{"id":216885,"date":"2025-01-02T12:22:19","date_gmt":"2025-01-02T17:22:19","guid":{"rendered":"https:\/\/ibkrcampus.com\/campus\/?p=216885"},"modified":"2025-01-02T12:22:53","modified_gmt":"2025-01-02T17:22:53","slug":"long-options-payoff-profiles","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/long-options-payoff-profiles\/","title":{"rendered":"Long Options Payoff Profiles"},"content":{"rendered":"\n<p><em>The post &#8220;Long Options Payoff Profiles&#8221; first appeared on <a href=\"https:\/\/robotwealth.com\/long-options-payoff-profiles\/\">Robot Wealth<\/a> blog.<\/em><\/p>\n\n\n\n<p>In this article, we explore the payoff to holding long options positions.<\/p>\n\n\n\n<p><em>Read the previous parts of this 101 series on options:<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/robotwealth.com\/options-are-just-options-why-are-they-valuable\/\" target=\"_blank\" rel=\"noreferrer noopener\">Options are just options \u2013 why are they valuable?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/robotwealth.com\/options-101-understanding-the-basics-of-financial-options\/\" target=\"_blank\" rel=\"noreferrer noopener\">Options 101: Understanding the basics<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/robotwealth.com\/the-value-of-an-option-at-expiration\/\" target=\"_blank\" rel=\"noreferrer noopener\">The value of an option at expiration<\/a><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-payoff-profile-of-a-long-call\">Payoff Profile of a Long Call<\/h2>\n\n\n\n<p><a href=\"https:\/\/robotwealth.com\/the-value-of-an-option-at-expiration\/\">So fa<\/a><a href=\"https:\/\/robotwealth.com\/the-value-of-an-option-at-expiration\/\" target=\"_blank\" rel=\"noreferrer noopener\">r<\/a>, we\u2019ve plotted the value of an option at expiration.<\/p>\n\n\n\n<p>This is useful (as we\u2019ll see later), but it doesn\u2019t represent our profit and loss from being long that option.<\/p>\n\n\n\n<p>For that, we need to subtract the amount we paid from the option from the payoff for all values of the underlying asset price.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-example-long-a-call-option-expiring-in-the-money\">Example: Long a call option expiring \u201cin the money\u201d<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Pam pays $2 for $100 strike calls on product X.<\/li>\n\n\n\n<li>At the expiration date, product X is trading in the market at $130.<\/li>\n<\/ul>\n\n\n\n<p><em>What is Pam\u2019s net P&amp;L?<\/em><\/p>\n\n\n\n<p>We follow this process:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Calculate the value of the option<\/li>\n\n\n\n<li>Subtract the value we paid for it.<\/li>\n<\/ol>\n\n\n\n<p>If the product is trading at $130 at expiry, and Pam holds calls with a $100 strike then Pam\u2019s calls are \u201cin the money\u201d. Pam can exercise her call option at $100 and then sell the product in the market at $130. So:<\/p>\n\n\n\n<p><code>Value of call option at expiration = $130 - $100 = $30<\/code><\/p>\n\n\n\n<p>Now we subtract the amount Pam paid for the call option to get her net p&amp;l in the trade.<\/p>\n\n\n\n<p><code>Net Profit = $30 - $2 = $28<\/code><\/p>\n\n\n\n<p><em>Nice trade, Pam.<\/em><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-example-long-a-call-option-expiring-out-of-the-money\">Example: Long a call option expiring \u201cout of the money\u201d<\/h3>\n\n\n\n<p>Let\u2019s take a similar example\u2026<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Pam pays $2 for $100 strike calls on product X<\/li>\n\n\n\n<li>At the expiration date, product X is trading in the market at $90<\/li>\n<\/ul>\n\n\n\n<p><em>What is Pam\u2019s net P&amp;L now?<\/em><\/p>\n\n\n\n<p>We follow this process:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Calculate the value of the option<\/li>\n\n\n\n<li>Subtract the value we paid for it.<\/li>\n<\/ol>\n\n\n\n<p>The product is trading below the value of Pam\u2019s call strike. So there\u2019s no point exercising these options. Pam is a smart lady and she wouldn\u2019t pay more than she had to for product X.<br>So Pam\u2019s options expire \u201cout of the money\u201d.&nbsp; They expire worthlessly.<\/p>\n\n\n\n<p><code>Value of call option at expiration = $0<\/code><\/p>\n\n\n\n<p>Now, we subtract the amount Pam paid for the call option to get her net P&amp;L in the trade.<\/p>\n\n\n\n<p><code>Net Profit = $0 - $2 = -$2<\/code><\/p>\n\n\n\n<p>It was a loss \u2013 but because she bought call options, Pam\u2019s loss was limited to the amount she paid for the calls.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-payoff-profile-for-long-call-position\">Payoff Profile for Long Call Position<\/h2>\n\n\n\n<p>We can plot the P&amp;L of a long options position held to expiration as a function of the price of the underlying asset.<\/p>\n\n\n\n<p>To do this we:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>calculate the value of the option at expiry for a range of underlying asset prices<\/li>\n\n\n\n<li>subtract the value paid for the option (\u201cthe premium\u201d) from each point.<\/li>\n<\/ul>\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=\"\">min_price &lt;- 50\nmax_price &lt;- 150\nstrike &lt;- 100 \npremium &lt;- 2 \ncall_payoffs &lt;- tibble(price = c(min_price, strike, max_price)) %&gt;%\n  mutate(callvalue = case_when(price &lt; strike ~ 0, TRUE ~ price - strike)) %&gt;%\n  mutate(payoff = callvalue - premium) \ncall_payoffs %&gt;%\n  ggplot(aes(x = price, y = payoff)) + \n    geom_line() + \n    ggtitle(paste0('Payoff profile for long $100 call - Premium $2))<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/01\/payoff100call-1-robot-wealth.png\" alt=\"\" class=\"wp-image-216891 lazyload\" style=\"width:950px;height:auto\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>We see that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Our maximum loss is the amount we paid for the call options<\/li>\n\n\n\n<li>Our profit is unlimited<\/li>\n\n\n\n<li>We break even at the price where the value of the call option at expiration is equal to the amount we paid for the calls. This is&nbsp;<code>premium + strike = $102<\/code><\/li>\n<\/ul>\n\n\n\n<p><em>Now let\u2019s do the same for a Put Option.<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-payoff-profile-of-a-long-put-option\">Payoff Profile of a Long Put Option<\/h2>\n\n\n\n<p>This is exactly the same deal as before.<\/p>\n\n\n\n<p>We calculate the put\u2019s value at expiration at various prices of the underlying asset and then subtract the price we paid for the option.<\/p>\n\n\n\n<p>We just need to remember that a put option is an option on being able to sell the underlying asset at the strike price. So this time around, all the action happens when the underlying asset price is below our strike.<\/p>\n\n\n\n<p>At the risk of being a bit tedious, we\u2019ll run through some examples.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Example: Long a put option expiring \u201cin the money\u201d<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fabio pays $1 for $95 strike puts on product Z.<\/li>\n\n\n\n<li>At the expiration date, product Z is trading in the market at $90.<\/li>\n<\/ul>\n\n\n\n<p><em>What is Fabios\u2019s net P&amp;L?<\/em><\/p>\n\n\n\n<p>We follow this process:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Calculate the value of the option<\/li>\n\n\n\n<li>Subtract the value we paid for it.<\/li>\n<\/ol>\n\n\n\n<p>If the product is trading at $90 at expiry, and Fabio holds puts with a $95 strike, then Fabio\u2019s calls are \u201cin the money\u201d. Fabio can exercise his put option and sell product X at $95, then immediately buy back the product in the market at $90. So:<\/p>\n\n\n\n<p><code>Value of put option at expiration = $95 - $90 = $5<\/code><\/p>\n\n\n\n<p>Now we subtract the amount Fabio paid for the put option to get his net p&amp;l in the trade.<\/p>\n\n\n\n<p><code>Net Profit = $5 - $1 = $4<\/code><\/p>\n\n\n\n<p><em>Nice trade, Fabio.<\/em><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-example-long-a-call-option-expiring-out-of-the-money-0\">Example: Long a call option expiring \u201cout of the money\u201d<\/h3>\n\n\n\n<p>Let\u2019s take a similar example\u2026<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fabio pays $1 for $95 strike calls on product X<\/li>\n\n\n\n<li>At the expiration date, product X is trading in the market at $100<\/li>\n<\/ul>\n\n\n\n<p><em>What is Fabio\u2019s net P&amp;L now?<\/em><\/p>\n\n\n\n<p>We follow this process:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Calculate the value of the option<\/li>\n\n\n\n<li>Subtract the value we paid for it.<\/li>\n<\/ol>\n\n\n\n<p>The product is trading above the value of Fabio\u2019s call strike. So there\u2019s no point exercising these options. Fabio is a smart man and he wouldn\u2019t sell product Z for less than he could get in the market.<\/p>\n\n\n\n<p>So Fabio\u2019s options expire \u201cout of the money\u201d.&nbsp; They expire worthlessly.<\/p>\n\n\n\n<p><code>Value of put option at expiration = $0<\/code><\/p>\n\n\n\n<p>Now we subtract the amount Fabio paid for the call option to get her net p&amp;l in the trade.<\/p>\n\n\n\n<p><code>Net Profit = $0 - $1 = -$1<\/code><\/p>\n\n\n\n<p>It was a loss \u2013 but because he bought options, Fabio\u2019s loss was limited to the amount he paid for the puts.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-payoff-profile-for-long-put-position\">Payoff Profile for Long Put Position<\/h2>\n\n\n\n<p>We can plot the P&amp;L of a long options position held to expiration as a function of the price of the underlying asset.<\/p>\n\n\n\n<p>To do this we:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>calculate the value of the option at expiry for a range of underlying asset prices<\/li>\n\n\n\n<li>subtract the value paid for the option (\u201cthe premium\u201d) from each point.<\/li>\n<\/ul>\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=\"\">min_price &lt;- 50\nmax_price &lt;- 150\nstrike &lt;- 95 \npremium &lt;- 1\nput_payoffs &lt;- tibble(price = c(min_price, strike, max_price)) %&gt;% \n  mutate(putvalue = case_when(price &gt; strike ~ 0, TRUE ~ strike - price)) %&gt;%\n  mutate(payoff = putvalue - premium)\nput_payoffs %&gt;% \n  ggplot(aes(x = price, y = payoff)) + \n  geom_line() + \n  ggtitle(paste0('Payoff profile for long $95 put - Premium $1'))<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/01\/put95payoff-robot-wealth.png\" alt=\"\" class=\"wp-image-216894 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>We see that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Our maximum loss is the amount we paid for the put options<\/li>\n\n\n\n<li>Our profit is unlimited<\/li>\n\n\n\n<li>We break even at the price where the value of the put option at expiration is equal to the amount we paid for the puts. This is&nbsp;<code>strike - premium = $95 - $1 = $94<\/code><\/li>\n<\/ul>\n\n\n\n<p><em>Now we understand the basics, we will next look at a simple applied use of put options: portfolio hedging.<\/em><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this article, we explore the payoff to holding long options positions.<\/p>\n","protected":false},"author":271,"featured_media":169075,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,343,338,341,9563,342],"tags":[1006,1926,494],"contributors-categories":[13676],"class_list":{"0":"post-216885","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-programing-languages","9":"category-ibkr-quant-news","10":"category-quant-development","11":"category-options-quant","12":"category-r-development","13":"tag-fintech","14":"tag-options","15":"tag-quant","16":"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 v27.3) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ 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