{"id":17567,"date":"2021-07-11T21:19:29","date_gmt":"2021-07-11T21:19:29","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17567"},"modified":"2026-06-25T20:05:36","modified_gmt":"2026-06-25T20:05:36","slug":"explain-the-calculation-and-use-of-option-adjusted-spreads","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/explain-the-calculation-and-use-of-option-adjusted-spreads\/","title":{"rendered":"Option-adjusted Spreads"},"content":{"rendered":"<p><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\/Option-adjusted-spread-OAS.jpg\",\r\n  \"caption\": \"Option-Adjusted Spread (OAS)\",\r\n  \"width\": 1590,\r\n  \"height\": 1195,\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<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\/Binomial-Interest-Rate-Tree-1.jpg\",\r\n  \"caption\": \"Binomial Interest Rate Tree\",\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<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\/Value-of-the-Bond-1024x784.jpg\",\r\n  \"caption\": \"Value of the Bond\",\r\n  \"width\": 1024,\r\n  \"height\": 784,\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<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\/Value-of-the-Bond-with-0.5bps-Constant-Spread-1024x784.jpg\",\r\n  \"caption\": \"Value of the Bond with 0.5 bps Constant Spread\",\r\n  \"width\": 1024,\r\n  \"height\": 784,\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> <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\": \"Relative to bond Y, bond X is most likely:\",\r\n    \"text\": \"The following information relates to two risky callable bonds with similar characteristics and credit quality:\\n\\nBond X: OAS = 30 bps\\nBond Y: OAS = 20 bps\\n\\nRelative to bond Y, bond X is most likely:\\n\\nA. Underpriced.\\nB. Fairly priced.\\nC. Overpriced.\",\r\n    \"answerCount\": 3,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is A. An option-adjusted spread (OAS) that is higher than that of a comparable bond indicates the bond is likely underpriced. Since bond X has a higher OAS than bond Y, bond X is most likely underpriced relative to bond Y.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\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>A workable approach employed in constructing a suitable yield curve for risky bonds involves adding a fixed Z-spread to the one-year forward rates derived from the default-free spot yield curve.<\/p>\r\n<p>Recall that <em><strong>Z-spread<\/strong><\/em> is the basis point spread that would need to be added to the default-free spot curve to equate the present value of cash flows to the market price of a risky bond.<\/p>\r\n<p>The same approach can be used to value risky bonds with embedded options. Option-adjusted spread, which is the spread after adjusting for the options risk, is used in this case.<\/p>\r\n<p><em><strong>Option-adjusted spread<\/strong><\/em> (OAS) is the fixed spread added to the one-year forward rates on the interest rate tree that equates the arbitrage-free value and the market price of a risky bond with embedded options.<\/p>\r\n<p>Note that the Z-spread for a straight bond is its option-adjusted spread assuming volatility of zero.<\/p>\r\n<p>$$ \\text{Option adjusted spread (OAS)} = \\text{Z-Spread} \u2013 \\text{Option cost.} $$<\/p>\r\n<p>The above relationship is illustrated in the following figure.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26603\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Option-adjusted-spread-OAS.jpg\" alt=\"Option adjusted spread (OAS)\" width=\"1590\" height=\"1195\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Option-adjusted-spread-OAS.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Option-adjusted-spread-OAS-300x225.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Option-adjusted-spread-OAS-1024x770.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Option-adjusted-spread-OAS-768x577.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Option-adjusted-spread-OAS-1536x1154.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Option-adjusted-spread-OAS-400x301.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>OAS can be used to assess bond relative values. Two bonds with the <strong>same characteristics<\/strong> and <strong>credit quality<\/strong> must have the same OAS. Otherwise, the bond with the largest OAS is likely to be underpriced (cheap) relative to the bond with the smallest OAS.<\/p>\r\n<p>Further, an OAS equal to that of a comparable bond indicates that the bond is fairly priced.<\/p>\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 option-adjusted spread calculations with our CFA Free Trial.<\/a><\/div>\r\n<h4>Example: Option-adjusted Spread<\/h4>\r\n<p>The following binomial interest rate tree has been calibrated, assuming an interest rate volatility of 15%.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26604\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1.jpg\" alt=\"Binomial Interest Rate Tree\" width=\"1590\" height=\"1611\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1-296x300.jpg 296w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1-1011x1024.jpg 1011w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1-768x778.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1-1516x1536.jpg 1516w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1-400x405.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>The OAS of a three-year 5% annual coupon risky bond, callable at par one year and two years from now priced at $102.05 can be computed as follows:<\/p>\r\n<p>The value of the bond using the above interest rate tree is $103.00, as shown in the following figure:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26605\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-scaled.jpg\" alt=\"Value of the Bond\" width=\"2048\" height=\"1567\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-scaled.jpg 2048w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-300x230.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-1024x784.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-768x588.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-1536x1175.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-400x306.jpg 400w\" sizes=\"auto, (max-width: 2048px) 100vw, 2048px\" \/>We need to compute the OAS that justifies the current market price of $102.05 by trial and error.<\/p>\r\n<p>If we add a constant spread (OAS) of 50 bps to each one-period forward rate, we obtain a value of $102.05, which is equal to the current market price, as shown below:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26606\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-with-0.5bps-Constant-Spread.jpg\" alt=\"Value of the Bond with 0.5bps Constant Spread\" width=\"1590\" height=\"1217\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-with-0.5bps-Constant-Spread.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-with-0.5bps-Constant-Spread-300x230.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-with-0.5bps-Constant-Spread-1024x784.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-with-0.5bps-Constant-Spread-768x588.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-with-0.5bps-Constant-Spread-1536x1176.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-with-0.5bps-Constant-Spread-400x306.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>The following information relates to two risky callable bonds with similar characteristics and credit quality.<\/p>\r\n<p>$$ \\begin{array}{c|c} \\textbf{Bond} &amp; \\textbf{OAS} \\\\ \\hline X &amp; 30\\text{bps} \\\\ \\hline Y &amp; 20\\text{bps} \\end{array} $$<\/p>\r\n<p>Relative to bond Y, bond X is <em>most likely<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>Underpriced.<\/li>\r\n\t<li>Fairly priced.<\/li>\r\n\t<li>Overpriced.<\/li>\r\n<\/ol>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>An OAS higher than that of a comparable bond indicates that the bond is likely underpriced relative to the other.<\/p>\r\n<p>In this case, bond X has the largest OAS, indicating that it is likely to be underpriced (cheap) relative to the bond Y with the smallest OAS.<\/p>\r\n<\/blockquote>\r\n<p>Reading 30: Valuation and Analysis of Bonds with Embedded Options<\/p>\r\n<p><em>LOS 30 (g) Explain the calculation and use of option-adjusted spreads.<\/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<h2>Why Option-Adjusted Spreads Matter in CFA Level II<\/h2>\r\n\r\n<p>Option-adjusted spreads are core CFA Level II Fixed Income concepts. Candidates may be tested on embedded options, bond valuation using interest rate trees, and how OAS helps compare callable or putable bonds with similar credit risk.<\/p>\r\n\r\n<p>Build confidence with the <a href=\"https:\/\/analystprep.com\/cfa-level-2\/\" target=\"_blank\">CFA Level II prep course<\/a> featuring guided lessons, practice questions, and full mock exams.<\/p>\r\n<p>\r\n<\/p>\r\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-block; padding: 12px 24px; 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>\r\n<p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Strengthen your CFA Level II fixed-income skills with exam-style practice on option-adjusted spreads, embedded options, and bond valuation.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>A workable approach employed in constructing a suitable yield curve for risky bonds involves adding a fixed Z-spread to the one-year forward rates derived from the default-free spot yield curve. Recall that Z-spread is the basis point spread that would&#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-17567","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>Option-Adjusted Spread (OAS) Explained<\/title>\n<meta name=\"description\" content=\"Learn how option-adjusted spread (OAS) is calculated and how it helps compare bond values when embedded options are present.\" \/>\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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