Option Pricing Theory

<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">Option Pricing Theory</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="#Black-Scholes theory">Black-Scholes theory</a><p class="word_list"> <a href="#derivatives pricing theory">derivatives pricing theory</a><p class="word_list"> <a href="#differential equations approach">differential equations approach</a><p class="word_list"> <a href="#financial engineers">financial engineer</a><p class="word_list"> <a href="#financial engineering">financial engineering</a><p class="word_list"> <a href="#Option pricing theory">option pricing theory</a><p class="word_list"> <a href="#quants">quant</a><p class="word_list"> <a href="#rocket scientists">rocket scientist</a><p class="word_list"> <a href="#risk neutral valuation">risk neutral valuation</a><p class="word_list"> <a href="#stochastic calculus approach">stochastic calculus approach</a><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"> <a href="../articles/option.htm">Options</a> have existed—at least in concept—since antiquity. It wasn't until publication of the <a href="../articles/black_scholes_1973.htm">Black-Scholes (1973) option pricing formula</a> that a theoretically consistent framework for pricing options became available. That framework was a direct result of work by Robert Merton as well as Black and Scholes. In 1997, Scholes and Merton won the Nobel Prize in economics for this work. Black had died in 1995, but otherwise would have shared the prize. </strong></p> <p class="body"><a name="Option pricing theory"><b>Option pricing theory</b></a>—also called <b><a name="Black-Scholes theory">Black-Scholes theory</a></b> or <b><a name="derivatives pricing theory">derivatives pricing theory</a></b>—traces its roots to <a href="../articles/brownian_motion.htm">Bachelier</a> (<a href="#Bachelier">1900</a>) who invented <a href="../articles/brownian_motion.htm">Brownian motion</a> to model options on French government bonds. This work anticipated by five years Einstein's independent use of Brownian motion in physics.</p> <p class="body">Research picked up in the 1960's. Typical of efforts during this period is Samuelson (<a href="#Samuelson">1965</a>). He considered long-term <a href="../articles/common_stock.htm">equity</a> options, and <strong> <span style="font-weight: 400">u</span></strong>s<strong><span style="font-weight: 400">e</span></strong>d<span style="font-weight: 400"><strong> </strong></span>g<strong><span style="font-weight: 400">e</span></strong>o<strong><span style="font-weight: 400">m</span></strong>e<strong><span style="font-weight: 400">t</span></strong>r<strong><span style="font-weight: 400">i</span></strong>c<strong><span style="font-weight: 400"> </span></strong>B<strong><span style="font-weight: 400">r</span></strong>o<strong><span style="font-weight: 400">w</span></strong>n<strong><span style="font-weight: 400">i</span></strong>a<strong><span style="font-weight: 400">n</span></strong> <strong><span style="font-weight: 400">m</span></strong>o<strong><span style="font-weight: 400">t</span></strong>i<strong><span style="font-weight: 400">o</span></strong>n<strong><span style="font-weight: 400"> </span></strong>t<strong><span style="font-weight: 400">o</span></strong> <strong><span style="font-weight: 400">m</span></strong>o<strong><span style="font-weight: 400">d</span></strong>e<strong><span style="font-weight: 400">l</span></strong> <strong><span style="font-weight: 400">t</span></strong>h<strong><span style="font-weight: 400">e</span></strong> <strong><span style="font-weight: 400">r</span></strong>a<strong><span style="font-weight: 400">n</span></strong>d<strong><span style="font-weight: 400">o</span></strong>m<strong><span style="font-weight: 400"> </span></strong>b<strong><span style="font-weight: 400">e</span></strong>h<strong><span style="font-weight: 400">a</span></strong>v<strong><span style="font-weight: 400">i</span></strong>o<strong><span style="font-weight: 400">r</span></strong> <strong><span style="font-weight: 400">o</span></strong>f<strong><span style="font-weight: 400"> </span></strong>t<strong><span style="font-weight: 400">h</span></strong>e<strong><span style="font-weight: 400"> </span></strong>u<strong><span style="font-weight: 400">n</span></strong>d<strong><span style="font-weight: 400">e</span></strong>r<strong><span style="font-weight: 400">l</span></strong>y<strong><span style="font-weight: 400">i</span></strong>ng<strong><span style="font-weight: 400"> </span></strong><a href="../articles/common_stock.htm">s<strong><span style="font-weight: 400">t</span></strong>o<strong><span style="font-weight: 400">c</span></strong>k</a><strong><span style="font-weight: 400">.</span></strong> <strong><span style="font-weight: 400">B</span></strong>a<strong><span style="font-weight: 400">s</span></strong>e<strong><span style="font-weight: 400">d</span></strong> <strong><span style="font-weight: 400">u</span></strong>p<strong><span style="font-weight: 400">o</span></strong>n<strong><span style="font-weight: 400"> </span></strong>t<strong><span style="font-weight: 400">h</span></strong>i<strong><span style="font-weight: 400">s</span></strong>,<strong><span style="font-weight: 400"> </span></strong>h<strong><span style="font-weight: 400">e</span></strong> <strong><span style="font-weight: 400">m</span></strong>o<strong><span style="font-weight: 400">d</span></strong>e<strong><span style="font-weight: 400">l</span></strong>e<span style="font-weight: 400">d<strong> </strong></span>t<strong><span style="font-weight: 400">h</span></strong>e<strong><span style="font-weight: 400"> </span></strong>r<strong><span style="font-weight: 400">a</span></strong>n<strong><span style="font-weight: 400">d</span></strong>o<strong><span style="font-weight: 400">m</span></strong> <strong><span style="font-weight: 400">value of the option at <a href="../articles/option.htm">exercise</a>. The model required two assumptions. The first was the <a href="../articles/mean.htm">expected</a> <a href="../articles/return.htm">rate of return</a> <font face="Times New Roman">α</font> for the stock price. The second was the rate <font face="Times New Roman">β at which the option's value at exercise should be discounted back to the pricing date. These two factors depended upon the unique risk characteristics of, respectively, the <a href="../articles/derivative_instrument.htm">underlying</a> stock and the option. Neither factor was observable in the market place. Depending upon their degree of <a href="../articles/risk_aversion.htm">risk aversion</a>, different observers might propose different values for the factors. Accordingly, Samuelson's formula was largely arbitrary. It offered no means for a buyer and seller with different risk aversions to agree on a price for an option. </font> </span></strong>Black and Scholes got around the problem with a completely new approach.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table6"> <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">Consider an options trader who is about to sell an option. She intends to <a href="../articles/dynamic_hedging.htm">dynamically hedge</a> the exposure until the option expires. What price should she charge for the option? Black and Scholes propose that she charge the cost of dynamically hedging the short option. Significantly, given certain simplifying assumptions, they proposed that this cost could be known in advance.</p> <p class="body"><strong style="font-weight: 400"><font face="Times New Roman">In the real world, dynamic hedging is an uncertain undertaking. Depending upon the <a href="../articles/derivative_instrument.htm">underlier</a> and your hedging strategy, you might adjust your hedge a few times a week, or a few times a day. Between those adjustments, a lot can happen. If the underlier moves, you might rehedge immediately, or you might wait. If you wait, the underlier might move back, saving you the need to rehedge, or the underlier might keep moving in the same direction, causing a large loss. Real world dynamic hedging entails risk.</font></strong></p> <p class="body"><strong style="font-weight: 400"><font face="Times New Roman">As a practical matter, there is a limit to how frequently a trader can rehedge, however, as the frequency of rehedging increases, dynamic hedging becomes more predictable. Using stochastic calculus and certain simplifying assumptions, Black and Scholes took the limiting case as the frequency of rehedging approaches infinity. In that limiting case, the cost of dynamic hedging is independent of the actual path taken by the underlier's price. It depends only upon the price's <a href="../articles/volatility.htm">volatility</a>. If that volatility is constant and known in advance, the cost of dynamic hedging a <a href="../articles/short_sale.htm">short</a> option is certain. Being certain, it entails no risk, so it can be discounted at a <a href="../articles_old/glossaryriskfreerate.htm">risk free rate</a> to obtain the price of the option.</font></strong></p> <p class="body">Based upon this approach, Black and Scholes derived a partial differential equation for valuing claims contingent on a traded underlier. The equation is general. By applying different boundary conditions, it can be solved to price any such contingent claim. Black and Scholes applied the boundary conditions for a <a href="../articles/option.htm">European call option</a> on a <a href="../articles/common_stock.htm">non-dividend-paying</a> stock and obtained their famous (<a href="#Black">1973</a>) option pricing formula.</p> <p class="body"> <span style="font-size: 12.0pt; font-family: Times New Roman; color: black">John C. Cox and Stephen A. Ross</span> made an important contribution with their method of <b><a name="risk neutral valuation">risk neutral valuation</a></b>. Consider again Samuelson's (<a href="#Samuelson">1965</a>) approach to pricing options, where he modeled an underlying stock price with some expected <a href="../articles/return.htm">return</a> <font face="Times New Roman"> α and discounted option values at exercise back to the pricing date with some rate β. There was nothing theoretically wrong with such an approach. Indeed, if it were based upon the same assumptions as the Black-Scholes approach, it would produce consistent option prices. This lead Cox and Rubinstein to an interesting conclusion. The two approaches were equivalent, yet one required α and β as inputs whereas the other did not. They concluded that the effects of α and β must somehow cancel. As long as α and β reflect the same degree of risk aversion, they do not affect the option price—<i>and this must be true no matter what degree of risk aversion they reflect</i>. If they can reflect any degree of risk aversion and still yield correct option prices, then they can be based upon an assumption of no risk aversion whatsoever. If an investor were <a href="../articles/risk_aversion.htm">risk neutral</a>, he would require no excess return for taking risk. He would discount all cash flows—irrespective of their risk—at the risk free rate. The factors α and β would both equal the risk free rate. This brilliant insight was the Cox and Rubinstein risk neutral approach to option pricing. </font></p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table7"> <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">The risk neutral approach opened the door to a host of option valuation techniques that used binomial trees or the Monte Carlo method to model future asset values. Rather than attempt to ascribe "realistic" expected returns and "realistic" discount rates in the analyses, users could treat all financial assets as having expected returns equal to the risk free rate. They could discount all cash flows at the risk free rate. The risk neutral assumption is not reflective of the real world. Real investors are not risk neutral, but this doesn't matter. Correctly implemented, the risk neutral assumption produces correct option prices.</p> <p class="body"> <font face="Times New Roman"> Cox and Ross didn't immediately perceive how profound risk neutral valuation would be. They buried it in the middle of a </font>(<a href="#Cox, John C. and Stephen A. Ross">1976</a>) paper on pricing options with jump processes. But three years later, they teamed up with <span style="font-size: 12.0pt; font-family: Times New Roman; color: black"> Mark Rubinstein to publish a (<a href="#Cox, John C., Stephen A. Ross and Mark Rubinstein">1979</a>) paper that used risk neutral valuation to develop the method of binomial trees. The mathematics of risk neutral valuation was formalized in continuous time by other authors to become the method of equivalent martingale measures. Today, this is the predominant </span> <span style="font-family: Times New Roman">methodology for derivatives pricing in complete markets.</span></p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table8"> <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">Work on financial engineering spawned the  field of <a name="financial engineering"> <b>financial engineering</b></a>. Practitioners, called <b><a name="financial engineers">financial engineers</a></b>, design and implement derivatives pricing models. Top financial engineers are highly paid professionals who typically hold advanced degrees in mathematics or physics. Financial engineers are informally called <b> <a name="quants">quants</a></b> or <b><a name="rocket scientists">rocket scientists</a></b>.</p> <p class="body">The Black Scholes approach and generalizations employ partial differential equations, so they are sometimes called the <b> <a name="differential equations approach">differential equations approach</a></b>. Those differential equations often have <a href="../articles/closed_form_solution.htm">closed-form solutions</a>, leading to simple pricing formulas such as the original Black-Scholes (<a href="#Black">1973</a>) formula. Other times, the differential equations need to be solved numerically using techniques such as the <a href="../articles/monte_carlo_method.htm">Monte Carlo method</a>. </p> <p class="body">The risk neutral approach tends to entail extensive use of stochastic calculus with changes of measure between a "real world" and a "risk neutral" world. For this reason, it (and analogous approaches) tend to be called the <b><a name="stochastic calculus approach">stochastic calculus approach</a></b>. It can lead to closed form solutions, but <a href="../articles/closed_form_solution.htm">numerical solutions</a> tend to be the norm. It is more flexible than the Black-Scholes approach. Sometimes, it can be used to price derivatives that the Black-Scholes approach cannot.</p> <p class="body">Techniques of financial engineering have been extended to fixed income derivatives, which generally require the modeling of entire <a href="../articles/fixed_income_term_structure.htm">term structures</a>. They have also been extended to commodities markets, where risk neutral valuation becomes problematic.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table9"> <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="table10"> <tr> <td width="100%"> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><span class="verdana10"><strong style="font-weight: 400"><a href="../articles/arbitrage_free_pricing.htm">arbitrage-free pricing</a></strong></span><strong style="font-weight: 400"> The approach to pricing instruments that underlies essentially all of financial engineering.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/black_1976.htm">Black (1976)</a> option pricing formula Used to price options on forwards.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/black_scholes_1973.htm">Black-Scholes (1973) option pricing formula</a> Used for pricing options on non-dividend paying stocks.</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">derivative instrument</a> 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"><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"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/EMM.htm">fundamental theorem of asset pricing</a> A theorem that relates the existence of an equivalent martingale measure to the no-arbitrage condition and completeness of markets.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/garman_kohlhagen_1983.htm">Garman and Kohlhagen (1983) option pricing formula</a> Used for pricing foreign exchange options.</p> <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 used to measure risk exposures related to options or other derivatives.</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/law_of_one_price.htm">law of one price</a> The notion that, if two assets have identical cash flows, they should have the same market value.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/merton_1973.htm">Merton (1973) option pricing formula</a> Used to price European options on dividend paying stocks or stock indexes.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/model_risk.htm">model risk</a> The risk that models are applied to tasks for which they are inappropriate or are otherwise implemented incorrectly.</p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="http://www.riskglossary.com/articles/option.htm"><strong style="font-weight: 400">option</strong></a><strong style="font-weight: 400"> A type of derivative instrument.</strong></p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/option_adjusted_spread.htm">option-adjusted spread</a> Yield spread not attributable to embedded options.</p> <p class="explanatory_text" align="left"> <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.</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 target="main" href="../articles/put_call_parity.htm">put-call parity</a> 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/time_value_and_intrinsic_value.htm">time value and intrinsic value</a> </strong> The two components that comprise an option's market value.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/valuation.htm">valuation</a> Article about book value and market value accounting.</p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/volatility.htm">volatility</a> A metric of  variability in a stochastic process.</strong></p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/volatility_skew.htm">volatility skew</a> A condition where implied volatilities vary by strike.</strong></p> </td> </tr> </table> </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="table11"> <tr> <td width="100%"> <p class="explanatory_text">To learn about the historical origins of option pricing theory, Bernstein (<a target="_blank" href="http://www.riskbook.com/link/bernstein_p_(1993).htm">1993</a>) offers a nice overview of 20th century finance. Mehrling (<a target="_blank" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm">2005</a>) and Lehman (<a target="_blank" href="http://www.riskbook.com/link/lehmann_bruce_(2005).htm">2004</a>) offer more detailed accounts of Fischer Black and his associates' contributions. Haug (<a target="_blank" href="http://www.riskbook.com/link/haug_e_(1997).htm">1997</a>) is a handy encyclopedia of published option pricing formulas. Chriss (<a target="_blank" href="http://www.riskbook.com/link/chriss_n_(1997).htm">1997</a>) is an excellent introduction to option pricing theory and financial engineering. It is a bit dated, presenting the subject more as it was first developed by Black, Scholes, Merton and contemporaries. Once you have read Chriss, proceed to more modern treatments by Back (<a target="_blank" href="http://www.riskbook.com/link/back_kerry_(2005).htm">2005</a>) and Bingham and Kiesel (<a target="_blank" href="http://www.riskbook.com/link/bingham_and_kiesel_(2004).htm">2004</a>).</p> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table12"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap559"> <area target="_blank" href="http://www.riskbook.com/link/bernstein_p_(1993).htm" shape="rect" coords="0, 0, 81, 124"> </map> <img border="0" src="../images/book_bernstein_peter_(1993)_125.jpg" usemap="#FPMap559" width="82" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table13"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/bernstein_p_(1993).htm"> Capital Ideas</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Peter L. Bernstein</b></td> </tr> <tr> <td width="10%"> <map name="FPMap560"> <area target="_blank" coords="2, 1, 89, 15" shape="rect" href="http://www.riskbook.com/main/ratings.htm"> </map> <img border="0" src="../images/star_6.gif" usemap="#FPMap560" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <map name="FPMap561"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_0.gif" usemap="#FPMap561" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">1993</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap562"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/bernstein_p_(1993).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap562" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table14"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap739"> <area target="_blank" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_mehrling_perry_(2005)_125.jpg" usemap="#FPMap739" width="83" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table15"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm"> Fischer Black</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Perry Mehrling</b></td> </tr> <tr> <td width="10%"><map name="FPMap737"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/star_4.gif" usemap="#FPMap737" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"><map name="FPMap738"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_1.gif" usemap="#FPMap738" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2005</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap666"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/mehrling_perry_(2005).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap666" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table16"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap740"> <area target="_blank" href="http://www.riskbook.com/link/lehmann_bruce_(2005).htm" shape="rect" coords="0, 0, 83, 124"> </map> <img border="0" src="../images/book_lehmann_bruce_(2005)_125.gif" usemap="#FPMap740" width="84" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table17"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/lehmann_bruce_(2005).htm"> The Legacy of Fischer Black</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Bruce N. Lehmann (ed.)</b></td> </tr> <tr> <td width="10%"> <map name="FPMap741"> <area target="_blank" href="http://www.riskbook.com/link/bodie_kane_marcus_(2002).htm" shape="rect" coords="0, 0, 98, 124"> </map> <img border="0" src="../images/star_5.gif" usemap="#FPMap741" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <map name="FPMap742"> <area target="_blank" coords="2, 1, 89, 15" shape="rect" href="http://www.riskbook.com/main/ratings.htm"> </map> <img border="0" src="../images/tech_2.gif" usemap="#FPMap742" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2004</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap667"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/lehmann_bruce_(2005).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap667" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table18"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap496"> <area target="_blank" href="http://www.riskbook.com/link/haug_e_(1997).htm" shape="rect" coords="0, 0, 100, 124"> </map> <img border="0" src="../images/book_haug_espen_(1997)_125.gif" usemap="#FPMap496" width="101" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table19"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/haug_e_(1997).htm"> <font size="2">Complete Guide to </font><br> Option Pricing Formulas </a></td> </tr> <tr> <td colspan="3"> <p class="author"><b><span class="verdana10bold">Espen G. Haug </span></b></td> </tr> <tr> <td width="10%"><map name="FPMap563"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/star_5.gif" usemap="#FPMap741" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"><map name="FPMap564"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_2.gif" usemap="#FPMap742" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">1997</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap366"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/haug_e_(1997).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap366" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table20"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap536"> <area target="_blank" href="http://www.riskbook.com/link/chriss_n_(1997).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_chriss_n_(1997)_125.jpg" usemap="#FPMap536" width="83" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table21"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/chriss_n_(1997).htm"> Black-Scholes and Beyond</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Neil A. Chriss</b></td> </tr> <tr> <td width="10%"> <map name="FPMap537"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"></map> <img border="0" src="../images/star_6.gif" usemap="#FPMap537" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"><map name="FPMap538"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_2.gif" usemap="#FPMap538" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">1997</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap539"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/chriss_n_(1997).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap539" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table22"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap664"> <area target="_blank" href="http://www.riskbook.com/link/back_kerry_(2005).htm" shape="rect" coords="0, 0, 82, 124"> </map> <img border="0" src="../images/book_back_kerry_(2005)_125.gif" usemap="#FPMap664" width="83" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table23"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/back_kerry_(2005).htm"> A Course in Derivative Securities</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>Kerry Back</b></td> </tr> <tr> <td width="10%"><map name="FPMap661"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/star_5.gif" usemap="#FPMap661" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"><map name="FPMap662"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_4.gif" usemap="#FPMap662" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2005</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap665"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/back_kerry_(2005).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap665" width="146" height="20"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="4" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table24"> <tr> <td width="100" align="left" valign="top"> <map name="FPMap663"> <area target="_blank" href="http://www.riskbook.com/link/bingham_and_kiesel_(2004).htm" shape="rect" coords="0, 0, 83, 124"> </map> <img border="0" src="../images/book_bingham_and_kiesel_125.gif" usemap="#FPMap663" width="84" height="125"></td> <td align="left" valign="top"> <table border="0" cellpadding="1" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table25"> <tr> <td width="90%" colspan="3"> <p class="title"> <a target="_blank" href="http://www.riskbook.com/link/bingham_and_kiesel_(2004).htm"> Risk Neutral Valuation</a></td> </tr> <tr> <td colspan="3"> <p class="author"><b>N. H. Bingham and Rudiger Kiesel</b></td> </tr> <tr> <td width="10%"> <map name="FPMap645"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"></map> <img border="0" src="../images/star_6.gif" usemap="#FPMap645" width="90" height="16"></td> <td width="10%" class="author">quality</td> <td width="70%"> <p class="author"> </td> </tr> <tr> <td width="10%"> <map name="FPMap646"> <area target="_blank" href="http://www.riskbook.com/main/ratings.htm" shape="rect" coords="0, 0, 89, 15"> </map> <img border="0" src="../images/tech_5.gif" usemap="#FPMap646" width="90" height="16"></td> <td width="10%" class="author">technical</td> <td width="70%">  </td> </tr> <tr> <td width="10%"> <p class="author">2004</td> <td width="10%"> </td> <td width="70%"> <p align="center"><map name="FPMap37"> <area target="_blank" coords="0, 0, 145, 19" shape="rect" href="http://www.riskbook.com/link/bingham_and_kiesel_(2004).htm"> </map> <img border="0" src="../images/pointer_review.gif" usemap="#FPMap37" width="146" height="20"></td> </tr> </table> </td> </tr> </table> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Cited Papers</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="table26"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Bachelier">Bachelier</a>, Louis (1964). Theory of Speculation, <i>The Random Character of Stock Prices</i>, Paul H. Cootner (editor), Cambridge: MIT. Translated from the 1900 doctoral thesis.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Samuelson">Samuelson</a>, Paul A. (1965). Rational theory of warrant pricing, <i>Industrial Management Review</i>, 6, 13-31.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Black">Black</a>, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Merton">Merton</a>, Robert C. (1973). Theory of rational option pricing, <i>Bell Journal of Economics and Management Science</i>, 4 (1), 141-183. Available in Merton (1990).</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Cox, John C. and Stephen A. Ross">Cox, John C. and Stephen A. Ross</a> (1976). The valuation of options for alternative stochastic processes, <i>Journal of Financial Economics</i>, 3, 145-166.<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Cox, John C., Stephen A. Ross and Mark Rubinstein">Cox, John C., Stephen A. Ross and Mark Rubinstein</a> (1979). Option Pricing: A Simplified Approach, <i>Journal of Financial Economics</i> 7, 229-263.</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; 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="I2"> <![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 height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Forum Discussions</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="table27"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_bullhorn_narrow.gif" align="left" width="27" height="22"><a target="_top" href="http://www.contingencyanalysis.com/archive/link_thread/06-2/00000384.htm">Delta risk of forward strike options</a> <i>12 Apr 2006</i><br> An interesting option pricing puzzle.<p class="explanatory_text"> <img border="0" src="../images/bullet_bullhorn_narrow.gif" align="left" width="27" height="22"><a target="_top" href="http://www.contingencyanalysis.com/archive/link_thread/03-2/00004789.htm">Valuation of Options</a> <i>01 Apr 2003</i><br> Modeling bond options.<p class="explanatory_text"> <img border="0" src="../images/bullet_bullhorn_narrow.gif" align="left" width="27" height="22"><a target="_top" href="http://www.contingencyanalysis.com/archive/link_thread/02-3/000003d9.htm">transition from fundamental to quant analysis</a> <i>11 Aug 2002</i><br> Career advice: transitioning from equity analysis to financial engineering.<p class="explanatory_text"> <img border="0" src="../images/bullet_bullhorn_narrow.gif" align="left" width="27" height="22"><a target="_top" href="http://www.contingencyanalysis.com/archive/link_thread/00-3/00000981.htm">Pricing European Options - 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