{"id":17454,"date":"2021-07-09T23:13:42","date_gmt":"2021-07-09T23:13:42","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17454"},"modified":"2025-12-30T08:13:13","modified_gmt":"2025-12-30T08:13:13","slug":"describe-the-process-of-calibrating-a-binomial-interest-rate-tree-to-match-a-specific-term-structure","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-process-of-calibrating-a-binomial-interest-rate-tree-to-match-a-specific-term-structure\/","title":{"rendered":"Calibrating a Binomial Interest Rate Tree"},"content":{"rendered":"<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\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1536x1526.jpg\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1536x1526.jpg\",\r\n  \"caption\": \"Binomial interest rate tree calibrated to match a specific term structure (CFA Level II)\",\r\n  \"width\": 1536,\r\n  \"height\": 1526,\r\n  \"copyrightNotice\": \"\u00a9 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  },\r\n  \"isPartOf\": {\r\n    \"@type\": \"WebPage\",\r\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-process-of-calibrating-a-binomial-interest-rate-tree-to-match-a-specific-term-structure\/\"\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\": \"What is the higher one-year forward rate using the lognormal interest rate model?\",\r\n    \"text\": \"Chen Cheng, a portfolio manager at ABC Investment Bank, is training Zhang Wang, a junior investment analyst, on calibrating a binomial interest rate model using Excel. Cheng starts the process by guessing a lower one-year forward rate, i\u2081,d, of 3.820%. Assuming that Cheng uses a volatility assumption of 20%, the higher one-year forward rate using the lognormal model of interest rates is closest to:\\n\\nA. 2.56%\\nB. 4.67%\\nC. 5.70%\",\r\n    \"answerCount\": 3,\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"A. 2.56%\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"B. 4.67%\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"C. 5.70%\"\r\n      }\r\n    ],\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"C. 5.70%\",\r\n      \"commentary\": \"Under the lognormal model of interest rates, the higher forward rate is calculated by multiplying the lower forward rate by e raised to two times the volatility. Using i\u2081,d = 3.820% and volatility = 20%, the higher rate is 3.820% \u00d7 e^(2 \u00d7 0.20) \u2248 5.70%, making option C the correct answer.\",\r\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-process-of-calibrating-a-binomial-interest-rate-tree-to-match-a-specific-term-structure\/\"\r\n    },\r\n    \"author\": {\r\n      \"@type\": \"Organization\",\r\n      \"name\": \"AnalystPrep\"\r\n    }\r\n  }\r\n}\r\n<\/script>\r\n\r\n\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/dqDjJ-eiqNI\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>The following steps should be followed when calibrating binomial interest rate trees to match a particular term structure:<\/p>\r\n<ul>\r\n\t<li><em><strong>Step 1<\/strong>:<\/em> Estimate the appropriate spot and forward rates for a known par value curve.<\/li>\r\n\t<li><em><strong>Step 2<\/strong>:<\/em> Construct the interest rate tree using the assumed volatility and the interest rate model.<\/li>\r\n\t<li><em><strong>Step 3<\/strong>:<\/em> Determine the appropriate values for the zero-coupon bonds at each node using backward induction.<\/li>\r\n\t<li><em><strong>Step 4<\/strong>:<\/em> Calibrate the tree to ensure it is arbitrage-free. The value of a bond produced by the interest rate tree must be equal to its market price.<\/li>\r\n<\/ul>\r\n<p>\u00a0When constructing an interest rate tree as per step 2 above, it&#8217;s important to remember that:<\/p>\r\n<ul>\r\n\t<li>At each node, the forward interest rates can either go up (higher rate) or down (lower rate).<\/li>\r\n\t<li>You can determine the lower rate iteratively or by solving simultaneous equations depending on:\r\n\r\n<ul>\r\n\t<li>The relationship: \\(i_{1,d}=i_{1,u}\\times e^{-2\\sigma}\\).<\/li>\r\n\t<li>Known spot and forward rates.<\/li>\r\n\t<li>Features of the coupon bond, particularly its maturity.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>All this can be complicated to perform manually, but some Excel tools, such as Solver, can be employed to make it easy.<\/li>\r\n\t<li>Adjacent forward rates (for the same period) are two standard deviations apart. this means that if you know one of the forward rates for a particular nodal period, you can easily compute the other forward rates for that period in the tree.<\/li>\r\n<\/ul>\r\n<p>The following related formulas are important to remember:<\/p>\r\n<p>$$i_{1,u}=i_{1,d}e^{2\\sigma }$$<\/p>\r\n<p>$$i_{2,uu}=i_{2,dd}e^{4\\sigma }$$<\/p>\r\n<p>$$i_{2,uu}=i_{2,du}e^{2\\sigma }$$<\/p>\r\n<p>$$i_{2,du}=i_{2,dd}e^{2\\sigma }$$<\/p>\r\n<h4>Example: Binomial Interest Rate Tree<\/h4>\r\n<p>To calibrate a binomial interest rate tree, a portfolio manager collects the following information relating to the spot rate curve and forward rates:<\/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><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26496\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree.jpg\" alt=\"Binomial Interest Rate Tree\" width=\"1590\" height=\"1580\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-300x298.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1024x1018.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-150x150.jpg 150w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-768x763.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-1536x1526.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Binomial-Interest-Rate-Tree-400x397.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Determine the forward rates A, B, C, and D.<\/p>\r\n<p><strong>Forward rate A<\/strong><\/p>\r\n<p>$$ \\begin{align*} \\text{The higher rate is determined as } i_{1u}&amp;=i_{1d}\\times e^{2\\sigma} \\\\ &amp;=5.20\\%\\times e^{2\\times0.15}=7.019\\% \u2003\\end{align*} $$<\/p>\r\n<p><strong>Forward rate C<\/strong><\/p>\r\n<p>This is the middle forward rate approximated as \\(f (2,1)\\). Recall from the previous reading the relationship between forward and spot rates:<\/p>\r\n<p>$$ \\begin{align*} \\left[1+f\\left(t,T-t\\right)\\right]^{T-t} &amp;=\\left[\\frac{\\left(1+S_T\\right)^T}{\\left(1+S_t\\right)^t}\\right] \\\\ (1+f\\left(2,1\\right) &amp;=\\left[\\frac{\\left(1+S_3\\right)^3}{\\left(1+S_2\\right)^2}\\right] \\\\ f\\left(2,1\\right) &amp;=\\frac{{1.06}^3}{{1.05}^2}-1=8.029\\% \\end{align*} $$<\/p>\r\n<p><strong>Forward rate B<\/strong><\/p>\r\n<p>$$ \\begin{align*} i_{2,uu} &amp;=i_{2du } e^{2\\sigma} \\\\ i_{2,uu} &amp;=8.029\\%\\times e^{2\\times0.15 }=10.84\\% \\end{align*} $$<\/p>\r\n<p><strong>Forward rate D<\/strong><\/p>\r\n<p>$$ \\begin{align*} i_{2,dd} &amp;=i_{2du} e^{-2\\sigma} \\\\ i_{2,dd} &amp;=8.029\\%\\times e^{-2\\times0.15 }=5.95\\% \\end{align*} $$<\/p>\r\n<p>It is key to note that the change in the volatility assumption affects implied forward rates. A change of volatility to a lower value makes the potential implied forward rates to collapse on the tree and vice versa.<\/p>\r\n<blockquote style=\"font: normal;\">\r\n<h2>Question<\/h2>\r\n<p>Chen Cheng, a portfolio manager at ABC Investment Bank, is training Zhang Wang, a junior investment analyst, on calibrating binomial interest rate models using Excel. Cheng starts the process by guessing a lower one-year forward rate, \\(i_{1,d}\\), of 3.820%. Assuming that Cheng uses a volatility assumption of 20%, the higher one-year forward rate using the lognormal model of interest rates is <em>closest to<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>2.56%.<\/li>\r\n\t<li>4.67%.<\/li>\r\n\t<li>5.70%.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is C.<\/strong><\/p>\r\n<p>Based on the lognormal model of interest rates, the higher one-year forward rate is \\((i_{1,u})=i_{1,d}e^{2\\sigma}\\).<\/p>\r\n<p>$$ \\begin{align*} i_{1,d} &amp;=3.820\\% \\\\ (i_{1,u}) &amp;=3.820\\%\\times e^{2\\times0.20}=5.70\\% \u2003\\end{align*} $$<\/p>\r\n<\/blockquote>\r\n<p>Reading 29: The Arbitrage-Free Valuation Framework<\/p>\r\n<p><em>LOS 29 (d) Describe the process of calibrating a binomial interest rate tree to match a specific term structure.<\/em><\/p>\r\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                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