Numerical Solution, Closed-Form Solution

<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"><font style="font-size: 17pt">Numerical Solution, Closed-Form Solution</font></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="#closed-form solutions">closed-form solution</a><p class="word_list"> <a href="#numerical methods">numerical method</a><p class="word_list"> <a href="#closed form expression">closed form expression</a><p class="word_list"> <a href="#numerical solution">numerical solution</a><p class="word_list"> <a href="#quadratic formula">quadratic formula</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">In quantitative fields—such as finance or  physics—practitioners construct mathematical models. Models take many forms, but they generally comprise a <a href="../articles/set.htm">set</a> of assumptions that are formalized with mathematical equations. Practitioners use models to make predictions. In finance, a model might be used to predict market volatility. In physics, it might be used to predict a force of impact. </p> <p class="body">In some cases, a model will imply a specific formula that may be used to calculate such outputs. The practitioner plugs inputs into the formula and directly obtains the desired output. Such formulas are called <b> <a name="closed-form solutions">closed-form solutions</a></b>. A closed-form solution (or <b><a name="closed form expression">closed form expression</a></b>) is any formula that can be evaluated in a finite number of standard operations.</p> <p class="body">Consider the formula</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/closed_form_solution_1.gif" width="70" height="44"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">It is of closed-form because it is expressed as a finite number of operations: a logarithm, an addition, a division and a square root. An obvious question is: what is a "standard operation?" There is no general agreement on this. For example, someone might consider a Fourier  transform to be a standard operation, and someone else might not. In this regard, the concept of a closed-form solution is informal. </p> <p class="body">The <a name="[2]">formula</a></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/closed_form_solution_2.gif" width="40" height="50"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">is an infinite sum. Because it entails an infinite number of standard operations, it is not a closed-form solution. Suppose we need to evaluate [2]. We can't possibly perform an infinite number of calculations! One solution is to use our knowledge of calculus to rearrange [2] <a name="[3]">as</a></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"> <i>2x</i></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">Formulas [2] and [3] are equivalent. Formula [3] is a closed-form solution for [2]. In this case, we were fortunate that [2] had a particularly simple form, which allowed us to find a closed-form solution for it. Often, we will be confronted with more complicated formulas, and it will be difficult or impossible to discover a closed-form solution. In that case, we may seek a <b><a name="numerical solution"> numerical solution</a></b> instead. What is a numerical solution? <a name="[4]">Consider</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="../images/closed_form_solution_4.gif" width="55" height="50"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">It is a numerical solution for [<a href="#[2]">2</a>]. It approximates [<a href="#[2]">2</a>] by adding up the first 10,000 terms of that infinite sum. </p> <p class="body">A numerical solution is any approximation that can be evaluated in a finite number of standard operations. Closed form solutions and numerical solutions are similar in that they both can be evaluated with a finite number of standard operations. They differ in that a closed-form solution is exact whereas a numerical solution is only approximate.</p> <p class="body">Compare numerical solution [<a href="#[4]">4</a>] with closed form solution [<a href="#[3]">3</a>]. Obviously, [<a href="#[4]">4</a>] will be more difficult to value than [<a href="#[3]">3</a>]. Numerical solutions often involve extensive numerical calculations. Before the advent of computers, this could be prohibitive, and practitioners would strive to find closed-form solutions. Today, computers make it easy to implement numerical solutions that would have been unimaginable in years past.</p> <p class="body">Let's consider a practical example. We want to solve the polynomial equation:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <img border="0" src="../images/closed_form_solution_5.gif" width="105" height="25"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">where <i>a</i>, <i>b</i> and <i>c</i> are known constants. As any high school student knows, there is a closed-form solution to this problem called the <b><a name="quadratic formula">quadratic formula</a></b>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table11"> <tr> <td width="100%" align="center"> <img border="0" src="../images/closed_form_solution_6.gif" width="133" height="51"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">If she has a good teacher, the student will also know that there are generalizations of the quadratic formula that solve third order polynomial equations</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="../images/closed_form_solution_7.gif" width="144" height="25"></td> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">and fourth order polynomial equations</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="../images/closed_form_solution_8.gif" width="182" height="25"></td> <td width="5%" align="right">[8]</td> </tr> </table> <p class="body">Those solutions are complicated formulas, easily filling an entire page, but they exist and are exact, so they are closed-form solutions. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table14"> <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 class="body">What about fifth order polynomial equations? Is there a closed-form solution for them? The answer is "no." This is not a matter of mathematicians looking for but not yet having found a solution. Mathematicians have <i>proven</i> that no closed-form solutions exist for fifth or higher order polynomial equations. Higher-order polynomial equations are solved using numerical methods. A standard solution is Newton's method. This numerical method starts with an initial "guess" for a solution. It then uses that value to come up with a second, better "guess" at a solution. It then performs the same operation on the second "guess" to come up with a third, even better "guess." Continuing iteratively in this manner, Newton's method rapidly converges to an excellent approximation for a solution.</p> <p class="body">If you are confronted with a math problem and can't find a closed-form solution, how do you find a numerical solution? Fortunately, there is a vast literature on this topic that covers a number of standard approaches for constructing numerical solutions. Those general approaches are called <b><a name="numerical methods">numerical methods</a></b>. For example, Newton's method is a numerical method. When it is applied to a specific polynomial equation, it becomes a numerical solution for that particular equation. </p> <p class="body">The <a href="../articles/monte_carlo_method.htm">Monte Carlo method</a> is another standard numerical method. In finance, it is widely used by <a href="../articles/risk_management.htm">risk managers</a> to solve problems related to quantifying <a href="../articles/risk.htm"> risk</a>. It is also used by <a href="../articles/option_pricing_theory.htm">financial engineers</a> to price <a href="../articles/derivative_instrument.htm">derivatives</a>. Other numerical methods used by financial engineers are binomial trees and finite differences. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table15"> <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="table16"> <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/monte_carlo_method.htm">Monte Carlo method</a> Any numerical method that employs statistical sampling to solve problems.</strong></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_3.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%">  <p class="ad_legend" style="margin-left: 0; margin-bottom:2"> <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=82436;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=82436;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82436"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82436;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </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%" bgcolor="#FFFFFF"> <p align="center" style="margin-top: 18"><map name="FPMap365"> <area target="_top" href="http://www.riskglossary.com" shape="rect" coords="0, 0, 164, 102"> </map><img border="0" src="../images/logo.jpg" usemap="#FPMap365" width="165" height="103"></p> <p align="center"><span class="verdana10"> <a href="http://www.riskglossary.com/main/disclaimer.htm" target="_self"> Disclaimer</a></span></p> <p align="center">website: <span class="verdana10"> <a target="_top" href="http://www.contingencyanalysis.com"> http://www.contingencyanalysis.com</a></span><br> glossary direct link: <span class="verdana10"> <a target="_top" href="http://www.riskglossary.com"> http://www.riskglossary.com</a></span><br> copyright © Contingency Analysis, 2005<p align="center"><map name="FPMap1"> <area target="_top" href="http://www.contingencyanalysis.com" shape="rect" coords="0, 0, 240, 88"> </map><img border="0" src="../images/logo_glossary_white.gif" usemap="#FPMap1" width="241" height="89"></td> </tr> </table> </td> </tr> </table> </td> <td>  </td> </tr> </table> </div> </body>