta_3.gif" align="baseline" width="227" height="28"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body"><strong style="font-weight: 400">where <i>a</i>, <i>b</i> and <i>c</i> are constants determined to achieve the best fit. Gamma equals 2<i>c</i>. As it turns out, for the best-fitting parabola, the constant <i>b</i> is the portfolio's delta, and <i>a</i> can be solved for based upon the portfolio's current market value<i>.</i></strong></p> <p class="body"><strong style="font-weight: 400">Gamma not only tells us the magnitude of curvature, but its direction as well. Positive gamma corresponds to curvature that opens upward. Negative gamma corresponds to curvature that opens downward.</strong></p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table13"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Positive vs. Negative Gamma</span><br> <span class="exhibit_subheader">Exhibit 4</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/ex4_delta_and_gamma.gif" width="350" height="154"></td> </tr> <tr> <td> <p class="exhibit_legend">Positive gamma corresponds to curvature that opens upward. Negative gamma corresponds to curvature that opens downward.</td> </tr> </table> </center> </div> <p class="body">For a formal definition of gamma, we again turn to calculus. <b><a name="Gamma">Gamma</a></b> is the second partial derivative<strong style="font-weight: 400"> of a portfolio's value<i> </i><img border="0" src="../images/delta_0p.gif" align="absbottom" width="14" height="18"> with respect to the value <img border="0" src="../images/delta_0s.gif" align="absbottom" width="12" height="18"><i> </i>of the underlier:</strong></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center"> <img border="0" src="../images/delta_4.gif" width="109" height="50"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">By incorporating gamma, we can improve our approximation [<a href="#[2]">2</a>] for how the portfolio's value should change in response to small changes in the underlier's value:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table15"> <tr> <td width="100%" align="center"> <img border="0" src="../images/delta_5.gif" width="200" height="41"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body"><strong style="font-weight: 400">This is called the </strong><strong><a name="delta-gamma approximation">delta-gamma approximation</a></strong><strong style="font-weight: 400">.</strong></p> <p class="body"><strong style="font-weight: 400">While sensitivity to the value of an underlier may be a significant determinant of a derivative portfolio's  risk, other sensitivities are also important. These include sensitivities to implied volatilities, interest rates, and the passage of time. There are Greek factor sensitivities for each of these, called: <a href="../articles/vega.htm">vega</a>, <a href="../articles/rho.htm">rho</a>, and <a href="../articles/theta.htm">theta</a>, respectively. </strong></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"><a name="return1">Exercises</a></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%"> <p class="body" style="margin-left: 10; margin-top: 8; margin-bottom: 16"> <img border="0" src="../images/bullet_exercise.gif" align="absbottom" width="19" height="17">A trader has a portfolio of options on crude oil. The portfolio’s delta is 1,300 barrels. Suppose the price of crude oil immediately rose by USD 1.20, and all other variables remained unchanged. Use [<a href="#[2]">2</a>] to approximate what profit/loss the trader will realize from the <a name="return2">move</a>. [<a href="../solutions/delta_and_gamma_1.htm">solution</a>]<p class="body" style="margin-left: 10; margin-top: 8; margin-bottom: 8"> <img border="0" src="../images/bullet_exercise.gif" align="absbottom" width="19" height="17">The following graphs depict, for three hypothetical portfolios, each portfolio's market value as a function of some underlier's value. In each graph, the underlier's current value is indicated with a gold triangle and a vertical line. For each graph, decide if <p style="margin-left: 10; margin-top: 6; margin-bottom: 6">1. the portfolio’s delta is positive, zero or negative, and </p> <p style="margin-left: 10; margin-top: 6; margin-bottom: 6">2. the portfolio’s gamma is positive, zero or negative.</p> <p style="margin-left: 10; margin-top: 6; margin-bottom: 6"> <img border="0" src="../images/ex_exercise_2_delta_and_gamma.gif" width="350" height="661"><br> [<a href="../solutions/delta_and_gamma_2.htm">solution</a>]</p> </td> </tr> </table> </td> </tr> <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="table18"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/delta_hedge.htm">delta hedge</a> A type of hedge that is widely used by derivative dealers to reduce or eliminate a portfolio's exposure to some underlier.</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/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 align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/directional_strategy.htm">directional strategy</a> A trading or investment strategy that entails taking net long or short positions in a market.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/duration_and_convexity.htm">duration and convexity</a> Risk metrics employed in fixed income markets.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/dynamic_hedging.htm">dynamic hedging</a> A technique that is widely used by derivatives dealers to hedge gamma or vega exposures.<p class="explanatory_text" align="left"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/greeks.htm"><strong style="font-weight: 400">Greeks</strong></a><strong style="font-weight: 400"> A set of factor sensitivities, which includes delta and gamma.</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> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/options_spread.htm">option spreads</a> Positions combining one or more options in a single underlier.</p> <p class="explanatory_text" align="left"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/put_call_parity.htm">p</a></strong><a href="../articles/put_call_parity.htm" target="main"><strong style="FONT-WEIGHT: 400">ut-call parity</strong></a><strong style="FONT-WEIGHT: 400"> A formula that relates the price of a put to the price of a corresponding call.</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/rho.htm">rho</a> Factor sensitivity measuring a portfolio's first order (linear) sensitivity to the risk-free rate.</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/theta.htm">theta</a> Factor sensitivity measuring a portfolio's first order (linear) sensitivity to the passage of time</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/vega.htm">vega</a> Factor sensitivity measuring a portfolio's first order (linear) sensitivity to the implied volatility of an underlier</strong></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_1.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%">  <p class="ad_legend" style="margin-left: 0; 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