{"id":1499,"date":"2020-04-14T17:33:00","date_gmt":"2020-04-14T17:33:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1499"},"modified":"2026-02-10T06:15:04","modified_gmt":"2026-02-10T06:15:04","slug":"put-call-parity-european-options","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/put-call-parity-european-options\/","title":{"rendered":"Put-Call Parity for European Options"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n\n  \"name\": \"Basics of Derivative Pricing and Valuation (2025 Level I CFA\u00ae Exam \u2013 Derivative \u2013 Module 2)\",\n\n  \"description\": \"This video lesson covers Topic 7 \u2013 Derivatives, Module 2 \u2013 Basics of Derivative Pricing and Valuation. It explores core concepts like arbitrage, replication, and risk neutrality in pricing derivatives. Key highlights include the valuation of forward and futures contracts, factors affecting options pricing, put\u2013call parity, and the binomial model. It also differentiates between European and American options and explains their respective value determinants.\",\n\n  \"uploadDate\": \"2022-06-09T00:00:00+00:00\",\n\n  \"thumbnailUrl\": \"https:\/\/analystprep.com\/path-to-thumbnail\/derivative-pricing-thumbnail.jpg\",\n\n  \"contentUrl\": \"https:\/\/youtu.be\/0Geaej45v7w\",\n\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/0Geaej45v7w\",\n\n  \"duration\": \"PT1H08M27S\",\n\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/path-to-logo\/logo.jpg\",\n      \"width\": 600,\n      \"height\": 60\n    }\n  }\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\": \"Using put-call parity, what is the price of the European call option?\",\n    \"text\": \"European put and call options both have an exercise price of $50 that expires in 120 days. The underlying asset is priced at $52 and makes no cash payments during the life of the option. The risk-free rate is 4.5% and the put is selling for $3.80. According to the put-call parity, the price of the call option should be closest to:\\n\\nA. $6.52\\nB. $6.32\\nC. $7.12\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A. $6.52. Using put-call parity, c\u2080 = p\u2080 + S\u2080 \u2212 X\/(1+r)\u1d40 = 3.80 + 52 \u2212 (50 \/ 1.045^(120\/365)) = 6.52.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/put-call-parity-european-options\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e.png\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e.png\",\n  \"caption\": \"Put\u2013Call Parity for European Options\",\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\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/put-call-parity-european-options\/\"\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/0Geaej45v7w?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>Although parity means equivalence, puts and calls are not equivalent. However, there&nbsp;is a relationship between the price of a call and its corresponding put option. This is referred to as put-call parity.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Protective Puts and Fiduciary Calls<\/h2>\n\n\n\n<p>First, let&#8217;s consider a protective put strategy where an investor who holds the underlying asset purchases a protective put option. At inception, the investor commits S<sub>0<\/sub>, the underlying asset cost, and p<sub>0<\/sub>, the put premium.<\/p>\n\n\n\n<p>$$ \\text{Value at inception} = p_0 + S_0 $$<\/p>\n\n\n\n<!-- TOP CTA \u2013 Full Width Outline Button -->\n<div style=\"margin:24px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:block;\n       width:100%;\n       padding:14px 0;\n       border:2px solid #3b6fd8;\n       border-radius:50px;\n       font-size:18px;\n       font-weight:500;\n       text-align:center;\n       text-decoration:none;\n       color:#3b6fd8;\n       background-color:#f4f6f9;\n       box-sizing:border-box;\n     \">\n     Practice put call parity questions with free trial access.\n  <\/a>\n<\/div>\n\n\n\n<p>A protective put gives the holder a limited downside but reduces his upside a bit because of the cost paid to buy the put option. Graphically, it can\u00a0 be represented as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-10041\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e.png\" alt=\"protective-puts-and-fiduciary-calls\" width=\"974\" height=\"641\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e-300x197.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e-768x505.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e-400x263.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/p>\n<p>A strategy where an investor enters into a fiduciary call means an investor buys a call option on an underlying asset and a risk-free zero-coupon bond with a face value equal to the exercise price.<\/p>\n<p>$$ \\text{Value at inception} = c_0 + \\frac{X}{(1+r)^T} $$<\/p>\n<p>The payoff graph from this strategy will be the exact same as that of the protective put strategy.<\/p>\n<h2><strong>Put-Call Parity Arbitrage<\/strong><\/h2>\n<p>By examing the payoff profiles of a protective put and a fiduciary call, we note that they are identical. Therefore, if two combinations of assets or portfolios of assets have the same payoff, their acquisition cost must be identical. In any other case, there is an arbitrage opportunity. In other words, someone can make a risk-free profit by buying the cheaper one and selling the most expensive one.<\/p>\n<p>As a result of the no-arbitrage principle, we can set the value at the\u00a0inception of the protection call equal to the value at the\u00a0inception of the protective put as follows:<\/p>\n<p>$$ c_0 +\u00a0\\frac{X}{(1+r)^T} = p_0 + S_0 $$<\/p>\n<p>This equation is a key concept in derivatives pricing called\u00a0<strong>put-call parity.\u00a0<\/strong>This formula equates the value of calls and puts through equivalent portfolios. It must be assumed that since these are European options, they have the same strike, same expiry date, and the same underlying asset.<\/p>\n<p>By re-arranging the prior equation:<\/p>\n<p>$$ c_0\u00a0 = p_0 + S_0 &#8211;\u00a0\\frac{X}{(1+r)^T} $$<\/p>\n<p>The right side of this equation is equivalent to a call option and is referred to as a <strong>synthetic call<\/strong>. It consists of a long put, a long position in the underlying asset, and a short position in the risk-free bond.<\/p>\n<p>Another re-arrangement can be made to isolate the put option:<\/p>\n<p>$$\u00a0 p_0 =\u00a0c_0 +\\frac{X}{(1+r)^T} &#8211; S_0\u00a0 $$<\/p>\n<p>The right side of this equation is now referred to as a <strong>synthetic put<\/strong>\u00a0which consists of a long call, a short position in the underlying, and a long position in the risk-free bond.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>European put and call options both have an exercise price of $50 that expires in 120 days. The underlying asset is priced at $52 and makes no cash payments during the life of the option. The risk-free rate is 4.5% and the put is selling for $3.80. According to the put-call parity, the price of the call option should be <em>closest<\/em> to:<\/p>\n<p>A. $6.52<\/p>\n<p>B. $6.32<\/p>\n<p>C. $7.12<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is A.<\/p>\n<p>c<sub>0<\/sub>\u00a0=\u00a0p<sub>0<\/sub>\u00a0+\u00a0S<sub>0<\/sub> \u2013 X\/(1+r)<sup>T<\/sup><\/p>\n<p>c<sub>0<\/sub> = 3.80 + 52 \u2013 (50\/1.045<sup>120\/365<\/sup>) = 6.52<\/p>\n<\/blockquote>\n\n\n<!-- BOTTOM CTA \u2013 Refined Version -->\n<div style=\"text-align:center; background-color:#f4f6f9; padding:35px 20px; border-radius:12px; margin-top:40px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:inline-block;\n       padding:14px 34px;\n       background-color:#3b6fd8;\n       color:#ffffff;\n       border-radius:50px;\n       font-size:16px;\n       font-weight:600;\n       text-decoration:none;\n       margin-bottom:18px;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:700px; margin:0 auto; font-size:16px; line-height:1.6; color:#333;\">\n    Master European option pricing and put call parity relationships with exam-style derivatives questions, step by step solutions, and timed practice inside AnalystPrep\u2019s free trial.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Although parity means equivalence, puts and calls are not equivalent. However, there&nbsp;is a relationship between the price of a call and its corresponding put option. This is referred to as put-call parity. Protective Puts and Fiduciary Calls First, let&#8217;s consider&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10],"tags":[],"class_list":["post-1499","post","type-post","status-publish","format-standard","hentry","category-derivatives","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Put-Call Parity for European Options | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Put-call parity ensures no-arbitrage pricing by linking the cost of portfolios with identical payoffs. 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