{"id":14469,"date":"2021-04-29T11:27:46","date_gmt":"2021-04-29T11:27:46","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=14469"},"modified":"2026-04-16T15:22:12","modified_gmt":"2026-04-16T15:22:12","slug":"binomial-option-valuation-model","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/binomial-option-valuation-model\/","title":{"rendered":"Binomial Option Valuation Model"},"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\/04\/Two-Period-Binomial-Option-Valuation-Model-1024x782.jpg\",\r\n  \"caption\": \"Two-Period Binomial Option Valuation Model\",\r\n  \"width\": 1024,\r\n  \"height\": 782,\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\/04\/One-period-Binomial-Put-Option-Payoffs.jpg\",\r\n  \"caption\": \"One-Period Binomial Put Option Payoffs\",\r\n  \"width\": 1549,\r\n  \"height\": 1131,\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\/04\/One-period-Binomial-Call-Option-Payoffs.jpg\",\r\n  \"caption\": \"One-Period Binomial Call Option Payoffs\",\r\n  \"width\": 1549,\r\n  \"height\": 1131,\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\/04\/One-Period-Binomial.jpg\",\r\n  \"caption\": \"One-Period Binomial Tree\",\r\n  \"width\": 1549,\r\n  \"height\": 1131,\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\": \"What is the price of a one-year European call option using a one-period binomial model?\",\r\n    \"text\": \"Suppose that a one-year European call option has a strike price of \u00a360. The underlying non-dividend-paying stock is currently trading at \u00a360. Over one year, the stock price can either jump up to \u00a390 or jump down to \u00a350. The annual risk-free interest rate is 4%. Using a one-period binomial option valuation model, the price of the call option is closest to:\\n\\nA. \u00a32.44\\nB. \u00a39.04\\nC. \u00a315.64\",\r\n    \"answerCount\": 3,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is B. The call option price is approximately \u00a39.04.\\n\\nThe up and down factors are u = 90\/60 = 1.5 and d = 50\/60 = 0.83. The call payoffs are \u00a330 in the up state and \u00a30 in the down state. The risk-neutral probability is q = (1.04 \u2212 0.83) \/ (1.5 \u2212 0.83) = 0.3134. Discounting the expected payoff gives: (0.3134 \u00d7 30) \/ 1.04 \u2248 \u00a39.04.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\r\n\r\n<h2>One-Period Binomial Option Valuation Model<\/h2>\r\n<p>In the one-period binomial model, we start today (at time t=0) when the stock price is \\(S_{0}\\). Then, the stock price can either jump upwards or downwards over the one-period time interval to t=1. This is illustrated below:<\/p>\r\n<p>$$S_1=\\{\\begin{align*}S_{0}u&amp; , \\text{if the stock price jumps up} \\\\S_{0}d&amp; , \\text{if the stock price jumps down}\\end{align*}$$<\/p>\r\n<p>This can be shown in the following binomial tree:<\/p>\r\n\r\n\r\n\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26429\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-Period-Binomial.jpg\" alt=\"One-Period Binomial Tree\" width=\"1549\" height=\"1131\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-Period-Binomial.jpg 1549w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-Period-Binomial-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-Period-Binomial-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-Period-Binomial-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-Period-Binomial-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-Period-Binomial-400x292.jpg 400w\" sizes=\"auto, (max-width: 1549px) 100vw, 1549px\" \/>Where:<\/p>\r\n<p>$$u=\\frac{S_{0}u}{S_{0}}$$<\/p>\r\n<p>$$d=\\frac{S_{0}d}{S_{0}}$$<\/p>\r\n<h3 style=\"text-align: justify; line-height: 115%;\"><span style=\"font-size: 13.5pt; line-height: 115%; font-family: 'Century Gothic','sans-serif';\">One period Binomial option payoffs<\/span><\/h3>\r\n<p>Consider a call option that pays \\(c_{u}\\) if the price of the underlying asset jumps up and\u00a0 \\(c_{d}\\) if the price of the underlying asset jumps down.<\/p>\r\n<p>The value of the call option at expiry is expressed as:<\/p>\r\n<p>\\(c_{u}=Max(0,S_{0}u-K)\\),\u00a0if the price of the underlying jumps up<\/p>\r\n<p>and\u00a0<\/p>\r\n<p>\\(c_{d}=Max(0,S_{0}d-K), \\)if the stock price jumps down<\/p>\r\n<p>Where K is the strike price.<\/p>\r\n<p>This is shown in the following binomial tree:<\/p>\r\n\r\n\r\n\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26432\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Call-Option-Payoffs.jpg\" alt=\"One period Binomial Call Option Payoffs\" width=\"1549\" height=\"1131\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Call-Option-Payoffs.jpg 1549w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Call-Option-Payoffs-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Call-Option-Payoffs-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Call-Option-Payoffs-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Call-Option-Payoffs-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Call-Option-Payoffs-400x292.jpg 400w\" sizes=\"auto, (max-width: 1549px) 100vw, 1549px\" \/>Similarly, the value of a put option at expiration is given by:<\/p>\r\n<p>\\(p_{u}=Max(0,K-S_{0}u)\\), if the price of the underlying asset jumps up<\/p>\r\n<p>and<\/p>\r\n<p>\\(p_{d}=Max(0, K- S_{0}d)\\), if the price of the underlying asset jumps down.<\/p>\r\n\r\n\r\n\r\n<h3><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26433\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Put-Option-Payoffs.jpg\" alt=\"One period Binomial Put Option Payoffs\" width=\"1549\" height=\"1131\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Put-Option-Payoffs.jpg 1549w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Put-Option-Payoffs-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Put-Option-Payoffs-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Put-Option-Payoffs-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Put-Option-Payoffs-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/One-period-Binomial-Put-Option-Payoffs-400x292.jpg 400w\" sizes=\"auto, (max-width: 1549px) 100vw, 1549px\" \/>One period Binomial option values<\/h3>\r\n<p>The initial values of call and put options with a one period to expiry are determined using the following formulas:<\/p>\r\n<p>$$C_{0}=\\frac{qc_{u}+(1-q)c_{d}}{1+r}$$<\/p>\r\n<p>And<\/p>\r\n<p>$$p_{0}=\\frac{qp_{u}+(1-q)p_{d}}{1+r}$$<\/p>\r\n<p>Where:\u00a0<\/p>\r\n<p>$$q=\\frac{(1+r)-d}{u-d}$$<\/p>\r\n<p>Where:\u00a0<\/p>\r\n<p>\\(r\\) is the risk-free rate for a single period.<\/p>\r\n<p>\\(q\\) gives the risk-neutral probability of an upward move in price<\/p>\r\n<p>\\(1-q\\) gives the risk-neutral probability of a downward move<\/p>\r\n<div style=\"margin: 20px 0;\"><a style=\"display: block; width: 100%; text-align: center; padding: 10px; border: 2px solid #2f5bea; border-radius: 40px; font-size: 16px; color: #2f5bea; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Practice binomial option pricing questions with our free trial. <\/a><\/div>\r\n<h4>Example: Calculating the price of an option using the one-period binomial option valuation model<\/h4>\r\n<p>Consider a European put option with a strike price of $50 on a stock whose initial price is $50. The risk-free rate of interest is 4%, the up-move factor u = 1.20, and the down move factor d =0.83. The price of the put option can be determined using the one-period binomial model as follows:<\/p>\r\n<p>$$S_{0}u=50\\times1.20=$60$$<\/p>\r\n<p>$$S_{0}d=50\\times0.83=$41.50$$<\/p>\r\n<p>Recall that put payoff is given by:<\/p>\r\n<p>\\(p_{u}=\\text{Max}(0, K-S_{0}u)\\),\u00a0if the price of the underlying asset jumps up<\/p>\r\n<p>and\u00a0<\/p>\r\n<p>\\(p_{d}=\\text{Max}(0, K-S_{0}d)\\), if the price of the underlying asset jumps down.<\/p>\r\n<p>$$p_{u}=Max(0,50-60)=$0$$<\/p>\r\n<p>$$p_{d}=\\text{Max}(0,60-41.50)=$18.50$$<\/p>\r\n<p>The value of the put is then calculated using the formula:\u00a0<\/p>\r\n<p>$$p_{0}=\\frac{qp_{u}+(1-q)p_{d}}{1+r}$$<\/p>\r\n<p>Where:<\/p>\r\n<p>$$q=\\frac{(1.04)-0.83}{1.20-0.83}=0.5676$$<\/p>\r\n<p>$$p_{0}=\\frac{0.5676\\times$0+0.4324\\times$18.50}{1.04}=$7.69$$<\/p>\r\n<h2>Two-Period Binomial Option Valuation Model<\/h2>\r\n<p>The one-period binomial model can be extended into a multi-period context. The two-period binomial lattice can be seen as three-one period binomial lattices as shown below:<\/p>\r\n\r\n\r\n\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26434\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Two-Period-Binomial-Option-Valuation-Model.jpg\" alt=\"Two-Period Binomial Option Valuation Model\" width=\"1590\" height=\"1214\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Two-Period-Binomial-Option-Valuation-Model.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Two-Period-Binomial-Option-Valuation-Model-300x229.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Two-Period-Binomial-Option-Valuation-Model-1024x782.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Two-Period-Binomial-Option-Valuation-Model-768x586.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Two-Period-Binomial-Option-Valuation-Model-1536x1173.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Two-Period-Binomial-Option-Valuation-Model-400x305.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>The underlying asset can result in only three possible values:<\/p>\r\n<p>\\(S_{0}uu=\\) When price moves up twice<\/p>\r\n<p>\\(S_{0}ud=\\) When price either moves up then down or down then up<\/p>\r\n<p>\\(S_{0}dd=\\) When price moves down twice<\/p>\r\n<h3>Call Payoffs<\/h3>\r\n<p>A call option under the two-period binomial option model will have three possible payoffs at expiry as follows:<\/p>\r\n<p>$$C_{uu}=\\text{max}(0, S_{0}u^{2}-K)$$<\/p>\r\n<p>$$C_{ud}=Max(0,S_{0}ud-K)$$<\/p>\r\n<p>$$C_{dd}=Max(0,S_{0}d^{2}-K)$$<\/p>\r\n<h3>Put Payoffs<\/h3>\r\n<p>Similar to a call option, a put option will have three possible payoffs:<\/p>\r\n<p>$$p_{uu}=Max(o,K-S_{0}u^{2})$$<\/p>\r\n<p>$$p_{ud}=Max(0,K-S_{0}ud)$$<\/p>\r\n<p>$$p_{dd}=Max(0,K-S_{0}d^{2})$$<\/p>\r\n<h3>Option Values<\/h3>\r\n<p>A European call option\u2019s value can be determined using the two-step binomial valuation model using the following formula.<\/p>\r\n<p>$$c_{0}=\\frac{q^{2}c_{uu}+2q(1-q)c_{ud}+(1-q)^{2}c_{dd}}{(1+r)^{2}}$$<\/p>\r\n<p>The two-period European put value is given as:<\/p>\r\n<p>$$p_{o}=\\frac{q^{2}p_{uu}+2q(1-q)p_{ud}+(1-q)^{2}p_{dd}}{(1+r)^{2}}$$<\/p>\r\n<p>These concepts will be explained more with examples in the sections that follow.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Suppose that a one-year European call option has a strike price of \u00a360. The underlying non-dividend-paying stock is currently trading at \u00a360. Over one year, the stock price can either jump up to \u00a390 or jump down to \u00a350. The annual risk-free interest rate is 4%. Using a one-period binomial option valuation model, the price of the call option is closest to:<\/p>\r\n<ol style=\"list-style-type: upper-alpha;\">\r\n\t<li>\u00a32.44<\/li>\r\n\t<li>\u00a39.04<\/li>\r\n\t<li>\u00a315.64<\/li>\r\n<\/ol>\r\n<h3>Solution<\/h3>\r\n<p><strong>The correct answer is B:<\/strong><\/p>\r\n<p>The payoff of a European call option at expiration is given by:<\/p>\r\n<p>\\(c_{u}=Max(0,S_{0}u-K)\\), if the price of the underlying jumps up<\/p>\r\n<p>and<\/p>\r\n<p>\\(c_{d}=Max(0,S_{0}d-K)\\),\u00a0if the stock price jumps down<\/p>\r\n<p>$$u=\\frac{90}{60}=1.5$$<\/p>\r\n<p>$$d=\\frac{50}{60}=0.83$$<\/p>\r\n<p>$$c_{u}=Max(0,90-60)=\u00a330$$<\/p>\r\n<p>$$c_{d}=Max(0,50-60)=\u00a30$$<\/p>\r\n<p>The value of a call option is then calculated using the formula:<\/p>\r\n<p>$$c_{0}=\\frac{qc_{u}+(1-q)c_{d}}{1+r}$$<\/p>\r\n<p>Where:<\/p>\r\n<p>$$q=\\frac{1.04-0.83}{1.5-0.83}=0.3134$$<\/p>\r\n<p>$$c_{0}=\\frac{0.3134\\times\u00a330+0.6866\\times\u00a30}{1.04}=\u00a39.04$$<\/p>\r\n<\/blockquote>\r\n<p><em>Reading 38: Valuation of Contingent Claims<\/em><\/p>\r\n<p><em>LOS 38 (a): Describe and interpret the binomial option valuation model and its component terms<\/em><\/p>\r\n\r\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-block; padding: 10px 26px; background: #3f78d7; color: #fff; border-radius: 40px; font-size: 16px; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a>\r\n<p style=\"margin-top: 10px; max-width: 600px; margin-left: auto; margin-right: auto; font-size: 14px;\">Solve CFA Level II questions on one-period binomial models and option valuation techniques.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>One-Period Binomial Option Valuation Model In the one-period binomial model, we start today (at time t=0) when the stock price is \\(S_{0}\\). Then, the stock price can either jump upwards or downwards over the one-period time interval to t=1. This&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,302],"tags":[317,216,304],"class_list":["post-14469","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-binomial-option-valuation-model","tag-cfa-level-2","tag-derivatives","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Binomial Option Pricing Model and Valuation<\/title>\n<meta name=\"description\" content=\"Learn how the binomial option pricing model works, including price movements, valuation steps, and how options are priced over time.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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