{"id":215990,"date":"2024-12-05T11:33:21","date_gmt":"2024-12-05T16:33:21","guid":{"rendered":"https:\/\/ibkrcampus.com\/campus\/?p=215990"},"modified":"2024-12-05T11:33:15","modified_gmt":"2024-12-05T16:33:15","slug":"the-value-of-an-option-at-expiration","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/the-value-of-an-option-at-expiration\/","title":{"rendered":"The Value of an Option at Expiration"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>The article &#8220;The Value of an Option at Expiration&#8221; first appeared on <a href=\"https:\/\/robotwealth.com\/the-value-of-an-option-at-expiration\/\">Robot Wealth<\/a> blog.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Options have value because the future is uncertain.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One thing that is known and always unchanging, however, is the value of a given option contract at expiry. This is the most fundamental thing about an option.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-value-of-a-call-option-at-expiration\">Value of a Call Option at Expiration<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A call option gives the holder the right to\u00a0<strong>buy<\/strong>\u00a0the underlying at the strike price on or before the expiration date.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That means that the value,\u00a0Vcall, of the call option at expiration, is described by a step-wise function in the price at expiry,\u00a0\u00a0Pe\u00a0and the strike,\u00a0s:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"723\" height=\"76\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration.jpg\" alt=\"\" class=\"wp-image-215991 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration.jpg 723w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration-700x74.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration-300x32.jpg 300w\" data-sizes=\"(max-width: 723px) 100vw, 723px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 723px; aspect-ratio: 723\/76;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The option, at expiration, is:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>worthless if the price at expiration is less than or equal to the strike (\u201cout of the money\u201d)<\/li>\n\n\n\n<li>worth something if the price at expiration is more than the strike (\u201cin the money\u201d)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">If the asset is trading below the strike price then we say that the option is \u201cout of the money\u201d. The option gives you the option to buy the asset at the strike price \u2013 but nobody would willingly buy an asset at a price higher than they could buy it in the market. So an \u201cout of the money\u201d call option is worthless at expiry.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the option expires \u201cin the money\u201d then it is worth the price at expiration less the strike. That\u2019s because you can&nbsp;<em>exercise<\/em>&nbsp;the option to buy the stock at the strike price and you can immediately sell at the market price at expiration, pocketing the difference.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here\u2019s a visual example of the payoff function of a call option at expiry with a strike of 100:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"r\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">library(tidyverse) \nmin_price &lt;- 0\nmax_price &lt;- 200\nstrike &lt;- 100\ncall_payoffs &lt;- tibble(price = c(min_price, strike, max_price)) %&amp;gt;%\n  mutate(value = case_when(price &lt; strike ~ 0, TRUE ~ price - strike))\ncall_payoffs %&gt;%\n  ggplot(aes(x = price, y = value)) +\n  geom_line() +\n  ggtitle('Value of call option at expiration')<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"792\" height=\"360\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/call-at-expiry-1-robot-wealth.png\" alt=\"\" class=\"wp-image-215994 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/call-at-expiry-1-robot-wealth.png 792w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/call-at-expiry-1-robot-wealth-700x318.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/call-at-expiry-1-robot-wealth-300x136.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/call-at-expiry-1-robot-wealth-768x349.png 768w\" data-sizes=\"(max-width: 792px) 100vw, 792px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 792px; aspect-ratio: 792\/360;\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-value-of-a-put-option-at-expiration\">Value of a Put Option at Expiration<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A put option gives the holder the right to&nbsp;<strong>sell<\/strong>&nbsp;the underlying at the strike price on or before the expiration date.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That means that the value,&nbsp;&nbsp;Vput, of the put option at expiration is also described by a step-wise function:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"752\" height=\"72\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration-2.jpg\" alt=\"\" class=\"wp-image-215995 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration-2.jpg 752w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration-2-700x67.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/robot-wealth-value-option-expiration-2-300x29.jpg 300w\" data-sizes=\"(max-width: 752px) 100vw, 752px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 752px; aspect-ratio: 752\/72;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The option, at expiration, is:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>worth something if the price at expiration is less than the strike (\u201cin the money\u201d)<\/li>\n\n\n\n<li>worthless if the price at expiration is greater than or equal to the strike (\u201cout of the money\u201d)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">If the asset is trading above the strike price, then we say that the option is \u201cout of the money\u201d. The option gives you the option to sell the asset at the strike price \u2013 but nobody would willingly sell an asset at a price lower than they could sell it in the market. So an \u201cout of the money\u201d put option is worthless at expiry.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the option expires \u201cin the money\u201d it\u2019s worth the strike less the price at expiration. That\u2019s simply because when the option expires in the money, you acquire a short position in the stock at the (high) strike price and can immediately buy it back at the (lower) expiration price, pocketing the difference.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here\u2019s a visual example of the payoff function of a call option at expiry with a strike of 100:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"r\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">put_payoffs &lt;- tibble(price = c(min_price, strike, max_price)) %&gt;%\n  mutate(value = case_when(price &gt; strike ~ 0, TRUE ~ strike - price))\nput_payoffs %&gt;%\n  ggplot(aes(x = price, y = value)) +\n  geom_line() +\n  ggtitle('Value of put option at expiration')<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"792\" height=\"360\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/put-at-expiry-2-robot-wealth.png\" alt=\"\" class=\"wp-image-215996 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/put-at-expiry-2-robot-wealth.png 792w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/put-at-expiry-2-robot-wealth-700x318.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/put-at-expiry-2-robot-wealth-300x136.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/12\/put-at-expiry-2-robot-wealth-768x349.png 768w\" data-sizes=\"(max-width: 792px) 100vw, 792px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 792px; aspect-ratio: 792\/360;\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-woah-there-this-all-looks-like-a-pretty-good-deal\">Woah there! This all looks like a pretty good deal!<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">That \u201ckink\u201d at the strike price means that by buying options, we get the ability to participate in the upside of a move without participating in the downside.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Pretty sweet, eh?<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is better than a stop loss because we always keep our exposure to any upside until expiration. We\u2019re not \u201cstopped out.\u201d<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Needless to say, this is extremely attractive.&nbsp;And there are no free lunches in the market\u2014nothing comes for free.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Options contracts are derivative contracts. And for a derivatives contract to trade, there needs to be a buyer and a seller. So we need to find a price for an option where people are prepared to sell it&nbsp;<em>and&nbsp;<\/em>buy it.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Whereas the buyer of the options contract gets to participate in unlimited upside and limited downside, the seller of the options contract gets limited upside and unlimited downside.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><em>So the seller is going to want compensation for this risk. He will want to get paid for the option.<\/em><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Next, we\u2019ll look at the net P&amp;L from being long options.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>One thing that is known and always unchanging, however, is the value of a given option contract at expiry. This is the most fundamental thing about an option.<\/p>\n","protected":false},"author":271,"featured_media":200534,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,343,338,341,9563,342],"tags":[851,806,4922,2857,487,6591,1045],"contributors-categories":[13676],"class_list":["post-215990","post","type-post","status-publish","format-standard","has-post-thumbnail","category-data-science","category-programing-languages","category-ibkr-quant-news","category-quant-development","category-options-quant","category-r-development","tag-algo-trading","tag-data-science","tag-econometrics","tag-ggplot","tag-r","tag-rstats","tag-tidyverse","contributors-categories-robot-wealth"],"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v28.1) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>The Value of an Option at Expiration | IBKR Quant<\/title>\n<meta name=\"description\" content=\"One thing that is known and always unchanging, however, is the value of a given option contract at expiry.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/215990\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Value of an Option at Expiration\" \/>\n<meta property=\"og:description\" content=\"One thing that is known and always unchanging, however, is the value of a given option contract at expiry. This is the most fundamental thing about an option.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/the-value-of-an-option-at-expiration\/\" \/>\n<meta property=\"og:site_name\" content=\"IBKR Campus US\" \/>\n<meta property=\"article:published_time\" content=\"2024-12-05T16:33:21+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/12\/options-trading-red-green-featured-img.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1000\" \/>\n\t<meta property=\"og:image:height\" content=\"563\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Kris Longmore\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kris Longmore\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading 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