{"id":18563,"date":"2021-07-27T16:15:00","date_gmt":"2021-07-27T16:15:00","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=18563"},"modified":"2025-11-26T19:32:49","modified_gmt":"2025-11-26T19:32:49","slug":"calculate-the-no-arbitrage-values-of-european-and-american-options-using-a-two-period-binomial-model","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-the-no-arbitrage-values-of-european-and-american-options-using-a-two-period-binomial-model\/","title":{"rendered":"No-Arbitrage Values of Options"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Using a two-period binomial model, what is the early exercise premium of the American-style put option?\",\n    \"text\": \"Using a two-period binomial model, the early exercise premium of an American-style put option is closest to:\\n\\n1. $0.00\\n2. $0.06\\n3. $3.36\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is 2. The early exercise premium equals the American put value minus the European put value, which in this example is approximately $0.06.\"\n    }\n  }\n}\n<\/script><\/p>\n<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@graph\": [\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example.jpg\",\n      \"caption\": \"Binomial tree call option example\",\n      \"width\": 1590,\n      \"height\": 1161,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example.jpg\",\n      \"caption\": \"Binomial tree put option example\",\n      \"width\": 1590,\n      \"height\": 1212,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example-1024x740.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example-1024x740.jpg\",\n      \"caption\": \"Value of an American put option example\",\n      \"width\": 1024,\n      \"height\": 740,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example-768x574.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example-768x574.jpg\",\n      \"caption\": \"Value of a European put option example\",\n      \"width\": 768,\n      \"height\": 574,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2-768x556.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2-768x556.jpg\",\n      \"caption\": \"Value of an American put option example (tree continuation)\",\n      \"width\": 768,\n      \"height\": 556,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    }\n  ]\n}\n<\/script><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/jmbJLCdBEgs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Valuing European Options<\/h2>\n<p>A <em><strong>European option<\/strong><\/em> is an option that can only be exercised at expiry. Let&#8217;s consider a simple example to better illustrate the concept.<\/p>\n<h4>Example: The Value of a European Option<\/h4>\n<p>Consider a stock with an initial price of $70 and a risk-free rate of 1% per year. The asset price can move up by 10% or down by 10%. The price of a European (1) call option and (2) put option with two years to maturity and a strike price of $80 using a two-period binomial model is <em>closest to<\/em>:<\/p>\n<h4>Solution 1:\u00a0Call Option<\/h4>\n<p>Re<span class=\"primary-bg\">member<\/span> that the two-period binomial formula for pricing a call option from the previous section:<\/p>\n<p>$$ c_o=\\frac{q^2c_{uu} + 2q\\left(1-q\\right)c_{ud} +\\left(1 &#8211; q\\right)^2c_{dd}}{\\left(1 + r\\right)^2} $$<\/p>\n<p>Where:<\/p>\n<p>\\(u\\) = Up-move factor = 1.10<\/p>\n<p>\\(d\\) = Down-move factor = 0.90<\/p>\n<p>and the risk-neutral probability, \\(q\\), is given by:<\/p>\n<p>$$ q=\\frac{\\left(1+r\\right)-d}{u-d} $$<\/p>\n<p>The stock prices at each node and the call option payoffs are shown in the following binomial tree.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26436\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example.jpg\" alt=\"1 - Binomial Tree - Call Option Example\" width=\"1590\" height=\"1161\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/1-Binomial-Tree-Call-Option-Example-400x292.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>The risk-neutral probability, \\(q\\), is given by:<\/p>\n<p>$$ \\begin{align*} q &amp;=\\frac{\\left(1+r\\right)-d}{u-d} \\\\ q &amp;=\\frac{\\left(1.01\\right)-0.9}{\\left(1.1-0.9\\right)}=0.55 \\end{align*} $$<\/p>\n<p>The two-period binomial value of the call option:<\/p>\n<p>$$ \\begin{align*} c_o &amp;=\\frac{q^2c_{uu} + 2q\\left(1-q\\right)c_{ud}+\\left(1 &#8211; q\\right)^2c_{dd}}{\\left(1 + r\\right)^2} \\\\ c_0 &amp;=\\frac{{0.55}^2\\times$4.70+2\\times0.55\\left(1-0.55\\right)\\times0+\\left(1-0.55\\right)^2\\times0}{\\left(1.01\\right)^2} \\\\ c_0 &amp;=$1.39 \\end{align*} $$<\/p>\n<h4>Solution 2: Put Option<\/h4>\n<p>The stock prices and the put option payoffs are shown in the following two-period binomial tree:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26437\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example.jpg\" alt=\"2 - Binomial Tree - Put Option Example\" width=\"1590\" height=\"1212\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example-300x229.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example-1024x781.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example-768x585.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example-1536x1171.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/2-Binomial-Tree-Put-Option-Example-400x305.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>We determined \\(q\\) to be 0.55.<\/p>\n<p>Using the formula:<\/p>\n<p>$$ \\begin{align*} p_o &amp;=\\frac{q^2p_{uu} + 2q\\left(1-q\\right)p_{ud} +\\left(1 &#8211; q\\right)^2p_{dd}}{\\left(1 + r\\right)^2} \\\\ p_0 &amp;=\\frac{{0.55}^2\\times0+2\\times0.55\\times0.45\\times$10.70+{0.45}^2\\times$23.30}{{1.01}^2} \\\\ p_0 &amp;=$9.82 \\end{align*} $$<\/p>\n<h2>Valuing American Options<\/h2>\n<p>The difference between an American and a European option is that with an <em><strong>American option<\/strong><\/em>, the holder can exercise the option before the expiry date, not just on the expiry date as is the case with a European option. Thus, the value of an American option may be higher than that of a comparable European option due to the early exercise feature.<\/p>\n<p>The difference between the value of an American option and a comparable European option is referred to as the <strong><em>early exercise premium<\/em>.<\/strong><\/p>\n<p>$$ \\text{Early exercise premium} = \\text{American option value} \u2013 \\text{European option value} $$<\/p>\n<p>American calls on non-dividend paying stocks do not benefit from the early exercise feature. However, American puts may have an early exercise premium and must be checked for early exercise. An early exercise captures the options\u2019 intrinsic value and forgoes the time value.<\/p>\n<p>It is worth noting that deep in the money, American put options cannot be valued simply as the discounted value of the expected future option payouts. Therefore, we must establish whether or not early exercise is optimal at each node by working backward through the binomial tree.<\/p>\n<p>The unexercised American option values at time one is determined using the following formula:<\/p>\n<p>$$ \\begin{align*} p_u &amp; =\\frac{qp_{uu} +\\left(1 &#8211; q\\right)p_{ud}}{(1+r)} \\\\ p_d &amp; =\\frac{qp_{ud}\\ +\\left(1 &#8211; q\\right)p_{dd}}{(1+r)} \\end{align*} $$<\/p>\n<p>The next step is to check each node for early exercise.<\/p>\n<h4>Example: Calculating the Value of an American Put Option<\/h4>\n<p>Consider a non-dividend-paying stock with an initial price of $70 and a risk-free rate of 1% compounded annually. The asset price can either move up by 10% or down by 10%. The price of an American put option with two years to maturity and a strike price of $80 using a two-period binomial model is <em>closest<\/em> to:<\/p>\n<h4>Solution<\/h4>\n<p>The put option\u2019s payoffs at time two have been calculated as illustrated in the following binomial tree.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26438\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example.jpg\" alt=\"3 - Value of an American Put Option - Example\" width=\"1590\" height=\"1149\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example-300x217.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example-1024x740.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example-768x555.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example-1536x1110.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/3-Value-of-an-American-Put-Option-Example-400x289.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>At time one, the unexercised values of the option are determined as follows:<\/p>\n<p>$$ \\begin{align*} p_u &amp; =\\frac{qp_{uu} +\\left(1 &#8211; q\\right)p_{ud}}{(1+r)} \\\\ p_u &amp; =\\frac{0.55\\times$0 +\\left(1 &#8211; 0.55\\right)\\times$10.70}{(1.01)}=$4.77 \\\\ p_d &amp;=\\frac{qp_{ud} +\\left(1 &#8211; q\\right)p_{dd}}{(1+r)} \\\\ p_d &amp; =\\frac{0.55\\times$10.70 +\\left(1 &#8211; 0.55\\right)\\times$23.30}{1.01}=$16.21 \\\\ \\end{align*} $$<\/p>\n<p>The next step is to evaluate whether it is optimal for early exercise at each node:<\/p>\n<ul>\n<li>The value at the up-jump node \\(= max($4.77,$80-$77)=$4.77\\).<\/li>\n<li>The value at the down-jump node \\(= max($16.21,$80-$63) = $17\\).<\/li>\n<\/ul>\n<p>The value of the American put at \\(t=0\\) is then calculated as follows:<\/p>\n<p>$$ \\begin{align*} p_0 &amp;=\\frac{qp_u+\\left(1-q\\right)p_d}{1+r} \\\\ p_0 &amp;=\\frac{\\left(0.55\\times$4.77\\right)+\\left(1-0.55\\right)\\times$17}{1.01}=$10.17 \\end{align*} $$<\/p>\n<p>Remember that the value of a comparable European put option was determined as $9.82.<\/p>\n<p>In addition, notice that the American put is worth more than a similar European put.<\/p>\n<p>$$ \\text{Early exercise premium} = $10.17-$9.82 =$0.35 $$<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Consider the following information:<\/p>\n<ul>\n<li>\\(S_0\\) = $30<\/li>\n<li>K = $29<\/li>\n<li>u = \\(1.3333 (\\frac{4}{3})\\)<\/li>\n<li>d = 0.75<\/li>\n<li>n = 2 years<\/li>\n<li>r = 2% compounded annually<\/li>\n<li>No dividends<\/li>\n<\/ul>\n<p>Using a two-period binomial model, the early exercise premium of an American-style put option is closest <em>to<\/em>:<\/p>\n<ol type=\"A\">\n<li>$0.00.<\/li>\n<li>$0.06.<\/li>\n<li>$3.36.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p><strong>The correct answer is B.<\/strong><\/p>\n<p>$$ \\text{Early exercise premium} = \\text{American put value} \u2013 \\text{European put value} $$<\/p>\n<p><strong>European Put Value<\/strong><\/p>\n<p>We determine the value of the European put by constructing an appropriate binomial tree and applying the relevant formulas as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26440\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example.jpg\" alt=\"4 - Value of an European Put Option - Example\" width=\"1590\" height=\"1188\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example-300x224.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example-1024x765.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example-768x574.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example-1536x1148.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/4-Value-of-an-European-Put-Option-Example-400x299.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Formula:<\/p>\n<p>$$ p_o=\\frac{q^2p_{uu} + 2q\\left(1-q\\right)p_{ud} +\\left(1- q\\right)^2p_{dd}}{\\left(1 + r\\right)^2} $$<\/p>\n<p>Where:<\/p>\n<p>$$ \\begin{align*} q&amp; =\\frac{\\left(1+r\\right)-d}{u-d} \\\\ q &amp;=\\frac{1.02-0.75}{1.3333-0.75}=0.463 \\\\ p_0 &amp;=\\frac{{0.463}^2\\times0+2\\times0.463\\times0.537\\times$0+{0.537}^2\\times$12.125}{{1.02}^2} \\\\ p_0 &amp;=$3.361 \\end{align*} $$<\/p>\n<p><strong>American Put Value<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26441\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2.jpg\" alt=\"5 - Value of an American Put Option - Example 2\" width=\"1590\" height=\"1152\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2-300x217.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2-1024x742.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2-768x556.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2-1536x1113.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/5-Value-of-an-American-Put-Option-Example-2-400x290.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>At time one, the unexercised values of the option are determined as follows:<\/p>\n<p>$$ \\begin{align*} p_u &amp; =\\frac{qp_{uu} +\\left(1 &#8211; q\\right)p_{ud}}{(1+r)} \\\\ p_u &amp; =\\frac{0.463\\times$0 +0.537\\times0}{(1.02)}=$0 \\\\ p_d &amp; =\\frac{qp_{ud} +\\left(1 &#8211; q\\right)p_{dd}}{(1+r)} \\\\ p_d &amp; =\\frac{0.463\\times$0 +0.537\\times$12.125}{1.02}=$6.383 \\end{align*} $$<\/p>\n<p>The next step is to evaluate whether it is optimal for early exercise at each node:<\/p>\n<p>The value at the down-jump node \\(= max($6.383,$29-$22.50) = $6.50\\).<\/p>\n<p>The value of the American put at \\(t=0\\) is then calculated as follows:<\/p>\n<p>$$ \\begin{align*} p_0 &amp;=\\frac{qp_u+\\left(1-q\\right)p_d}{1+r} \\\\ p_0 &amp;=\\frac{\\left(0.463\\times$0\\right)+0.537\\times$6.50}{1.02}=$3.422 \\end{align*} $$<\/p>\n<p>The American put is worth more than a comparable European put as \\($3.422 &gt; $3.361\\).<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\text{Early exercise premium} = $3.422-$3.361=$0.061 $$<\/p>\n<\/blockquote>\n<p>Reading 34: Valuation of Contingent Claims<\/p>\n<p><em>LOS 34 (b) Calculate the no-arbitrage values of European and American options using a two-period binomial model.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Valuing European Options A European option is an option that can only be exercised at expiry. Let&#8217;s consider a simple example to better illustrate the concept. Example: The Value of a European Option Consider a stock with an initial price&#8230;<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,302],"tags":[216,304],"class_list":["post-18563","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-cfa-level-2","tag-derivatives","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>No-Arbitrage Values of Options - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Understand the concepts of early exercise premium, risk-neutral probabilities, and option payoffs. 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