{"id":43426,"date":"2022-12-12T07:37:15","date_gmt":"2022-12-12T07:37:15","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=43426"},"modified":"2026-02-27T19:10:34","modified_gmt":"2026-02-27T19:10:34","slug":"determining-the-value-at-expiration-and-profit-from-a-long-or-a-short-position-in-a-call-or-put-option","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/determining-the-value-at-expiration-and-profit-from-a-long-or-a-short-position-in-a-call-or-put-option\/","title":{"rendered":"Determining the Value at Expiration and Profit from a Long or a Short Position in a Call or Put Option"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Forward Commitment and Contingent Claim Features And Instruments (2024\/25 LI CFA\u00ae\u2013Derivatives M2)\",\n  \"description\": \"This video lesson covers the definitions and characteristics of forward contracts, futures, options (calls and puts), swaps, and credit derivatives. It explains the valuation and profit calculation for call and put options and contrasts forward commitments with contingent claims in derivatives markets.\",\n  \"uploadDate\": \"2022-11-25T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/yr98xRssWAs\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/yr98xRssWAs\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/yr98xRssWAs\",\n  \"duration\": \"PT55M35S\"\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What are the value at expiration and the profit to the seller of a put option?\",\n    \"text\": \"If a put option has a premium of $3 and the exercise price is $100, and the price of the underlying is $105, the value at expiration and the profit to the option seller are closest to:\\n\\nA. Value = \u2013$3; Profit = $0\\nB. Value = $0; Profit = $8\\nC. Value = $0; Profit = $3\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The value at expiration is $0 and the profit to the option seller is $3.\\n\\nBecause the underlying price ($105) is greater than the exercise price ($100), the put option expires worthless. The payoff at expiration is therefore zero. The option seller\u2019s profit equals the premium received, which is $3.\",\n      \"dateCreated\": \"2026-01-02\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@graph\": [\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Lower.png\",\n      \"caption\": \"Profit of a Put Option (Lower Price Scenario)\",\n      \"width\": 888,\n      \"height\": 569,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Higher.jpg\",\n      \"caption\": \"Profit of a Put Option (Higher Price Scenario)\",\n      \"width\": 888,\n      \"height\": 569,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-Mod-2.png\",\n      \"caption\": \"Profit of a Call Option (Modified Scenario)\",\n      \"width\": 974,\n      \"height\": 641,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option.png\",\n      \"caption\": \"Profit of a Call Option\",\n      \"width\": 974,\n      \"height\": 641,\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\n    }\n  ]\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/yr98xRssWAs?rel=0&#038;modestbranding=1&#038;playsinline=1&#038;vq=hd1080\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  allowfullscreen>\n<\/iframe>\n\n\n\n<p>Define the following:<\/p>\n\n\n\n<p>\\(c_T =\\) Value of the call at expiration.<\/p>\n\n\n\n<p>\\(p_T =\\) Value of a put option at expiration.<\/p>\n\n\n\n<p>\\(S_T =\\) Price of the underlying at time T.<\/p>\n\n\n\n<p>\\(X =\\) Exercise price.<\/p>\n\n\n\n<p>\\(c_0=\\) Call option premium.<\/p>\n\n\n\n<p>\\(p_0 =\\) Put option premium.<\/p>\n\n\n\n<p>\\(\\Pi =\\) Profit from an option strategy.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Payoff Profile of a Call Option<\/h3>\n\n\n\n<p>Recall that in call options, the buyer has the right but not the obligation to buy the underlying. Moreover, the call option will only be exercised&nbsp;if the payoff is positive; otherwise, the option expires worthless, and the option buyer incurs a loss equal to the option premium.<\/p>\n\n\n\n<p>Intuitively for a call option, the buyer would only exercise the option if \\(S_T&gt;X\\). As such, the payoff to the buyer at expiration is given by:<\/p>\n\n\n\n<p>$$\\text{C}_{\\text{T}}=\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X})$$<\/p>\n\n\n\n<p>Conversely, the payoff to the seller at expiration is:<\/p>\n\n\n\n<p>$$-\\text{C}_{\\text{T}}=-(\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X}))$$<\/p>\n\n\n\n<p>Note also that the option buyer pays the seller the call option premium (\\(c_0\\)) at time \\(t=0\\)&nbsp;for the right to buy the underlying \\(S_T\\)&nbsp;at an exercise price of \\(X\\) at time \\(t=T\\). Therefore, the profit the buyer will earn is calculated as follows:<\/p>\n\n\n\n<p>$$\\Pi =\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X})-\\text{C}_0$$<\/p>\n\n\n\n<p>For the call option seller\u2019s profit, it is given by:<\/p>\n\n\n\n<p>$$\\Pi =-\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X})+\\text{C}_0$$<\/p>\n\n\n\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"641\" data-id=\"43432\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option.png\" alt=\"\" class=\"wp-image-43432\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-300x197.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-768x505.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-400x263.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/figure>\n<\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"641\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-Mod-2.png\" alt=\"\" class=\"wp-image-43433\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-Mod-2.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-Mod-2-300x197.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-Mod-2-768x505.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Profit-of-a-call-option-Mod-2-400x263.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/figure>\n\n\n\n<p>From the above graphs, the following points can be deduced:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>The maximum loss <\/strong>for the long (buyer) of the call option is the <strong>premium<\/strong>, and the <strong>profit <\/strong>for the <strong>buyer <\/strong>is <strong>unlimited<\/strong>.<\/li>\n\n\n\n<li><strong>The maximum loss <\/strong>for the short (seller) of the call option is <strong>unlimited<\/strong>, and the <strong>greatest profit <\/strong>the <strong>seller <\/strong>can make is the premium.<\/li>\n\n\n\n<li>The <strong>breakeven point <\/strong>for both the long (buyer) and short (seller) is the strike<strong> price plus the premium<\/strong>.<\/li>\n\n\n\n<li>The sum of the profits between the long and the short equals zero since options trading is a <strong>zero-sum game<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Example: Calculating the Value (Payoff) of a Call Option at Expiration<\/h3>\n\n\n\n<p>Consider a one-year call option with a premium of $2 and a strike price of $30. If the price of the underlying at expiration is $40, the value at expiration is <em>closest <\/em>to:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Solution<\/h4>\n\n\n\n<p>At $40, the stock price is above the exercise price. Therefore, the option has a value of:<\/p>\n\n\n\n<p>$$\\begin{align*}c_T&amp;=\\text{max}(0, S_T-X)\\\\&amp;=\\$40-\\$30\\\\&amp;=\\$10\\end{align*}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Example: Calculating the Profit\/Loss of a Call Option at Expiration<\/h3>\n\n\n\n<p>Consider a one-year call option with a premium of $2 and a strike price of $30. If the price of the underlying at expiration is $40, the value and profit\/loss at expiration is <em>closest <\/em>to:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Solution<\/h4>\n\n\n\n<p>The buyer of the call option will exercise the option and make a profit of:<\/p>\n\n\n\n<p>$$\\begin{align*}\\Pi &amp;=\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X})-\\text{C}_0\\\\&amp;=\\$10-\\$2\\\\&amp;=\\$8\\end{align*}$$<\/p>\n\n\n\n<p>Intuitively the seller is at a loss of:<\/p>\n\n\n\n<p>$$\\begin{align*}\\Pi &amp;=-\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X})+\\text{C}_0\\\\&amp;-10+2\\\\&amp;-\\$8\\end{align*}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Payoff Profile of a Put Option<\/h3>\n\n\n\n<p>For a put option, the buyer has the right but not an obligation to exercise the option at expiry. Exercising the option means that at expiration, the buyer sells the underlying \\(S_T\\)&nbsp;at the exercise price X. As such, the put option is only exercisable if \\(S_T &lt;X\\).<\/p>\n\n\n\n<p>Therefore, the payoff to the buyer is given by:<\/p>\n\n\n\n<p>$$p_T=\\text{max}(0,X-S_T)$$<\/p>\n\n\n\n<p>Conversely, the payoff to the put option seller is:<\/p>\n\n\n\n<p>$$-p_T=-(\\text{max}(0,X-S_T))$$<\/p>\n\n\n\n<p>Recall that the put option buyer pays the seller a put option premium (\\(p_0\\)). &nbsp;Therefore, the profit to the option buyer is given by:<\/p>\n\n\n\n<p>$$\\Pi =\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X})-\\text{p}_0$$<\/p>\n\n\n\n<p>Conversely, the profit to the options\u2019 seller is given by:<\/p>\n\n\n\n<p>$$\\Pi =-\\text{max}(0,\\text{S}_{\\text{T}}-\\text{X})+\\text{p}_0$$<\/p>\n\n\n\n<p>The following graphs can represent the profit of a put option to both buyer and the seller:<\/p>\n\n\n\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-2 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"888\" height=\"569\" data-id=\"43438\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Higher.jpg\" alt=\"\" class=\"wp-image-43438\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Higher.jpg 888w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Higher-300x192.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Higher-768x492.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Higher-400x256.jpg 400w\" sizes=\"auto, (max-width: 888px) 100vw, 888px\" \/><\/figure>\n<\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"888\" height=\"569\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Lower.png\" alt=\"\" class=\"wp-image-43439\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Lower.png 888w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Lower-300x192.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Lower-768x492.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/12\/Lower-400x256.png 400w\" sizes=\"auto, (max-width: 888px) 100vw, 888px\" \/><\/figure>\n\n\n\n<p>From the above graphs and formulas, the following can be deduced:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The<strong> maximum loss <\/strong>for the long (buyer) of the put option is the <strong>premium<\/strong> paid, and the <strong>profit <\/strong>for the <strong>buyer <\/strong>is <strong>limited<\/strong> to the difference between the strike price and the underlying asset&#8217;s price (or strike price if the underlying price falls to 0) at expiration. The price of the underlying asset cannot drop below zero, so the maximum profit for the buyer cannot be unlimited.<\/li>\n\n\n\n<li>The<strong> maximum loss <\/strong>for the short (seller) of the put option is <strong>limited <\/strong>to the strike price,&nbsp;and the <strong>greatest profit <\/strong>the <strong>seller <\/strong>can make is the premium.<\/li>\n\n\n\n<li>The <strong>breakeven point <\/strong>for both the long (buyer) and short (seller) is the strike<strong> price minus the premium<\/strong>.<\/li>\n\n\n\n<li>The sum of the profits between the long and the short party equals zero. Options trading is a <strong>zero-sum game<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Example: Calculating the Payoff (Value) and Profit\/Loss of Put Options<\/h3>\n\n\n\n<p>Consider a one-year put option with a premium of $3 and a strike of $30. If the underlying price at expiration is $20, the value and profit\/loss at expiration is <em>closest <\/em>to:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Solution<\/h4>\n\n\n\n<p>At $20, the stock price is below the exercise price. Therefore, the options have a value of:<\/p>\n\n\n\n<p>$$\\begin{align*}p_T&amp;=\\text{max}(0,X-S_T)\\\\&amp;=\\$30-\\$20\\\\&amp;=\\$10\\end{align*}$$<\/p>\n\n\n\n<p>The buyer of the put option will exercise the option; therefore, he makes a profit of:<\/p>\n\n\n\n<p>$$\\begin{align*}\\Pi &amp;=\\text{max}(0,\\text{X}-\\text{S}_{\\text{T}})-\\text{p}_0\\\\&amp;=\\$10-\\$3\\\\&amp;=\\$7\\end{align*}$$<\/p>\n\n\n\n<p>The seller\/writer makes a loss of:<\/p>\n\n\n\n<p>$$\\begin{align*}\\Pi &amp;=-(\\text{max}(0,X-\\text{S}_{\\text{T}}))+\\text{p}_0\\\\&amp;=-\\$10+\\$3\\\\&amp;=-\\$7\\end{align*}$$<\/p>\n\n\n<blockquote>\n<h2>Question<\/h2>\n<p>If a put option has a premium of $3 and the exercise price is $100, and the price of the underlying is $105, the value at expiration and the profit to the option seller are <em>closest to<\/em>:<\/p>\n<p>A. Value\u00a0= -$3;\u00a0Profit = $0<\/p>\n<p>B. Value = $0; Profit = $8<\/p>\n<p>C. Value = $0; Profit = $3<\/p>\n<h4>Solution<\/h4>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>Note that the exercise price ($100) is less than the underlying price ($105), so we have a situation where \\(S_T\\geq\\text{X}\\). Therefore, the option expires worthless, so the value (payoff) at maturity is zero (\\(p_T=0\\)).<\/p>\n<p>Intuitively, the profit to the seller is equal to the option premium paid by the option buyer (\\(\\Pi =p_0\\)),\u00a0which is $3. From the perspective of the put buyer (long put),\u00a0\u00a0\\(p_T=0\\)\u00a0and\u00a0\\(\\Pi =-p_0\\)\u00a0or a loss of $3.<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Define the following: \\(c_T =\\) Value of the call at expiration. \\(p_T =\\) Value of a put option at expiration. \\(S_T =\\) Price of the underlying at time T. \\(X =\\) Exercise price. \\(c_0=\\) Call option premium. \\(p_0 =\\) Put&#8230;<\/p>\n","protected":false},"author":13,"featured_media":43439,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10],"tags":[36],"class_list":["post-43426","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-derivatives","tag-value-at-expiration-and-profit-from-a-long-or-a-short-position-in-a-call-or-put-option","blog-post","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin 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