{"id":18638,"date":"2021-07-28T14:33:27","date_gmt":"2021-07-28T14:33:27","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=18638"},"modified":"2025-11-26T08:37:10","modified_gmt":"2025-11-26T08:37:10","slug":"calculate-and-interpret-the-value-of-an-interest-rate-option-using-a-two-period-binomial-model","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-and-interpret-the-value-of-an-interest-rate-option-using-a-two-period-binomial-model\/","title":{"rendered":"Valuation of an Interest Rate Option"},"content":{"rendered":"\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@graph\": [\r\n    {\r\n      \"@type\": \"ImageObject\",\r\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-scaled.jpg\",\r\n      \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-scaled.jpg\",\r\n      \"caption\": \"Two-period Interest Rate Tree \u2013 Put Option\",\r\n      \"width\": 2048,\r\n      \"height\": 1543,\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    {\r\n      \"@type\": \"ImageObject\",\r\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-scaled.jpg\",\r\n      \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-scaled.jpg\",\r\n      \"caption\": \"Two-period Interest Rate Tree \u2013 Call Option\",\r\n      \"width\": 2048,\r\n      \"height\": 1543,\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  ]\r\n}\r\n<\/script>\r\n\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/jmbJLCdBEgs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p><em><strong>Interest rate options<\/strong><\/em> are options with an interest rate as the underlying. A call option on interest rates has a positive payoff when the current spot rate is greater than the exercise rate.<\/p>\r\n<p>$$ \\begin{align*} \\text{Call option payoff} &amp; =\\text{Notional amount }\\times \\\\ &amp; [\\text{max } (\\text{Current spot rate} &#8211; \\text{Exercise rate},0)] \\end{align*} $$<\/p>\r\n<p>On the other hand, a put option on interest rates has a positive payoff when the current spot rate is less than the exercise rate.<\/p>\r\n<p>$$ \\text{Put payoff} =\\text{Notional amount} \\times [\\text{max } (\\text{Exercise rate} &#8211; \\text{Current spot rate},0)] $$<\/p>\r\n<p>We can apply the binomial model to value such interest rate options.<\/p>\r\n<h4>Example: Calculating the Value of an Interest Rate European Option<\/h4>\r\n<p>Consider a two-year European-style call option with a one-year spot rate compounded annually as the underlying. The exercise rate is 6%. The two-period interest rate tree is given below:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26449\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-scaled.jpg\" alt=\"Two-period Interest Rate Tree - Call Option\" width=\"2048\" height=\"1543\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-scaled.jpg 2048w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-300x226.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-1024x771.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-768x579.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-1536x1157.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-400x301.jpg 400w\" sizes=\"auto, (max-width: 2048px) 100vw, 2048px\" \/>Assume that the notional principal of each option is $500,000, and the risk-neutral probability of an up jump is 0.5. The value of the European call option can be determined as follows:<\/p>\r\n<h4><strong>Payoffs at Time 2<\/strong><\/h4>\r\n<p>$$ \\begin{align*} c_{uu} &amp;=max(0, S_0u^2-K) \\\\ c_{uu} &amp;=max(0,0.11-0.06)=0.05 \\\\ C_{ud} &amp;=Max\\left(0,S_0ud-K\\right) \\\\ C_{ud} &amp;=max{\\left(0,0.07-0.06\\right)}=0.01 \\\\ c_{dd} &amp;=Max(0,S_0d^2-K) \\\\ c_{dd} &amp;=max{\\left(0,0.05-0.06\\right)}=0 \\end{align*} $$<\/p>\r\n<h4><strong>Value of Call at Time 1<\/strong><\/h4>\r\n<p>$$ \\begin{align*} c_u &amp; =PV_{1,2}\\left[qc_{uu}\\ +\\left(1 &#8211; q\\right)c_{ud}\\right] \\\\ c_u &amp;=\\frac{1}{1.06}\\left[0.5\\times0.05+\\left(1-0.5\\right)\\times0.01\\right]=0.028302 \\\\ c_d &amp; =PV_{1,2}\\left[qc_{ud} +\\left(1 &#8211; q\\right)c_{dd}\\right] \\\\ c_d &amp;=\\frac{1}{1.04}[0.5\\times0.01+\\left(1-0.5\\right)\\times0)]=0.004808 \\end{align*} $$<\/p>\r\n<h4><strong>Value of Call at Time 0<\/strong><\/h4>\r\n<p>$$ \\begin{align*} c_0 &amp; =PV_{0,1}\\left[qc_u +\\left(1 &#8211; q\\right)c_d\\right] \\\\ c_0 &amp;=\\frac{1}{1.03} \\left[0.5\\times0.028302+\\left(1-0.5\\right)\\times0.004808 \\right]=0.01607 \\end{align*} $$<\/p>\r\n<p>The call value at time 0 is then obtained by multiplying with the notional amount:<\/p>\r\n<p>$$ \\text{Call value} =0.01607\\times$500,000=$8,035 $$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Consider a two-year European-style put option with the annually compounded one-year spot interest rate as the underlying. The exercise rate is 6%. The two-period interest rate tree is <em>given below<\/em>:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26450\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-scaled.jpg\" alt=\"Two-period Interest Rate Tree - Put Option\" width=\"2048\" height=\"1543\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-scaled.jpg 2048w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-300x226.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-1024x771.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-768x579.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-1536x1157.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-400x301.jpg 400w\" sizes=\"auto, (max-width: 2048px) 100vw, 2048px\" \/>Assume that the notional principal of each option is $500,000, and the risk-neutral (RN) probability of an up jump is 0.5. The value of the European put option is <em>closest<\/em> to<em>:<\/em><\/p>\r\n<ol type=\"A\">\r\n\t<li>$1,167.<\/li>\r\n\t<li>$5,835.<\/li>\r\n\t<li>$8,035.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<h4><strong>Put Payoffs at time 2<\/strong><\/h4>\r\n<p>Similar to a call option, a put option will have three possible payoffs:<\/p>\r\n<p>$$ \\begin{align*} p_{uu}&amp; = Max(0,K \u2013 S_0u^2) \\\\ p_{uu} &amp; = Max(0,0.06 \u2013 0.11)=0 \\\\ p_{ud} &amp; = Max(0,K \u2013 S_0ud) \\\\ p_{ud} &amp; = Max(0,0.06 \u2013 0.07)=0 \\\\ p_{dd} &amp; =Max(0,K \u2013 S_0dd) \\\\ p_{dd} &amp; = Max(0,0.06 \u2013 0.05)=0.01 \\end{align*} $$<\/p>\r\n<h4><strong>Value of Put at Time 1<\/strong><\/h4>\r\n<p>$$ \\begin{align*} p_u &amp; =PV_{1,2}\\left[qp_{uu} +\\left(1 &#8211; q\\right)p_{ud}\\right] \\\\ p_u &amp;=\\frac{1}{1.06}\\left[0.5\\times0+\\left(1-0.5\\right)\\times0\\right]=0 \\\\ p_d &amp; =PV_{1,2}\\left[qp_{ud} +\\left(1 &#8211; q\\right)c_{dd}\\right] \\\\ c_d &amp;=\\frac{1}{1.04} \\left[0.5\\times0+\\left(1-0.5\\right)\\times0.01) \\right]=0.004808 \\end{align*} $$<\/p>\r\n<h4><strong>Value of Put at Time 0<\/strong><\/h4>\r\n<p>$$ \\begin{align*} p_o &amp; =PV_{0,1}\\left[qp_u +\\left(1- q\\right)p_d\\right] \\\\ c_0 &amp;=\\frac{1}{1.03} \\left[0.5\\times0+\\left(1-0.5\\right)\\times0.004808 \\right]=0.00233398 \\end{align*} $$<\/p>\r\n<p>The put value at time 0 is then obtained by multiplying by the notional amount:<\/p>\r\n<p>$$ \\text{put value} =0.002334\\times$500,000=$1,167 $$<\/p>\r\n<\/blockquote>\r\n<p>Reading 34: Valuation of Contingent Claims<\/p>\r\n<p><em>LOS 34 (d) Calculate and interpret the value of an interest rate option using a two-period binomial model.<\/em><\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Interest rate options are options with an interest rate as the underlying. A call option on interest rates has a positive payoff when the current spot rate is greater than the exercise rate. $$ \\begin{align*} \\text{Call option payoff} &amp; =\\text{Notional&#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":[102,302],"tags":[216,304],"class_list":["post-18638","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-cfa-level-2","tag-derivatives","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>Valuation of an Interest Rate Option - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Understand call and put options on interest rates, calculate their payoffs, and determine their values based on spot rates and exercise rates\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-and-interpret-the-value-of-an-interest-rate-option-using-a-two-period-binomial-model\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Valuation of an Interest Rate Option - 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