{"id":17534,"date":"2021-07-10T22:47:45","date_gmt":"2021-07-10T22:47:45","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17534"},"modified":"2026-06-25T20:26:36","modified_gmt":"2026-06-25T20:26:36","slug":"describe-how-the-arbitrage-free-framework-can-be-used-to-value-a-bond-with-embedded-options","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-how-the-arbitrage-free-framework-can-be-used-to-value-a-bond-with-embedded-options\/","title":{"rendered":"Using Arbitrage Free Framework to Value Bonds with Embedded Options"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Exhibit 1 gives a binomial interest rate tree constructed assuming an interest rate volatility of 0%. Based on Exhibit 1, the value of a two-year 8% annual coupon default-free bond, callable at $102 one year from today at zero volatility is closest to:\",\r\n    \"answerCount\": 3,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is B. $105.25. At the upper node at time 1, the bond value is $100.93, so the bond is not called at $102. At the lower node at time 1, the bond value is $102.86, so the bond is called at $102. The value at time 0 is calculated by discounting the expected node values plus coupon payments: V0 = 0.5 \u00d7 [(100.93 + 8) \/ 1.04 + (102 + 8) \/ 1.04] = $105.25.\"\r\n    },\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"$104.29.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"$105.25.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"$105.67.\"\r\n      }\r\n    ]\r\n  }\r\n}\r\n<\/script>\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates.jpg\",\r\n  \"caption\": \"Exhibit-1-One-Year-Forward-Rates\",\r\n  \"width\": 1590,\r\n  \"height\": 1611,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/5zZuTYmjxW8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>Recall that given either spot rates, forward rates, or par rates, one can determine the value of a straight bond by discounting its cashflows.<\/p>\r\n<p>However, valuing the embedded option requires one-period forward rates. This is because we work out the value of the bond at different points in time in the future to see if the embedded option is worth exercising at those respective points in time. Thus, the binomial tree interest rates framework applies here.<\/p>\r\n\r\n<div style=\"margin: 18px 0;\"><a style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #2F5BFF; border-radius: 18px; color: #ffffff ; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8 ;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">Practice valuing bonds with embedded options using our CFA Free Trial.<\/a><\/div>\r\n\r\n<h4>Example: Value of Callable and Putable Bonds<\/h4>\r\n<p>We&#8217;ve been given the following yield curve:<\/p>\r\n<p>$$ \\begin{array}{c|c|c} \\textbf{Maturity (Years)} &amp; \\textbf{Spot Rate} &amp; \\textbf{One-Year Forward Rate} \\\\ \\hline 1 &amp; 1.20\\% &amp; 1.20\\% \\\\ \\hline 2 &amp; 2.31\\% &amp; 3.44\\% \\\\ \\hline 3 &amp; 3.35\\% &amp; 5.45\\% \\end{array} $$<\/p>\r\n<p>We want to value a 5% annual coupon bond that matures in 3 years assuming that it&#8217;s:<\/p>\r\n<ol type=\"1\">\r\n\t<li>Callable at par either one year or two years from today.<\/li>\r\n\t<li>Putable at par either one year or two years from today.<\/li>\r\n\t<li>A straight bond with zero volatility.<\/li>\r\n<\/ol>\r\n<ol type=\"1\">\r\n\t<li>Callable Bond\r\n\r\n<p>$$ V_{\\text{2Callable}}=\\frac{105}{1.0545}=$99.57 $$<\/p>\r\n<p>Therefore, the bond is not called in year two as \\(99.57 &lt; 100\\).<\/p>\r\n<p>$$ V_{\\text{1Callable}}=\\frac{99.57+5}{1.0344}=$101.09 $$<\/p>\r\n<p>The bond is called in year one as 101.09&gt;100. The investor will only receive $100. Thus, the value of the bond in one year at this node is $100.<\/p>\r\n<p>$$ V_{\\text{0Callable}}=\\frac{100+5}{1.0120}=$103.75 $$<\/p>\r\n<\/li>\r\n\t<li>Putable Bond\r\n\r\n<p>$$ V_{\\text{2Putable}}=\\frac{105}{1.0545}=$99.57 $$<\/p>\r\n<p>Therefore, the put option will be exercised in year two as \\(99.57 &lt; 100\\).<\/p>\r\n<p>Therefore, the 99.57 will be replaced by the exercise price of $100.<\/p>\r\n<p>$$ V_{\\text{1Putable}}=\\frac{100+5}{1.0344}=$101.51 $$<\/p>\r\n<p>The put option will not be exercised in year one as 101.51&gt;100.<\/p>\r\n<p>$$ V_{\\text{0Putable}}= \\frac{101.51+5}{1.0120}=$105.25 $$<\/p>\r\n<\/li>\r\n\t<li>Straight bond\r\n\r\n<p>$$ \\begin{align*} V_2 &amp;=\\frac{105}{1.0545}=99.57 \\\\ V_1 &amp;=\\frac{99.57+5}{1.0344}=101.09 \\\\ V_0 &amp;=\\frac{101.09+5}{1.0120}=104.83 \\end{align*} $$<\/p>\r\n<\/li>\r\n\t<li>Value of issuer call option\r\n\r\n<p>$$ \\begin{align*} V_{\\text{Call}}&amp; = \\text{Value}_{\\text{Straight}}\\ \u2013 V_{\\text{Callable}} \\\\ &amp; =$104.83-$103.75=$1.08 \\end{align*} $$<\/p>\r\n<\/li>\r\n\t<li>Value of investor put option\r\n\r\n<p>$$ \\begin{align*} V_{\\text{Put}}&amp; = V_{\\text{Putable}} \u2013 V_{\\text{Straight}} \\\\ &amp;=$105.25-$104.83=$0.42 \\end{align*} $$<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p>The above calculations are shown in the following table:<\/p>\r\n<p>$$ \\begin{array}{l|c|c|c|c} &amp; \\textbf{Today} &amp; \\textbf{Year 1} &amp; \\textbf{Year 2} &amp; \\textbf{Year 3} \\\\ \\hline \\text{Cash flow} &amp; &amp; 5 &amp; 5 &amp; 105 \\\\\\hline \\text{One year Forward rate} &amp; &amp; 1.20\\% &amp; 3.44\\% &amp; 5.45\\% \\\\ \\hline \\text{Value of the callable} &amp; 103.75 &amp; 101.10 &amp; 99.57 &amp; \\\\ \\text{bond} &amp; &amp; &amp; \\\\ \\hline &amp; &amp; {\\text{Called at }$100} &amp; \\text{Not Called} &amp; \\\\ \\hline \\text{Value of the putable} &amp; 105.25 &amp; 101.51 &amp; 99.57 &amp; \\\\ \\text{bond} &amp; &amp; &amp; \\\\ \\hline &amp; &amp; \\text{Not put} &amp; {\\text{Put at } $100} \\\\ \\hline \\text{Value of the straight} &amp; 104.84 &amp; 101.10 &amp; 99.57 &amp; \\\\ \\text{bond} &amp; &amp; &amp; \\\\ \\end{array} $$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Exhibit 1 gives a binomial interest rate tree constructed assuming an interest rate volatility of 0%.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26595\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates.jpg\" alt=\"Exhibit 1: One Year Forward Rates\" width=\"1590\" height=\"1611\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates-296x300.jpg 296w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates-1011x1024.jpg 1011w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates-768x778.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates-1516x1536.jpg 1516w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-One-Year-Forward-Rates-400x405.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Based on Exhibit 1, the value of a two-year 8% annual coupon default-free bond, callable at $102 one year from today at zero volatility is <em>closest to<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>$104.29<\/li>\r\n\t<li>$105.25<\/li>\r\n\t<li>$105.67<\/li>\r\n<\/ol>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<p><strong>The correct answer is B.<\/strong><\/p>\r\n<p>The bond\u2019s value at the upper node for time 1,<\/p>\r\n<p>$$ V_{1u}=0.5\\left[\\frac{108}{1.07}+\\frac{108}{1.07}\\right]=$100.93 $$<\/p>\r\n<p>Thus, the bond is not called at 102.<\/p>\r\n<p>The bond\u2019s value at the lower node for time 1,<\/p>\r\n<p>$$ V_{1d}=0.5\\left[\\frac{108}{1.05}+\\frac{108}{1.05}\\right]=$102.86 $$<\/p>\r\n<p>The bond is called at $102.<\/p>\r\n<p>The value of the bond at node 0:<\/p>\r\n<p>$$ V_0=0.5\\left[\\frac{100.93+8}{1.04}+\\frac{102+8}{1.04}\\right]=$105.25 \u2003$$<\/p>\r\n<\/blockquote>\r\n<p>Reading 30: Valuation and Analysis of Bonds with Embedded Options<\/p>\r\n<p><em>LOS 30 (c) Describe how the arbitrage-free framework can be used to value a bond with embedded options.<\/em><\/p>\r\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                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\n                data-elfsight-pricing-table-version=\"2.6.1\"\n                data-elfsight-widget-id=\"elfsight-pricing-table-3\">\n            <\/div>\n            \r\n\r\n\r\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a> <p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Review how the arbitrage-free framework is used to value bonds with embedded options with CFA Level 2 study notes, practice questions, mock exams, and video lessons designed to strengthen your exam preparation.<\/p>\r\n <\/div>","protected":false},"excerpt":{"rendered":"<p>Recall that given either spot rates, forward rates, or par rates, one can determine the value of a straight bond by discounting its cashflows. However, valuing the embedded option requires one-period forward rates. This is because we work out the&#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":[102,472],"tags":[],"class_list":["post-17534","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-fixed-income","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Arbitrage-Free Valuation of Bonds with Options | CFA L2<\/title>\n<meta name=\"description\" content=\"Learn how to value callable and putable bonds using the arbitrage-free framework, binomial interest rate trees, and forward rates.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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