{"id":3890,"date":"2020-04-14T17:33:00","date_gmt":"2020-04-14T17:33:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=3890"},"modified":"2026-03-31T14:12:43","modified_gmt":"2026-03-31T14:12:43","slug":"one-period-binomial-model","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/one-period-binomial-model\/","title":{"rendered":"One Period Binomial Model"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n\n  \"name\": \"Basics of Derivative Pricing and Valuation (2025 Level I CFA\u00ae Exam \u2013 Derivative \u2013 Module 2)\",\n\n  \"description\": \"This video lesson covers Topic 7 \u2013 Derivatives, Module 2 \u2013 Basics of Derivative Pricing and Valuation, exploring key principles like arbitrage, replication, and risk neutrality in derivative pricing. It differentiates between the value and price of forward and futures contracts, explains the forward contract lifecycle, and analyses monetary and nonmonetary impacts on pricing. The lesson also delves into forward rate agreements, swap contracts, and option valuation, including concepts like put-call parity, binomial models, and the differences between European and American options.\",\n\n  \"uploadDate\": \"2022-06-29T00:00:00+00:00\",\n\n  \"thumbnailUrl\": \"https:\/\/analystprep.com\/path-to-thumbnail\/derivative-pricing-thumbnail.jpg\",\n\n  \"contentUrl\": \"https:\/\/youtu.be\/0Geaej45v7w\",\n\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/0Geaej45v7w\",\n\n  \"duration\": \"PT1H08M27S\",\n\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/path-to-logo\/logo.jpg\",\n      \"width\": 600,\n      \"height\": 60\n    }\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Which factors are the most relevant to determine an option\u2019s value using a binomial pricing model?\",\n    \"text\": \"Which factors are the most relevant to determine an option\u2019s value using a binomial pricing model?\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The risk-free rate, the volatility of the underlying, and the exercise price.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"The probability that the underlying will go up or down, the risk-free rate, and the initial value of the option.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"The probability that the underlying will go up or down, the risk-free rate, and the risk-neutral probability.\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/0Geaej45v7w\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  referrerpolicy=\"strict-origin-when-cross-origin\"\n  allowfullscreen>\n<\/iframe>\n\n\n<p>As the underlying value determines the option payoff, if we know the outcome of the underlying, we know the value of the option. If the underlying is above the exercise price at expiration, then the payoff is S<sub>T<\/sub> &#8211; X for calls and zero for puts. The converse is true if the underlying is below the exercise price at expiration. The derivation of an option pricing model requires the specification of a model of random processes that describe the movements in the underlying.<\/p>\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 model questions with our free trial. <\/a><\/div>\n<h2><strong>The Binomial Model for Stocks<\/strong><\/h2>\n<p>A model with two possible outcomes\u00a0is a binomial model. We start with the underlying at S<sub>0<\/sub> and let the price move up to S<sub>1<\/sub><sup>+<\/sup> and down to S<sub>1<\/sub><sup>&#8211;<\/sup>. We don&#8217;t know which outcome will occur, but we can assign probabilities. Assuming the probability of the move to S<sub>1<\/sub><sup>+<\/sup> is q, then the probability of moving to S<sub>1<\/sub><sup>&#8211;<\/sup>\u00a0is 1 &#8211; q.<\/p>\n<p>$$<br \/>\\begin{array}<br \/>\\hline<br \/>{} &amp; {\\small q } &amp; { S }_{ 1 }+ \\\\<br \/>{ S }_{ 0 } &amp; {\\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} } &amp; {} \\\\<br \/>{} &amp; {\\small 1-q} &amp; { S }_{ 1 }- \\\\<br \/>\\end{array} $$<\/p>\n<p>We then specify the returns implied by these moves up and down as factors <em>u<\/em> and <em>d<\/em> where <em>u<\/em> = S<sub>1<\/sub><sup>+<\/sup>\/S<sub>0<\/sub> and <em>d<\/em> =\u00a0S<sub>1<\/sub><sup>&#8211;<\/sup>\/S<sub>0<\/sub>.<\/p>\n<p>$$<br \/>\\begin{array}<br \/>\\hline<br \/>{} &amp; {\\small q } &amp; { S }_{ 0 }u \\\\<br \/>{ S }_{ 0 } &amp; {\\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} } &amp; {} \\\\<br \/>{} &amp; {\\small 1-q} &amp; { S }_{ 0 }d \\\\<br \/>\\end{array} $$<\/p>\n<h2>Deriving the Value of a Call Option Using a Binomial Model<\/h2>\n<p>We now consider a European call option with price c<sub>o<\/sub> today and price c<sub>1<\/sub><sup>+<\/sup> and c<sub>1<\/sub><sup>&#8211;<\/sup> at expiration. Assume we sell a call and buy <em>h<\/em> units of the underlying asset with portfolio value at inception V<sub>0<\/sub> = hS<sub>0<\/sub> &#8211; c<sub>o<\/sub>. At time 1, the portfolio will either be worth:<\/p>\n<p>$$ V_1^+ = hS_1^+ &#8211; c_1^+; or $$<\/p>\n<p>$$ V_1^- = hS_1^-\u00a0 &#8211; c_1^-$$<\/p>\n<p>If the portfolio was hedged then:<\/p>\n<p>$$ V_1^+ =\u00a0V_1^- $$<\/p>\n<p>Which could be re-written as:<\/p>\n<p>$$ hS_1^+ &#8211; c_1^+= hS_1^-\u00a0 &#8211; c_1^- $$<\/p>\n<p>Where\u00a0\\(h = \\frac{c_1^+ &#8211; c_1^- }{S_1^+ &#8211; S_1^-}\\)<\/p>\n<p>We also know that a perfectly hedged portfolio will earn the risk-free rate so:<\/p>\n<p>$$\u00a0V_1^+ or \\quad V_1^- = V_0(1+r)$$<\/p>\n<p>We can finally obtain the formula for the option price as:<\/p>\n<p>$$ c_0 =\u00a0\\frac{\u03c0c_1^+ + (1-\u03c0)c_1^-}{1+r}$$<\/p>\n<p>Where \\(\u03c0 = \\frac{1 + r &#8211; d}{u &#8211; d}\\)<\/p>\n<h3>How do we Interpret this Equation?<\/h3>\n<p>Having worked through all of the above, we have arrived at an equation for the value of a call option today, which takes the form of an expected future value (the numerator) discounted at the risk-free rate (the denominator). The volatility of the underlying is an important factor in determining the value of the option. If the volatility increases, the difference between S<sub>1<\/sub><sup>+<\/sup>\u00a0and S<sub>1<\/sub><sup>&#8211;\u00a0<\/sup>increases which widen the range between <em>c<sub>1<\/sub><sup>+\u00a0<\/sup><\/em>and <em>c<sub>1<\/sub><sup>&#8211;<\/sup><\/em>\u00a0leading to a higher option value.<\/p>\n<h3>How did <em>q<\/em> become\u00a0<em>\u03c0<\/em>?<\/h3>\n<p>We note that our actual probabilities of q and (1 &#8211; q) are not used, but instead, we have \u03c0 and (1 &#8211; \u03c0), which are called risk-neutral probabilities. If the option is trading at a price too high relative to the formula, investors can sell the call, buy <em>h<\/em> shares of the underlying and earn a return in excess of the risk-free rate while funding the transaction by borrowing at the risk-free rate. This action will put downward pressure on the call price until it conforms with the model price once more.<\/p>\n<h2>The Value of a Put Option Using a Binomial Model<\/h2>\n<p>Following the same methodology as above, we can derive a model for a put option as follows:<\/p>\n<p>$$p_0\u00a0=\u00a0\\frac{\u03c0p_1^+ + (1-\u03c0)p_1^-}{1+r}$$<\/p>\n<p>Where \\(\u03c0 = \\frac{1 + r &#8211; d}{u &#8211; d}\\)<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Which factors are the <em>most<\/em> relevant to determine an option&#8217;s value using a binomial pricing model?<\/p>\n<p>A. The probability that the underlying will go up or down, the risk-free rate, and the initial value of the option<\/p>\n<p>B. The risk-free rate, the volatility of the underlying, and the exercise price<\/p>\n<p>C.\u00a0The probability that the underlying will go up or down, the risk-free rate, and the risk-neutral probability<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>The probability that the underlying will go up or down is not a factor in determining the price of an option using a binomial model because we derive it from the formula \\(\u03c0 = \\frac{1 + r &#8211; d}{u &#8211; d}\\).<\/p>\n<p>The volatility of the underlying asset is an important factor, as is the risk-free rate, the risk-neutral probability, and the exercise price.<\/p>\n<\/blockquote>\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>\n<p style=\"margin-top: 10px; max-width: 600px; margin-left: auto; margin-right: auto; font-size: 14px;\">Solve CFA-style derivatives questions and master the binomial option pricing model.<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>As the underlying value determines the option payoff, if we know the outcome of the underlying, we know the value of the option. If the underlying is above the exercise price at expiration, then the payoff is ST &#8211; X&#8230;<\/p>\n","protected":false},"author":18,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10],"tags":[],"class_list":["post-3890","post","type-post","status-publish","format-standard","hentry","category-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>One-Period Binomial Model for Options | CFA Level 1<\/title>\n<meta name=\"description\" content=\"The one-period binomial model estimates option prices by modeling the random movements of an underlying asset using defined probabilities.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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