{"id":17526,"date":"2021-07-10T22:27:35","date_gmt":"2021-07-10T22:27:35","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17526"},"modified":"2026-01-24T07:07:01","modified_gmt":"2026-01-24T07:07:01","slug":"explain-the-relationships-between-the-values-of-a-callable-or-putable-bond-the-underlying-option-free-straight-bond-and-the-embedded-option","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/explain-the-relationships-between-the-values-of-a-callable-or-putable-bond-the-underlying-option-free-straight-bond-and-the-embedded-option\/","title":{"rendered":"Valuing Embedded Options"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/price-yield-curve-callable-bond\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond.jpg\",\r\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond.jpg\",\r\n  \"caption\": \"Price\u2013yield curve of a callable bond showing the impact of the call option\",\r\n  \"width\": 1590,\r\n  \"height\": 1562,\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    \"url\": \"https:\/\/analystprep.com\/\"\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  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/price-yield-curve-putable-bond\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond.jpg\",\r\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond.jpg\",\r\n  \"caption\": \"Price\u2013yield curve of a putable bond illustrating downside protection from the put option\",\r\n  \"width\": 1590,\r\n  \"height\": 1142,\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    \"url\": \"https:\/\/analystprep.com\/\"\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\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Value of the Issuer Call Option\",\r\n    \"text\": \"The value of an option-free, 5% annual coupon bond that matures in three years is $106.80. If the value of a callable bond with similar terms is $105.50, the value of the issuer call option is closest to:\\n\\nA. -$1.30.\\n\\nB. $0.00.\\n\\nC. $1.30.\",\r\n    \"answerCount\": 3,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"C. $1.30.\"\r\n    },\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"A. -$1.30.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"B. $0.00.\"\r\n      }\r\n    ]\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>According to the arbitrage-free framework, the value of a bond with embedded options is the sum of the arbitrage-free value of the option-free bond (straight bond) and the arbitrage-free values of any embedded options.<\/p>\r\n<h2>Embedded Calls<\/h2>\r\n<p>A callable bond is similar to a portfolio with a long position on a straight bond and a short position on a call option. The issuer makes the decision to exercise the call option.<\/p>\r\n<p>$$ \\begin{align*} \\text{Value of callable bond}&amp; =\\text{Value of straight bond} \\\\ &amp; \u2013\\text{Value of issuer call option} \\\\ \\text{Value of issuer call option}&amp; =\\text{Value of straight bond} \\\\ &amp; \u2013 \\text{Value of callable bond} \\\\ \\end{align*} $$<\/p>\r\n<p>i.e.,<\/p>\r\n<p>$$ \\begin{align*} V_{\\text{Callable}} &amp;= V_{\\text{Straight}} \u2013 V_{\\text{Call}} \\\\ V_{\\text{Call}} &amp;= V_{\\text{Straight}} \u2013 V_{\\text{Callable}} \\end{align*} $$<\/p>\r\n<p>This has been illustrated in the following diagram:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26505\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond.jpg\" alt=\"Price-Yield Curve of a Callable Bond\" width=\"1590\" height=\"1162\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond-1536x1123.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Callable-Bond-400x292.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>It is worth noting that callable bonds are riskier, have higher spreads, and generate higher yields relative to put bonds.<\/p>\r\n<h2>Embedded Puts<\/h2>\r\n<p>The investor has a long position in both the straight bond and the put option.<\/p>\r\n<p>$$ \\begin{align*} \\text{Value of putable bond} &amp; =\\text{Value of straight bond} \\\\ &amp; +\\text{Value of investor put option} \\\\ \\text{Value of investor put option} &amp; =\\text{Value of putable bond} \\\\ &amp; \u2013 \\text{Value of straight bond} \\end{align*} $$<\/p>\r\n<p>i.e.,<\/p>\r\n<p>$$ \\begin{align*} V_{\\text{Putable}}= V_{\\text{Straight}}+ V_{\\text{Put}} \\\\ V_{\\text{Put}}=V_{\\text{Putable}} \u2013 V_{\\text{Straight}} \\end{align*} $$<\/p>\r\n<p>The concept of embedded puts is illustrated in the following diagram:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26506\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond.jpg\" alt=\"\" width=\"1590\" height=\"1142\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond-300x215.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond-1024x735.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond-768x552.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond-1536x1103.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Price-Yield-Curve-of-a-Putable-Bond-400x287.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>The value of an option free, 5% annual coupon bond that matures in three years is $106.80. If the value of a callable bond with similar terms is $105.50, the value of the issuer call option is <em>closest to<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>-$1.30.<\/li>\r\n\t<li>$0.00.<\/li>\r\n\t<li>$1.30.<\/li>\r\n<\/ol>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<p><strong>The correct answer is C.<\/strong><\/p>\r\n<p>$$ \\begin{align*} \\text{Value of issuer call option} &amp;=\\text{Value of straight bond} \\\\ &amp; &#8211; \\text{Value of callable bond} \\\\ V_{\\text{Callable}} &amp;=$105.50 \\\\ V_{\\text{Straight}} &amp;=$106.80 \\\\ V_{\\text{Call}} &amp;=$106.80-$105.50=$1.30 \\end{align*} $$<\/p>\r\n<\/blockquote>\r\n<p>Reading 30: Valuation and Analysis of Bonds with Embedded Options<\/p>\r\n<p><em>LOS 30 (b) Explain the relationships between the values of a callable or putable bond, the underlying option free (straight) bond, and the embedded option.<\/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","protected":false},"excerpt":{"rendered":"<p>According to the arbitrage-free framework, the value of a bond with embedded options is the sum of the arbitrage-free value of the option-free bond (straight bond) and the arbitrage-free values of any embedded options. Embedded Calls A callable bond is&#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-17526","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Valuing Embedded Options in Callable and Putable Bonds<\/title>\n<meta name=\"description\" content=\"Learn how callable and putable bonds relate to option-free bonds and embedded options, including portfolio decomposition and valuation relationships.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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