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class="vector-toc-numb">2</span> <span>Intrinsic value and time value</span> </div> </a> <ul id="toc-Intrinsic_value_and_time_value-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moneyness_terms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moneyness_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Moneyness terms</span> </div> </a> <button aria-controls="toc-Moneyness_terms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Moneyness terms subsection</span> </button> <ul id="toc-Moneyness_terms-sublist" class="vector-toc-list"> <li id="toc-At_the_money" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#At_the_money"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>At the money</span> </div> </a> <ul id="toc-At_the_money-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_the_money" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_the_money"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>In the money</span> </div> </a> <ul id="toc-In_the_money-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Out_of_the_money" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Out_of_the_money"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Out of the money</span> </div> </a> <ul id="toc-Out_of_the_money-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spot_versus_forward" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spot_versus_forward"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Spot versus forward</span> </div> </a> <ul id="toc-Spot_versus_forward-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Use" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Use"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Use</span> </div> </a> <ul id="toc-Use-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Definition</span> </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Moneyness_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moneyness_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Moneyness function</span> </div> </a> <ul id="toc-Moneyness_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conventions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conventions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Conventions</span> </div> </a> <ul id="toc-Conventions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Simple examples</span> </div> </a> <ul id="toc-Simple_examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Black–Scholes_formula_auxiliary_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Black–Scholes_formula_auxiliary_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Black–Scholes formula auxiliary variables</span> </div> </a> <ul id="toc-Black–Scholes_formula_auxiliary_variables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of 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role="note" class="hatnote navigation-not-searchable">"In the money" redirects here. For the poker term, see <a href="/wiki/In_the_money_(poker)" class="mw-redirect" title="In the money (poker)">In the money (poker)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Difference in the price of an underlying asset and its derivative's strike price</div> <p>In <a href="/wiki/Finance" title="Finance">finance</a>, <b>moneyness</b> is the relative position of the current <a href="/wiki/Price" title="Price">price</a> (or future price) of an underlying <a href="/wiki/Financial_asset" title="Financial asset">asset</a> (e.g., a <a href="/wiki/Stock" title="Stock">stock</a>) with respect to the <a href="/wiki/Strike_price" title="Strike price">strike price</a> of a <a href="/wiki/Derivative_(finance)" title="Derivative (finance)">derivative</a>, most commonly a <a href="/wiki/Call_option" title="Call option">call option</a> or a <a href="/wiki/Put_option" title="Put option">put option</a>. Moneyness is firstly a three-fold classification: </p> <ul><li>If the derivative would have positive <a href="/wiki/Intrinsic_value_(finance)" title="Intrinsic value (finance)">intrinsic value</a> if it were to <a href="/wiki/Expiration_(options)" title="Expiration (options)">expire</a> today, it is said to be <b>in the money</b> (<b>ITM</b>);</li> <li>If the derivative would be worthless if expiring with the underlying at its current price, it is said to be <b>out of the money</b> (<b>OTM</b>);</li> <li>And if the current underlying price and strike price are equal, the derivative is said to be <b>at the money</b> (<b>ATM</b>).</li></ul> <p>There are two slightly different definitions, according to whether one uses the current price (spot) or future price (forward), specified as "at the money spot" or "at the money forward", etc. </p><p>This rough classification can be quantified by various <a href="#Definition">definitions</a> to express the moneyness as a number, measuring how far the asset is in the money or out of the money with respect to the strike – or, conversely, how far a strike is in or out of the money with respect to the spot (or forward) price of the asset. This quantified notion of moneyness is most importantly used in defining the <i>relative</i> <a href="/wiki/Volatility_surface" class="mw-redirect" title="Volatility surface">volatility surface</a>: the implied <a href="/wiki/Volatility_(finance)" title="Volatility (finance)">volatility</a> in terms of moneyness, rather than absolute price. The most basic of these measures is <b>simple moneyness</b>, which is the ratio of spot (or forward) to strike, or the reciprocal, depending on convention. A particularly important measure of moneyness is the likelihood that the derivative will expire in the money, in the <a href="/wiki/Risk-neutral_measure" title="Risk-neutral measure">risk-neutral measure</a>. It can be measured in percentage <a href="/wiki/Probability" title="Probability">probability</a> of expiring in the money, which is the forward value of a <a href="/wiki/Binary_call_option" class="mw-redirect" title="Binary call option">binary call option</a> with the given strike, and is equal to the auxiliary <i>N</i>(<i>d</i><sub>2</sub>) term in the <a href="/wiki/Black%E2%80%93Scholes_formula" class="mw-redirect" title="Black–Scholes formula">Black–Scholes formula</a>. This can also be measured in <a href="/wiki/Standard_deviations" class="mw-redirect" title="Standard deviations">standard deviations</a>, measuring how far above or below the strike price the current price is, in terms of volatility; this quantity is given by <i>d</i><sub>2</sub>. (Standard deviations refer to the price fluctuations of the underlying instrument, not of the option itself.) Another measure closely related to moneyness is the <a href="/wiki/Delta_(finance)" class="mw-redirect" title="Delta (finance)">Delta</a> of a call or put option. There are other proxies for moneyness, with convention depending on market.<sup id="cite_ref-neftci_1-0" class="reference"><a href="#cite_note-neftci-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=1" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose the current stock price of <a href="/wiki/IBM" title="IBM">IBM</a> is $100. A <a href="/wiki/Call_option" title="Call option">call</a> or <a href="/wiki/Put_option" title="Put option">put option</a> with a strike of $100 is at-the-money. A <a href="/wiki/Call_option" title="Call option">call</a> with a strike of $80 is in-the-money (100 − 80 = 20 > 0). A <a href="/wiki/Put_option" title="Put option">put option</a> with a strike at $80 is out-of-the-money (80 − 100 = −20 < 0). Conversely, a call option with a $120 strike is out-of-the-money and a put option with a $120 strike is in-the-money. </p><p>The above is a traditional way of defining ITM, OTM and ATM, but some new authors find the comparison of strike price with current market price meaningless and recommend the use of Forward Reference Rate instead of Current Market Price. For example, a put option will be in the money if the strike price of the option is greater than the Forward Reference Rate.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Intrinsic_value_and_time_value">Intrinsic value and time value</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=2" title="Edit section: Intrinsic value and time value"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The intrinsic value (or "monetary value") of an option is its value assuming it were exercised immediately. Thus if the current (<a href="/wiki/Spot_price" class="mw-redirect" title="Spot price">spot</a>) price of the underlying security (or commodity etc.) is above the agreed (<a href="/wiki/Strike_price" title="Strike price">strike</a>) price, a <a href="/wiki/Call_option" title="Call option">call</a> has positive intrinsic value (and is called "in the money"), while a <a href="/wiki/Put_option" title="Put option">put</a> has zero intrinsic value (and is "out of the money"). </p><p>The <a href="/wiki/Option_time_value" title="Option time value">time value</a> of an option is the total value of the option, less the intrinsic value. It partly arises from the uncertainty of future price movements of the underlying. A component of the time value also arises from the unwinding of the discount rate between now and the expiry date. In the case of a European option, the option cannot be exercised before the expiry date, so it is possible for the time value to be negative; for an American option if the time value is ever negative, you exercise it (ignoring special circumstances such as the security going ex dividend): this yields a <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary condition</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Moneyness_terms">Moneyness terms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=3" title="Edit section: Moneyness terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="At_the_money">At the money</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=4" title="Edit section: At the money"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Option_(finance)" title="Option (finance)">option</a> is <b>at the money</b> (ATM) if the <a href="/wiki/Strike_price" title="Strike price">strike price</a> is the same as the current spot price of the underlying security. An at-the-money option has no intrinsic value, only time value.<sup id="cite_ref-at-the-money_3-0" class="reference"><a href="#cite_note-at-the-money-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>For example, with an "at the money" call stock option, the current share price and strike price are the same. Exercising the option will not earn the seller a profit, but any move upward in stock price will give the option value. </p><p>Since an option will rarely be exactly at the money, except for when it is written (when one may buy or sell an ATM option), one may speak informally of an option being <b>near the money</b> or <b>close to the money</b>.<sup id="cite_ref-iopntm_4-0" class="reference"><a href="#cite_note-iopntm-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Similarly, given standardized options (at a fixed set of strikes, say every $1), one can speak of which one is <b>nearest the money</b>; "near the money" may narrowly refer specifically to the nearest the money strike. Conversely, one may speak informally of an option being <b>far from the money</b>. </p> <div class="mw-heading mw-heading3"><h3 id="In_the_money">In the money</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=5" title="Edit section: In the money"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b>in the money</b> (ITM) option has positive intrinsic value as well as time value. A call option is in the money when the strike price is below the spot price. A put option is in the money when the strike price is above the spot price. </p><p>With an "in the money" call stock option, the current share price is greater than the strike price so exercising the option will give the owner of that option a profit. That will be equal to the market price of the share, minus the option strike price, times the number of shares granted by the option (minus any commission). </p> <div class="mw-heading mw-heading3"><h3 id="Out_of_the_money">Out of the money</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=6" title="Edit section: Out of the money"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b>out of the money</b> (OTM) option has no intrinsic value. A call option is out of the money when the strike price is above the spot price of the underlying security. A put option is out of the money when the strike price is below the spot price. </p><p>With an "out of the money" call stock option, the current share price is less than the strike price so there is no reason to exercise the option. The owner can sell the option, or wait and hope the price changes. </p> <div class="mw-heading mw-heading2"><h2 id="Spot_versus_forward">Spot versus forward</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=7" title="Edit section: Spot versus forward"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Moneyness" title="Special:EditPage/Moneyness">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">June 2008</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Assets can have a forward price (a price for delivery in future) as well as a spot price. One can also talk about moneyness with respect to the forward price: thus one talks about ATMF, "ATM Forward", and so forth. For instance, if the spot price for USD/JPY is 120, and the forward price one year hence is 110, then a call struck at 110 is ATMF but not ATM. </p> <div class="mw-heading mw-heading2"><h2 id="Use">Use</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=8" title="Edit section: Use"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Buying an ITM option is effectively lending money in the amount of the intrinsic value. Further, an ITM call can be replicated by entering a forward and buying an OTM put (and conversely). Consequently, ATM and OTM options are the main traded ones. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=9" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Moneyness_function">Moneyness function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=10" title="Edit section: Moneyness function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Intuitively speaking, moneyness and time to expiry form a two-dimensional <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> for valuing options (either in currency (dollar) value or in implied volatility), and changing from spot (or forward, or strike) to moneyness is a <a href="/wiki/Change_of_variables" title="Change of variables">change of variables</a>. Thus a <b>moneyness function</b> is a function <i>M</i> with input the spot price (or forward, or strike) and output a real number, which is called the <b>moneyness</b>. The condition of being a change of variables is that this function is monotone (either increasing for all inputs, or decreasing for all inputs), and the function can depend on the other parameters of the <a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a>, notably time to expiry, interest rates, and <a href="/wiki/Implied_volatility" title="Implied volatility">implied volatility</a> (concretely the ATM implied volatility), yielding a function: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(S,K,\tau ,r,\sigma ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>τ</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(S,K,\tau ,r,\sigma ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311339d2b4446435f854e16fcd4dd98624084bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.18ex; height:2.843ex;" alt="{\displaystyle M(S,K,\tau ,r,\sigma ),}"></span></dd></dl> <p>where <i>S</i> is the spot price of the underlying, <i>K</i> is the strike price, <i>τ</i> is the time to expiry, <i>r</i> is the <a href="/wiki/Risk-free_rate" title="Risk-free rate">risk-free rate</a>, and <i>σ</i> is the implied volatility. The forward price <i>F</i> can be computed from the spot price <i>S</i> and the risk-free rate <i>r.</i> All of these are observables except for the implied volatility, which can computed from the observable price using the Black–Scholes formula. </p><p>In order for this function to reflect moneyness – i.e., for moneyness to increase as spot and strike move relative to each other – it must be <a href="/wiki/Monotonic_function" title="Monotonic function">monotone</a> in both spot <i>S</i> and in strike <i>K</i> (equivalently forward <i>F,</i> which is monotone in <i>S</i>), with at least one of these strictly monotone, and have opposite direction: either increasing in <i>S</i> and decreasing in <i>K</i> (call moneyness) or decreasing in <i>S</i> and increasing in <i>K</i> (put moneyness). Somewhat different formalizations are possible.<sup id="cite_ref-hafner42_5-0" class="reference"><a href="#cite_note-hafner42-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Further axioms may also be added to define a "valid" moneyness. </p><p>This definition is abstract and notationally heavy; in practice relatively simple and concrete moneyness functions are used, and arguments to the function are suppressed for clarity. </p> <div class="mw-heading mw-heading3"><h3 id="Conventions">Conventions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=11" title="Edit section: Conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When quantifying moneyness, it is computed as a single number with respect to spot (or forward) and strike, without specifying a reference option. There are thus two conventions, depending on direction: call moneyness, where moneyness increases if spot increases relative to strike, and put moneyness, where moneyness increases if spot decreases relative to strike. These can be switched by changing sign, possibly with a shift or scale factor (e.g., the probability that a put with strike <i>K</i> expires ITM is one minus the probability that a call with strike <i>K</i> expires ITM, as these are complementary events). Switching spot and strike also switches these conventions, and spot and strike are often complementary in formulas for moneyness, but need not be. Which convention is used depends on the purpose. The sequel uses <i>call</i> moneyness – as spot increases, moneyness increases – and is the same direction as using call Delta as moneyness. </p><p>While moneyness is a function of both spot and strike, usually one of these is fixed, and the other varies. Given a specific option, the strike is fixed, and different spots yield the moneyness of that option at different market prices; this is useful in option pricing and understanding the <a href="/wiki/Black%E2%80%93Scholes_formula" class="mw-redirect" title="Black–Scholes formula">Black–Scholes formula</a>. Conversely, given market data at a given point in time, the spot is fixed at the current market price, while different options have different strikes, and hence different moneyness; this is useful in constructing an <a href="/wiki/Implied_volatility_surface" class="mw-redirect" title="Implied volatility surface">implied volatility surface</a>, or more simply plotting a <a href="/wiki/Volatility_smile" title="Volatility smile">volatility smile</a>.<sup id="cite_ref-neftci_1-1" class="reference"><a href="#cite_note-neftci-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Simple_examples">Simple examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=12" title="Edit section: Simple examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section outlines moneyness measures from simple but less useful to more complex but more useful.<sup id="cite_ref-hafner85_6-0" class="reference"><a href="#cite_note-hafner85-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Simpler measures of moneyness can be computed immediately from observable market data without any theoretical assumptions, while more complex measures use the implied volatility, and thus the Black–Scholes model. </p><p>The simplest (put) moneyness is <b>fixed-strike moneyness</b>,<sup id="cite_ref-hafner42_5-1" class="reference"><a href="#cite_note-hafner42-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> where <i>M</i>=<i>K,</i> and the simplest call moneyness is <b>fixed-spot moneyness</b>, where <i>M</i>=<i>S.</i> These are also known as <b>absolute moneyness</b>, and correspond to not changing coordinates, instead using the raw prices as measures of moneyness; the corresponding volatility surface, with coordinates <i>K</i> and <i>T</i> (tenor) is the <i>absolute volatility surface</i>. The simplest non-trivial moneyness is the ratio of these, either <i>S</i>/<i>K</i> or its reciprocal <i>K</i>/<i>S,</i> which is known as the (spot) <b>simple moneyness</b>,<sup id="cite_ref-hafner85_6-1" class="reference"><a href="#cite_note-hafner85-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> with analogous forward simple moneyness. Conventionally the fixed quantity is in the denominator, while the variable quantity is in the numerator, so <i>S</i>/<i>K</i> for a single option and varying spots, and <i>K</i>/<i>S</i> for different options at a given spot, such as when constructing a volatility surface. A volatility surface using coordinates a non-trivial moneyness <i>M</i> and time to expiry <i>τ</i> is called the <i>relative volatility surface</i> (with respect to the moneyness <i>M</i>). </p><p>While the spot is often used by traders, the forward is preferred in theory, as it has better properties,<sup id="cite_ref-hafner85_6-2" class="reference"><a href="#cite_note-hafner85-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> thus <i>F</i>/<i>K</i> will be used in the sequel. In practice, for low interest rates and short tenors, spot versus forward makes little difference.<sup id="cite_ref-hafner42_5-2" class="reference"><a href="#cite_note-hafner42-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In (call) simple moneyness, ATM corresponds to moneyness of 1, while ITM corresponds to greater than 1, and OTM corresponds to less than 1, with equivalent levels of ITM/OTM corresponding to reciprocals. This is linearized by taking the log, yielding the <b>log simple moneyness</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left(F/K\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left(F/K\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350c0287af5c0e33a5895d99fd20fa6d7832bbad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.365ex; height:2.843ex;" alt="{\displaystyle \ln \left(F/K\right).}"></span> In the log simple moneyness, ATM corresponds to 0, while ITM is positive and OTM is negative, and corresponding levels of ITM/OTM corresponding to switching sign. Note that once logs are taken, moneyness in terms of forward or spot differ by an additive factor (log of discount factor), as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left(F/K\right)=\ln(S/K)+rT.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left(F/K\right)=\ln(S/K)+rT.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab65287fe8b2d3c6fbe4f9e759d157e5d0954d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.465ex; height:2.843ex;" alt="{\displaystyle \ln \left(F/K\right)=\ln(S/K)+rT.}"></span> </p><p>The above measures are independent of time, but for a given simple moneyness, options near expiry and far from expiry behave differently, as options far from expiry have more time for the underlying to change. Accordingly, one may incorporate time to maturity <i>τ</i> into moneyness. Since dispersion of Brownian motion is proportional to the square root of time, one may divide the log simple moneyness by this factor, yielding:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left(F/K\right){\Big /}{\sqrt {\tau }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.623em" minsize="1.623em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left(F/K\right){\Big /}{\sqrt {\tau }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44034edff3c81715b9be10875ebffac173ac545d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.387ex; height:4.843ex;" alt="{\displaystyle \ln \left(F/K\right){\Big /}{\sqrt {\tau }}.}"></span> This effectively normalizes for time to expiry – with this measure of moneyness, volatility smiles are largely independent of time to expiry.<sup id="cite_ref-hafner85_6-3" class="reference"><a href="#cite_note-hafner85-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>This measure does not account for the volatility <i>σ</i> of the underlying asset. Unlike previous inputs, volatility is not directly observable from market data, but must instead be computed in some model, primarily using ATM implied volatility in the Black–Scholes model. Dispersion is proportional to volatility, so <a href="/wiki/Standardizing" class="mw-redirect" title="Standardizing">standardizing</a> by volatility yields:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {\ln \left(F/K\right)}{\sigma {\sqrt {\tau }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {\ln \left(F/K\right)}{\sigma {\sqrt {\tau }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc33b80cc0a436036894f34e9a4b5674cea76d68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.34ex; height:6.676ex;" alt="{\displaystyle m={\frac {\ln \left(F/K\right)}{\sigma {\sqrt {\tau }}}}.}"></span></dd></dl> <p>This is known as the <b>standardized moneyness</b> (forward), and measures moneyness in standard deviation units. </p><p>In words, the standardized moneyness is the number of <a href="/wiki/Standard_deviations" class="mw-redirect" title="Standard deviations">standard deviations</a> the current forward price is above the strike price. Thus the moneyness is zero when the forward price of the underlying equals the <a href="/wiki/Strike_price" title="Strike price">strike price</a>, when the option is <i>at-the-money-forward</i>. Standardized moneyness is measured in standard deviations from this point, with a positive value meaning an in-the-money call option and a negative value meaning an out-of-the-money call option (with signs reversed for a put option). </p> <div class="mw-heading mw-heading3"><h3 id="Black–Scholes_formula_auxiliary_variables"><span id="Black.E2.80.93Scholes_formula_auxiliary_variables"></span>Black–Scholes formula auxiliary variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=13" title="Edit section: Black–Scholes formula auxiliary variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standardized moneyness is closely related to the auxiliary variables in the Black–Scholes formula, namely the terms <i>d</i><sub>+</sub> = <i>d</i><sub>1</sub> and <i>d</i><sub>−</sub> = <i>d</i><sub>2</sub>, which are defined as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\pm }={\frac {\ln \left(F/K\right)\pm (\sigma ^{2}/2)\tau }{\sigma {\sqrt {\tau }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mrow> <mo>)</mo> </mrow> <mo>±</mo> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mi>τ</mi> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\pm }={\frac {\ln \left(F/K\right)\pm (\sigma ^{2}/2)\tau }{\sigma {\sqrt {\tau }}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59bf854043a0765432c4ca2f5c39174233ff78e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.58ex; height:6.843ex;" alt="{\displaystyle d_{\pm }={\frac {\ln \left(F/K\right)\pm (\sigma ^{2}/2)\tau }{\sigma {\sqrt {\tau }}}}.}"></span></dd></dl> <p>The standardized moneyness is the average of these: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {\ln(F/K)}{\sigma {\sqrt {\tau }}}}={\tfrac {1}{2}}\left(d_{-}+d_{+}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {\ln(F/K)}{\sigma {\sqrt {\tau }}}}={\tfrac {1}{2}}\left(d_{-}+d_{+}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9459e12c18636da92e819f03fbe0c3cff175a1a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.96ex; height:6.676ex;" alt="{\displaystyle m={\frac {\ln(F/K)}{\sigma {\sqrt {\tau }}}}={\tfrac {1}{2}}\left(d_{-}+d_{+}\right),}"></span></dd></dl> <p>and they are ordered as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{-}<m<d_{+},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo><</mo> <mi>m</mi> <mo><</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{-}<m<d_{+},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f18f54ffcd05350c36f73db199bf6654d62728e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.324ex; height:2.509ex;" alt="{\displaystyle d_{-}<m<d_{+},}"></span></dd></dl> <p>differing only by a step of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma {\sqrt {\tau }}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma {\sqrt {\tau }}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b72531cbcfeebef25d85844b90214848f2ea6a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.792ex; height:3.009ex;" alt="{\displaystyle \sigma {\sqrt {\tau }}/2}"></span> in each case. This is often small, so the quantities are often confused or conflated, though they have distinct interpretations. </p><p>As these are all in units of standard deviations, it makes sense to convert these to percentages, by evaluating the <a href="/wiki/Standard_normal" class="mw-redirect" title="Standard normal">standard normal</a> <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> <i>N</i> for these values. The <a href="/wiki/Black%E2%80%93Scholes#Interpretation" class="mw-redirect" title="Black–Scholes">interpretation</a> of these quantities is somewhat subtle, and consists of changing to a <a href="/wiki/Risk-neutral_measure" title="Risk-neutral measure">risk-neutral measure</a> with specific choice of <a href="/wiki/Num%C3%A9raire" title="Numéraire">numéraire</a>. In brief, these are interpreted (for a call option) as: </p> <ul><li><i>N</i>(<i>d</i><sub>−</sub>) is the (Future Value) price of a <a href="/wiki/Binary_call_option" class="mw-redirect" title="Binary call option">binary call option</a>, or the risk-neutral likelihood that the option will expire ITM, with numéraire cash (the risk-free asset);</li> <li><i>N</i>(<i>m</i>) is the percentage corresponding to standardized moneyness;</li> <li><i>N</i>(<i>d</i><sub>+</sub>) is the <a href="/wiki/Delta_(finance)" class="mw-redirect" title="Delta (finance)">Delta</a>, or the risk-neutral likelihood that the option will expire ITM, with numéraire asset.</li></ul> <p>These have the same ordering, as <i>N</i> is monotonic (since it is a CDF): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(d_{-})<N(m)<N(d_{+})=\Delta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(d_{-})<N(m)<N(d_{+})=\Delta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6296e67569f2d30a9441c17d95f1a3fa2e1fc898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.977ex; height:2.843ex;" alt="{\displaystyle N(d_{-})<N(m)<N(d_{+})=\Delta .}"></span></dd></dl> <p>Of these, <i>N</i>(<i>d</i><sub>−</sub>) is the (risk-neutral) "likelihood of expiring in the money", and thus the theoretically correct <b>percent moneyness</b>, with <i>d</i><sub>−</sub> the correct moneyness. The percent moneyness is the implied probability that the derivative will expire in the money, in the risk-neutral measure. Thus a moneyness of 0 yields a 50% probability of expiring ITM, while a moneyness of 1 yields an approximately 84% probability of expiring ITM. </p><p>This corresponds to the asset following <a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">geometric Brownian motion</a> with drift <i>r,</i> the risk-free rate, and diffusion <i>σ,</i> the implied volatility. Drift is the mean, with the corresponding median (50th <a href="/wiki/Percentile" title="Percentile">percentile</a>) being <i>r</i>−<i>σ</i><sup>2</sup>/2, which is the reason for the correction factor. Note that this is the <i>implied</i> probability, <i>not</i> the real-world probability. </p><p>The other quantities – (percent) standardized moneyness and Delta – are not identical to the actual percent moneyness, but in many practical cases these are quite close (unless volatility is high or time to expiry is long), and Delta is commonly used by traders as a measure of (percent) moneyness.<sup id="cite_ref-hafner42_5-3" class="reference"><a href="#cite_note-hafner42-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Delta is more than moneyness, with the (percent) standardized moneyness in between. Thus a 25 Delta call option has less than 25% moneyness, usually slightly less, and a 50 Delta "ATM" call option has less than 50% moneyness; these discrepancies can be observed in prices of binary options and <a href="/wiki/Vertical_spread" title="Vertical spread">vertical spreads</a>. Note that for puts, Delta is negative, and thus negative Delta is used – more uniformly, absolute value of Delta is used for call/put moneyness. </p><p>The meaning of the factor of (<i>σ</i><sup>2</sup>/2)<i>τ</i> is relatively subtle. For <i>d</i><sub>−</sub> and <i>m</i> this corresponds to the difference between the median and mean (respectively) of <a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">geometric Brownian motion</a> (the <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a>), and is the same correction factor in <a href="/wiki/It%C5%8D%27s_lemma#Geometric_Brownian_motion" class="mw-redirect" title="Itō's lemma">Itō's lemma for geometric Brownian motion</a>. The interpretation of <i>d</i><sub>+</sub>, as used in Delta, is subtler, and can be interpreted most elegantly as change of numéraire. In more elementary terms, the probability that the option expires in the money and the value of the underlying at exercise are not independent – the higher the price of the underlying, the more likely it is to expire in the money <i>and</i> the higher the value at exercise, hence why Delta is higher than moneyness. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moneyness&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-neftci-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-neftci_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-neftci_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">(<a href="#CITEREFNeftçi2008">Neftçi 2008</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IOEyEY3TM9AC&pg=PA458">458–460</a>, 11.2 How Can We Define Moneyness?)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFChugh2013" class="citation book cs1">Chugh, Aman (2013). <a rel="nofollow" class="external text" href="https://archive.today/20140818015727/http://icai.org/cmii_uploads1/c813b9f80c922bc3757e231a70dd1c7e.ppt"><i>Financial Derivatives- The Currency and Rates Factor</i></a> (First ed.). New Delhi: Dorling Kindersly (India) Pvt Ltd, licensees of Pearson Education in South Asia. p. 60. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-317-7433-5" title="Special:BookSources/978-81-317-7433-5"><bdi>978-81-317-7433-5</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://icai.org/cmii_uploads1/c813b9f80c922bc3757e231a70dd1c7e.ppt">the original</a> on August 18, 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">18 August</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Derivatives-+The+Currency+and+Rates+Factor&rft.place=New+Delhi&rft.pages=60&rft.edition=First&rft.pub=Dorling+Kindersly+%28India%29+Pvt+Ltd%2C+licensees+of+Pearson+Education+in+South+Asia&rft.date=2013&rft.isbn=978-81-317-7433-5&rft.aulast=Chugh&rft.aufirst=Aman&rft_id=http%3A%2F%2Ficai.org%2Fcmii_uploads1%2Fc813b9f80c922bc3757e231a70dd1c7e.ppt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMoneyness" class="Z3988"></span></span> </li> <li id="cite_note-at-the-money-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-at-the-money_3-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://atthemoney.com/what-is-at-the-money/">At the Money Definition</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120616065227/http://atthemoney.com/what-is-at-the-money/">Archived</a> 2012-06-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Cash Bauer 2012</span> </li> <li id="cite_note-iopntm-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-iopntm_4-0">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://www.investopedia.com/terms/n/near-the-money.asp">Near The Money</a>", Investopedia</span> </li> <li id="cite_note-hafner42-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-hafner42_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hafner42_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-hafner42_5-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-hafner42_5-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">(<a href="#CITEREFHäfner2004">Häfner 2004</a>, Definition 3.12, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=67OE8cKXWBQC&pg=PA42">42</a>)</span> </li> <li id="cite_note-hafner85-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-hafner85_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hafner85_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-hafner85_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-hafner85_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">(<a href="#CITEREFHäfner2004">Häfner 2004</a>, Section 5.3.1, Choice of Moneyness Measure, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=67OE8cKXWBQC&pg=PA85">85–87</a>)</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">(<a href="#CITEREFNatenberg1994">Natenberg 1994</a>, pp. 106–110)</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">(<a href="#CITEREFNatenberg1994">Natenberg 1994</a>)</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">(<a href="#CITEREFTompkins1994">Tompkins 1994</a>), who uses spot rather than forward.</span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHäfner2004" class="citation book cs1">Häfner, Reinhold (2004). <i>Stochastic Implied Votality: A Factor-Based Model</i>. Lecture Notes in Economics and Mathematical Systems (Paperback ed.). Berlin: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-22183-8" title="Special:BookSources/978-3-540-22183-8"><bdi>978-3-540-22183-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Implied+Votality%3A+A+Factor-Based+Model&rft.place=Berlin&rft.series=Lecture+Notes+in+Economics+and+Mathematical+Systems&rft.edition=Paperback&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=978-3-540-22183-8&rft.aulast=H%C3%A4fner&rft.aufirst=Reinhold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMoneyness" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcMillan2002" class="citation book cs1">McMillan, Lawrence G. (2002). <i>Options as a Strategic Investment</i> (4th ed.). New York : New York Institute of Finance. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7352-0197-8" title="Special:BookSources/0-7352-0197-8"><bdi>0-7352-0197-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Options+as+a+Strategic+Investment&rft.edition=4th&rft.pub=New+York+%3A+New+York+Institute+of+Finance&rft.date=2002&rft.isbn=0-7352-0197-8&rft.aulast=McMillan&rft.aufirst=Lawrence+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMoneyness" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNatenberg1994" class="citation book cs1">Natenberg, Sheldon (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/optionvolatility00shel"><i>Option Volatility & Pricing: Advanced Trading Strategies and Techniques</i></a></span>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-55738486-7" title="Special:BookSources/978-1-55738486-7"><bdi>978-1-55738486-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Option+Volatility+%26+Pricing%3A+Advanced+Trading+Strategies+and+Techniques&rft.pub=McGraw-Hill&rft.date=1994&rft.isbn=978-1-55738486-7&rft.aulast=Natenberg&rft.aufirst=Sheldon&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foptionvolatility00shel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMoneyness" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeftçi2008" class="citation book cs1"><a href="/wiki/Salih_Neft%C3%A7i" title="Salih Neftçi">Neftçi, Salih N.</a> (2008). <i>Principles of Financial Engineering</i> (2nd ed.). Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-373574-4" title="Special:BookSources/978-0-12-373574-4"><bdi>978-0-12-373574-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Financial+Engineering&rft.edition=2nd&rft.pub=Academic+Press&rft.date=2008&rft.isbn=978-0-12-373574-4&rft.aulast=Neft%C3%A7i&rft.aufirst=Salih+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMoneyness" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTompkins1994" class="citation book cs1">Tompkins, Robert (1994). <i>Options Explained<sup>2</sup></i>. Macmillan Business: Finance and Capital Markets (2nd ed.). Palgrave. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-33362807-2" title="Special:BookSources/978-0-33362807-2"><bdi>978-0-33362807-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Options+Explained%3Csup%3E2%3C%2Fsup%3E&rft.series=Macmillan+Business%3A+Finance+and+Capital+Markets&rft.edition=2nd&rft.pub=Palgrave&rft.date=1994&rft.isbn=978-0-33362807-2&rft.aulast=Tompkins&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMoneyness" class="Z3988"></span></li></ul> </div> <p><br /> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Terms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Delta_neutral" title="Delta neutral">Delta neutral</a></li> <li><a href="/wiki/Exercise_(options)" title="Exercise (options)">Exercise</a></li> <li><a href="/wiki/Expiration_(options)" title="Expiration (options)">Expiration</a></li> <li><a class="mw-selflink selflink">Moneyness</a></li> <li><a href="/wiki/Open_interest" title="Open interest">Open interest</a></li> <li><a href="/wiki/Pin_risk" title="Pin risk">Pin risk</a></li> <li><a href="/wiki/Risk-free_interest_rate" class="mw-redirect" title="Risk-free interest rate">Risk-free interest rate</a></li> <li><a href="/wiki/Strike_price" title="Strike price">Strike price</a></li> <li><a href="/wiki/Synthetic_position" title="Synthetic position">Synthetic position</a></li> <li><a href="/wiki/Greeks_(finance)" title="Greeks (finance)">the Greeks</a></li> <li><a href="/wiki/Volatility_(finance)" title="Volatility (finance)">Volatility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vanilla_option" class="mw-redirect" title="Vanilla option">Vanillas</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/American_option" class="mw-redirect" title="American option">American</a></li> <li><a href="/wiki/Bond_option" title="Bond option">Bond option</a></li> <li><a href="/wiki/Call_option" title="Call option">Call</a></li> <li><a href="/wiki/Employee_stock_option" title="Employee stock option">Employee stock option</a></li> <li><a href="/wiki/European_option" class="mw-redirect" title="European option">European</a></li> <li><a href="/wiki/Fixed_income" title="Fixed income">Fixed income</a></li> <li><a href="/wiki/Foreign_exchange_option" title="Foreign exchange option">FX</a></li> <li><a href="/wiki/Option_style" title="Option style">Option styles</a></li> <li><a href="/wiki/Put_option" title="Put option">Put</a></li> <li><a href="/wiki/Warrant_(finance)" title="Warrant (finance)">Warrants</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Exotic_option" title="Exotic option">Exotics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asian_option" title="Asian option">Asian</a></li> <li><a href="/wiki/Barrier_option" title="Barrier option">Barrier</a></li> <li><a href="/wiki/Basket_option" title="Basket option">Basket</a></li> <li><a href="/wiki/Binary_option" title="Binary option">Binary</a></li> <li><a href="/wiki/Callable_bull/bear_contract" title="Callable bull/bear contract">Callable bull/bear contract</a></li> <li><a href="/wiki/Chooser_option" title="Chooser option">Chooser</a></li> <li><a href="/wiki/Cliquet_option" title="Cliquet option">Cliquet</a></li> <li><a href="/wiki/Commodore_option" title="Commodore option">Commodore</a></li> <li><a href="/wiki/Compound_option" title="Compound option">Compound</a></li> <li><a href="/wiki/Forward_start_option" title="Forward start option">Forward start</a></li> <li><a href="/wiki/Interest_rate_option" title="Interest rate option">Interest rate</a></li> <li><a href="/wiki/Lookback_option" title="Lookback option">Lookback</a></li> <li>Mountain range</li> <li><a href="/wiki/Rainbow_option" title="Rainbow option">Rainbow</a></li> <li><a href="/wiki/Spread_option" title="Spread option">Spread</a></li> <li><a href="/wiki/Swaption" title="Swaption">Swaption</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Options_strategy" title="Options strategy">Strategies</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Backspread" title="Backspread">Backspread</a></li> <li><a href="/wiki/Box_spread_(options)" class="mw-redirect" title="Box spread (options)">Box spread</a></li> <li><a href="/wiki/Butterfly_(options)" title="Butterfly (options)">Butterfly</a></li> <li><a href="/wiki/Calendar_spread" title="Calendar spread">Calendar spread</a></li> <li><a href="/wiki/Collar_(finance)" title="Collar (finance)">Collar</a></li> <li><a href="/wiki/Condor_(options)" title="Condor (options)">Condor</a></li> <li><a href="/wiki/Covered_option" title="Covered option">Covered option</a></li> <li><a href="/wiki/Credit_spread_(options)" title="Credit spread (options)">Credit spread</a></li> <li><a href="/wiki/Debit_spread" title="Debit spread">Debit spread</a></li> <li><a href="/wiki/Diagonal_spread" title="Diagonal spread">Diagonal spread</a></li> <li><a href="/wiki/Fence_(finance)" title="Fence (finance)">Fence</a></li> <li><a href="/wiki/Intermarket_spread" title="Intermarket spread">Intermarket spread</a></li> <li><a href="/wiki/Iron_butterfly_(options_strategy)" title="Iron butterfly (options strategy)">Iron butterfly</a></li> <li><a href="/wiki/Iron_condor" title="Iron condor">Iron condor</a></li> <li><a href="/wiki/Jelly_roll_(options)" title="Jelly roll (options)">Jelly roll</a></li> <li><a href="/wiki/Ladder_(option_combination)" title="Ladder (option combination)">Ladder</a></li> <li><a href="/wiki/Naked_option" title="Naked option">Naked option</a></li> <li><a href="/wiki/Straddle" title="Straddle">Straddle</a></li> <li><a href="/wiki/Strangle_(options)" title="Strangle (options)">Strangle</a></li> <li><a href="/wiki/Protective_option" title="Protective option">Protective option</a></li> <li><a href="/wiki/Ratio_spread" title="Ratio spread">Ratio spread</a></li> <li><a href="/wiki/Risk_reversal" title="Risk reversal">Risk reversal</a></li> <li><a href="/wiki/Vertical_spread" title="Vertical spread">Vertical spread</a> (<a href="/wiki/Bear_spread" title="Bear spread">Bear</a>, <a href="/wiki/Bull_spread" title="Bull spread">Bull</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Valuation_of_options" title="Valuation of options">Valuation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bachelier_model" title="Bachelier model">Bachelier</a></li> <li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial</a></li> <li><a href="/wiki/Black_model" title="Black model">Black</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a> (<a href="/wiki/Black%E2%80%93Scholes_equation" title="Black–Scholes equation">equation</a>)</li> <li><a href="/wiki/Finite_difference_methods_for_option_pricing" title="Finite difference methods for option pricing">Finite difference</a></li> <li><a href="/wiki/Foreign_exchange_option#Valuation:_the_Garman–Kohlhagen_model" title="Foreign exchange option">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Lattice_model_(finance)" title="Lattice model (finance)">Lattices</a></li> <li><a href="/wiki/Margrabe%27s_formula" title="Margrabe's formula">Margrabe</a></li> <li><a href="/wiki/Put%E2%80%93call_parity" title="Put–call parity">Put–call parity</a></li> <li><a href="/wiki/Monte_Carlo_methods_for_option_pricing" title="Monte Carlo methods for option pricing">MC Simulation</a></li> <li><a href="/wiki/Real_options_valuation" title="Real options valuation">Real options</a></li> <li><a href="/wiki/Trinomial_tree" title="Trinomial tree">Trinomial</a></li> <li><a href="/wiki/Vanna%E2%80%93Volga_pricing" title="Vanna–Volga pricing">Vanna–Volga</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Swap_(finance)" title="Swap (finance)">Swaps</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amortising_swap" title="Amortising swap">Amortising</a></li> <li><a href="/wiki/Asset_swap" title="Asset swap">Asset</a></li> <li><a href="/wiki/Basis_swap" title="Basis swap">Basis</a></li> <li><a href="/wiki/Commodity_swap" title="Commodity swap">Commodity</a></li> <li><a href="/wiki/Conditional_variance_swap" title="Conditional variance swap">Conditional variance</a></li> <li><a href="/wiki/Constant_maturity_swap" title="Constant maturity swap">Constant maturity</a></li> <li><a href="/wiki/Correlation_swap" title="Correlation swap">Correlation</a></li> <li><a href="/wiki/Credit_default_swap" title="Credit default swap">Credit default</a></li> <li><a href="/wiki/Currency_swap" title="Currency swap">Currency</a></li> <li><a href="/wiki/Dividend_swap" title="Dividend swap">Dividend</a></li> <li><a href="/wiki/Equity_swap" title="Equity swap">Equity</a></li> <li><a href="/wiki/Foreign_exchange_swap" title="Foreign exchange swap">Forex</a></li> <li><a href="/wiki/Forward_rate_agreement" title="Forward rate agreement">Forward Rate Agreement</a></li> <li><a href="/wiki/Inflation_swap" title="Inflation swap">Inflation</a></li> <li><a href="/wiki/Interest_rate_swap" title="Interest rate swap">Interest rate</a></li> <li><a href="/wiki/Overnight_indexed_swap" title="Overnight indexed swap">Overnight indexed</a></li> <li><a href="/wiki/Total_return_swap" title="Total return swap">Total return</a></li> <li><a href="/wiki/Variance_swap" title="Variance swap">Variance</a></li> <li><a href="/wiki/Volatility_swap" title="Volatility swap">Volatility</a></li> <li><a href="/wiki/Year-on-year_inflation-indexed_swap" title="Year-on-year inflation-indexed swap">Year-on-year inflation-indexed</a></li> <li><a href="/wiki/Zero_coupon_swap" title="Zero coupon swap">Zero Coupon</a></li> <li><a href="/wiki/Zero-coupon_inflation_swap" title="Zero-coupon inflation swap">Zero-coupon inflation-indexed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Forward_contract" title="Forward contract">Forwards</a></li><li><a href="/wiki/Futures_contract" title="Futures contract">Futures</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contango" title="Contango">Contango</a></li> <li><a href="/w/index.php?title=Commodities_future&action=edit&redlink=1" class="new" title="Commodities future (page does not exist)">Commodities future</a></li> <li><a href="/wiki/Currency_future" title="Currency future">Currency future</a></li> <li><a href="/wiki/Dividend_future" title="Dividend future">Dividend future</a></li> <li><a href="/wiki/Forward_market" title="Forward market">Forward market</a></li> <li><a href="/wiki/Forward_price" title="Forward price">Forward price</a></li> <li><a href="/wiki/Forward_contract#Spot–forward_parity" title="Forward contract">Forwards pricing</a></li> <li><a href="/wiki/Forward_rate" title="Forward rate">Forward rate</a></li> <li><a href="/wiki/Futures_contract#Pricing" title="Futures contract">Futures pricing</a></li> <li><a href="/wiki/Interest_rate_future" title="Interest rate future">Interest rate future</a></li> <li><a href="/wiki/Margin_(finance)" title="Margin (finance)">Margin</a></li> <li><a href="/wiki/Normal_backwardation" title="Normal backwardation">Normal backwardation</a></li> <li><a href="/wiki/Perpetual_futures" title="Perpetual futures">Perpetual futures</a></li> <li><a href="/wiki/Single-stock_futures" title="Single-stock futures">Single-stock futures</a></li> <li><a href="/wiki/Slippage_(finance)" title="Slippage (finance)">Slippage</a></li> <li><a href="/wiki/Stock_market_index_future" title="Stock market index future">Stock market index future</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Exotic_derivative" title="Exotic derivative">Exotic derivatives</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Commodity_derivative" class="mw-redirect" title="Commodity derivative">Commodity derivative</a></li> <li><a href="/wiki/Energy_derivative" title="Energy derivative">Energy derivative</a></li> <li><a href="/wiki/Freight_derivative" class="mw-redirect" title="Freight derivative">Freight derivative</a></li> <li><a href="/wiki/Inflation_derivative" title="Inflation derivative">Inflation derivative</a></li> <li><a href="/wiki/Property_derivative" title="Property derivative">Property derivative</a></li> <li><a href="/wiki/Weather_derivative" title="Weather derivative">Weather derivative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other derivatives</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Collateralized_debt_obligation" title="Collateralized debt obligation">Collateralized debt obligation (CDO)</a></li> <li><a href="/wiki/Constant_proportion_portfolio_insurance" title="Constant proportion portfolio insurance">Constant proportion portfolio insurance</a></li> <li><a href="/wiki/Contract_for_difference" title="Contract for difference">Contract for difference</a></li> <li><a href="/wiki/Credit-linked_note" title="Credit-linked note">Credit-linked note (CLN)</a></li> <li><a href="/wiki/Credit_default_option" title="Credit default option">Credit default option</a></li> <li><a href="/wiki/Credit_derivative" title="Credit derivative">Credit derivative</a></li> <li><a href="/wiki/Equity-linked_note" title="Equity-linked note">Equity-linked note (ELN)</a></li> <li><a href="/wiki/Equity_derivative" title="Equity derivative">Equity derivative</a></li> <li><a href="/wiki/Foreign_exchange_derivative" title="Foreign exchange derivative">Foreign exchange derivative</a></li> <li><a href="/wiki/Fund_derivative" title="Fund derivative">Fund derivative</a></li> <li><a href="/wiki/Fund_of_funds" title="Fund of funds">Fund of funds</a></li> <li><a href="/wiki/Interest_rate_derivative" title="Interest rate derivative">Interest rate derivative</a></li> <li><a href="/wiki/Mortgage-backed_security" title="Mortgage-backed security">Mortgage-backed security</a></li> <li><a href="/wiki/Power_reverse_dual-currency_note" title="Power reverse dual-currency note">Power reverse dual-currency note (PRDC)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Market issues</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Consumer_debt" title="Consumer debt">Consumer debt</a></li> <li><a href="/wiki/Corporate_bond" title="Corporate bond">Corporate debt</a></li> <li><a href="/wiki/Government_debt" title="Government debt">Government debt</a></li> <li><a href="/wiki/Great_Recession" title="Great Recession">Great Recession</a></li> <li><a href="/wiki/Municipal_debt" class="mw-redirect" title="Municipal debt">Municipal debt</a></li> <li><a href="/wiki/Tax_policy" title="Tax policy">Tax policy</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Emblem-money.svg" class="mw-file-description"><img alt="icon" 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