{"id":947,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=947"},"modified":"2026-01-20T15:11:08","modified_gmt":"2026-01-20T15:11:08","slug":"binomial-stock-price-tree-question-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/binomial-stock-price-tree-question-example\/","title":{"rendered":"Binomial Stock Price Tree"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Binomial tree stock price after two up moves and one down move\",\n    \"text\": \"Use the binomial tree to calculate the stock price after three periods comprising two consecutive periods of stock price growth followed by a reduction in price.\\n\\nA. $30 with probability 0.49\\n\\nB. $30.6 with probability 0.147\\n\\nC. $30.6 with probability 0.49\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"B. $30.6 with probability 0.147.\\n\\nThe stock path is up\u2013up\u2013down (uud). The price is calculated as 30 \u00d7 1.02\u00b2 \u00d7 (1 \/ 1.02) = $30.6. The probability of this path is 0.7 \u00d7 0.7 \u00d7 0.3 = 0.147.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A. $30 with probability 0.49\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C. $30.6 with probability 0.49\"\n      }\n    ]\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2021 Level I CFA\u00ae Exam \u2013 Reading 9)\",\n  \"description\": \"This video lesson covers common probability distributions in quantitative methods. It explains key concepts like discrete vs. continuous variables, probability functions, cumulative distributions, binomial and normal distributions, confidence intervals, Monte Carlo simulations, and risk analysis\u2014using practical examples to help finance professionals apply statistical models to investment decisions.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/KbEfz3KiJDo\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=KbEfz3KiJDo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\",\n  \"duration\": \"PT45M31S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo?si=UaGUMeyevNdEf5AA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n<p>A binomial tree is used to predict stock price movements assuming there are two possible outcomes, each of which has a known probability of occurrence.<\/p>\n<p><!--more--><\/p>\n<p>$$<br \/>\\begin{array}{}<br \/>\\text{Binomial Stock Price Tree} \\\\<br \/>\\end{array}<br \/>$$<br \/>$$<br \/>\\begin{array}<br \/>\\hline<br \/>{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{uuuS} \\\\<br \/>{} &amp; {} &amp; {} &amp; {} &amp; \\text{uuS} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} \\\\<br \/>{} &amp; {} &amp; \\text{uS} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} &amp; {} &amp; \\text{uudS} \\\\<br \/>\\text{S} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} &amp; {} &amp; \\text{udS} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} \\\\<br \/>{} &amp; {} &amp; \\text{dS} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} &amp; {} &amp; \\text{uddS} \\\\<br \/>{} &amp; {} &amp; {} &amp; {} &amp; \\text{ddS} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} \\\\<br \/>{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{dddS} \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>The diagram above shows a series of Bernoulli trials that depict stock price movement as a binomial random variable. We define only two possible outcomes: the price can either go up or come down.<\/p>\n<p><em>S<\/em> denotes the stock price today while <em>u <\/em>denotes 1 plus the rate of return when the stock goes up. Yet, <em>d<\/em> denotes 1 plus the rate of return when the stock price comes down.<\/p>\n<p><em>uS <\/em>represents the price of the stock at the <strong>end<\/strong> of an \u201cup\u201d period. On the other hand, <em>dS <\/em>represents the stock price at the end of a \u201cdown\u201d period.<\/p>\n<p>The \u201cup transition probability\u201d is the probability of an \u201cup\u201d move, while the \u201cdown transition probability\u201d is the probability of a \u201cdown \u201c move.<\/p>\n<p>It\u2019s imperative to note that the tree recombines: <em>udS<\/em> = <em>duS <\/em><\/p>\n<h3><strong>Example: Binomial Tree<\/strong><\/h3>\n<p>Suppose the initial stock price is $30, <em>u <\/em>= 1.02, <em>d<\/em> = 1\/1.02, and the probability of an \u201cup\u201d move is 0.7, calculate the stock prices after 2 periods.<\/p>\n<p>$$ \\text{uuS} = 1.02^2 * 30 = 31.21 $$<\/p>\n<p>Then, we must factor in the probabilities of consecutive \u201cup\u201d movements:<\/p>\n<p>$$ = 0.7 * 0.7 = 0.49 $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\text{uus} = $31.21 \\text{ with probability } 0.49 $$<\/p>\n<p>Similarly,<\/p>\n<p>$$ \\text{uds} = 1.02 * \\cfrac{1}{1.02} * 30 = $30 \\text{ with a probability of } 0.21 (0.7 * 0.3) $$<\/p>\n<p>$$ \\text{dus} =\\cfrac {1}{1.02} * 1.02 * 30 = $30 \\text{ with a probability of } 0.21 (0.3 * 0.7) $$<\/p>\n<p>$$ \\text{dds} =\\cfrac {1}{1.02} * \\cfrac{1}{1.02} * 30 = $28.83 \\text{ with } 0.09 \\text{ probability } (0.3 * 0.3) $$<\/p>\n<p>Note to candidates:<\/p>\n<p>$$ \\text{dus} = \\text{uds} = $30 \\text{ since the tree} $$<\/p>\n<p><strong>recombines<\/strong>: the order of events does not matter.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Use the binomial tree above to calculate the stock price after 3 periods comprising 2 consecutive periods of stock price growth followed by a reduction in price.<\/p>\n<p>A. $30 with probability 0.49<\/p>\n<p>B. $30.6 with probability 0.147<\/p>\n<p>C. $30.6 with probability 0.49<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>In short, you have been asked to find the stock price after \u201cuud\u201d i.e. <em>uudS<\/em><\/p>\n<p>$$ \\text{uudS} = 1.02^2 * \\cfrac {1}{1.02} * 30 = $30.6 \\text{ with a probability of } 0.147 (0.7 * 0.7 * 0.3) $$<\/p>\n<\/blockquote>\n<p>You should also note that just like in an ordinary <a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/tree-diagram-example\/\">probability tree<\/a>, the sum of all probabilities at the end of a period must be 1. For example, the sum of probabilities after 2 periods = 0.49 + 0.21 + 0.21 + 0.09 = 1.<\/p>\n<p>The binomial model is applied when pricing derivatives in finance.<\/p>\n<div class=\"notes_inv\"><hr \/><\/div>","protected":false},"excerpt":{"rendered":"<p>A binomial tree is used to predict stock price movements assuming there are two possible outcomes, each of which has a known probability of occurrence.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-947","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","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 Stock Price Tree Example | CFA Level 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