{"id":17468,"date":"2021-07-19T21:21:36","date_gmt":"2021-07-19T21:21:36","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17468"},"modified":"2024-04-06T11:27:36","modified_gmt":"2024-04-06T11:27:36","slug":"compare-pricing-using-the-zero-coupon-yield-curve-with-pricing-using-an-arbitrage-free-binomial-lattice-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/compare-pricing-using-the-zero-coupon-yield-curve-with-pricing-using-an-arbitrage-free-binomial-lattice-2\/","title":{"rendered":"Pricing using the Zero-Coupon Yield Curve and an Arbitrage-Free Binomial Lattice"},"content":{"rendered":"<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/dqDjJ-eiqNI\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n\r\n<p>Valuing a fixed-rate coupon bond with no embedded options using the arbitrage-free lattice and the spot curve leads to the same bond value. This holds because the binomial interest rate tree is arbitrage-free. However, the spot curve will <em><strong>not<\/strong> <\/em>work for bonds with embedded options.<\/p>\r\n<h4>Example: Zero-Coupon Yield Curve<\/h4>\r\n<p>A three-year bond with no embedded options pays 5% annual coupons. Given the following spot curve, the bond price with a face value of $100 is <em>closest<\/em> to:<\/p>\r\n<p>$$ \\begin{array}{c|c} \\textbf{Term to Maturity} &amp; \\textbf{Spot Rate} \\\\ \\hline 1 &amp; 4.00\\% \\\\ \\hline 2 &amp; 5.00\\% \\\\ \\hline 3 &amp; 6.00\\% \\end{array} $$<\/p>\r\n<p><strong>Solution<\/strong><\/p>\r\n<p>$$ \\begin{align*} PV &amp;=\\frac{PMT}{\\left(1+S_1\\right)^1}+\\frac{PMT}{\\left(1+S_2\\right)^2}+\\ldots+\\frac{PMT+FV}{\\left(1+S_N\\right)^N} \\\\\u00a0 &amp;=\\frac{5}{1.04}+\\frac{5}{{1.05}^2}+\\frac{105}{{1.06}^3}=$97.50 \\end{align*} $$<\/p>\r\n<h4>Example: Arbitrage-Free Binomial Lattice<\/h4>\r\n<p>We can determine the price of the same bond using a binomial interest rate tree with the following forward rates:<\/p>\r\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26503\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Bond-Pricing.jpg\" alt=\"Bond Pricing\" width=\"1590\" height=\"999\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Bond-Pricing.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Bond-Pricing-300x188.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Bond-Pricing-1024x643.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Bond-Pricing-768x483.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Bond-Pricing-1536x965.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Bond-Pricing-400x251.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Solution<\/strong><\/p>\r\n<p>$$ \\begin{align*} V &amp;=0.5\\left[\\frac{V_u+C}{1+i}+\\frac{V_d+C}{1+i}\\right] \\\\ V_{2,uu} &amp; =\\frac{105}{1.10693}=94.857 \\\\ V_{2,ud} &amp;=\\frac{105}{1.07861}=97.348 \\\\ V_{2,dd} &amp;=\\frac{105}{1.05899}=99.151 \\\\ V_{1,u} &amp;=0.5\\left[\\frac{94.857+5}{1.06886}+\\frac{97.348+5}{1.06886}\\right]=94.589 \\\\ V_{1,d} &amp;=0.5\\left[\\frac{97.348+5}{1.05123}+\\frac{99.151+5}{1.05123}\\right]=98.218 \\\\ V_0&amp;=0.5\\left[\\frac{94.589+5}{1.04}+\\frac{98.218+5}{1.04}\\right]=$97.50 \\end{align*} $$<\/p>\r\n<p>We have calibrated the binomial interest rate tree to produce arbitrage-free values consistent with the spot rate curve. This has led to the same option-free bond value of $97.50 as using the spot yield curve.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>A fixed-rate coupon bond with no embedded options which is priced using the arbitrage-free binomial lattice leads to a value that&#8217;s:<\/p>\r\n<ol type=\"A\">\r\n\t<li>Higher than that generated using the benchmark spot rate curve.<\/li>\r\n\t<li>Lower than that generated using the benchmark spot rate curve.<\/li>\r\n\t<li>Similar to that generated using the benchmark spot rate curve.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is C.<\/strong><\/p>\r\n<p>Valuing a fixed-rate coupon bond with no embedded options using the arbitrage-free lattice and using the spot curve leads to the same bond value. This holds because the binomial interest rate tree is arbitrage-free.<\/p>\r\n<\/blockquote>\r\n<p>Reading 29: The Arbitrage-Free Valuation Framework<\/p>\r\n<p><em>LOS 29 (f) Compare pricing using the zero-coupon yield curve with pricing using an arbitrage-free binomial lattice.<\/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","protected":false},"excerpt":{"rendered":"<p>Valuing a fixed-rate coupon bond with no embedded options using the arbitrage-free lattice and the spot curve leads to the same bond value. This holds because the binomial interest rate tree is arbitrage-free. However, the spot curve will not work&#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-17468","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>Pricing using the Zero-Coupon Yield Curve and an Arbitrage-Free Binomial Lattice - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Valuing a fixed-rate coupon bond with no embedded options using the arbitrage-free lattice and the spot curve leads to the same bond value.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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