{"id":43925,"date":"2023-01-02T10:57:05","date_gmt":"2023-01-02T10:57:05","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=43925"},"modified":"2026-01-23T10:26:38","modified_gmt":"2026-01-23T10:26:38","slug":"risk-neutrality-in-derivative-pricing","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/risk-neutrality-in-derivative-pricing\/","title":{"rendered":"Risk-Neutrality in Derivative Pricing"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Valuing a Derivative Using a One-Period Binomial Model (2025 CFA\u00ae Level I Exam \u2013 Derivatives \u2013 Module 10)\",\n  \"description\": \"This lesson covers derivative valuation using a one-period binomial model for the 2025 CFA\u00ae Level I Derivatives curriculum. It explains how to construct a binomial price tree, compute derivative payoffs, and determine present values using the risk-free rate. The video also introduces the concept of risk neutrality and shows how risk-neutral probabilities are applied to price derivatives in an exam-focused framework.\",\n  \"uploadDate\": \"2022-12-31T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/P3O1yAGCPe4\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/P3O1yAGCPe4\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/P3O1yAGCPe4\",\n  \"duration\": \"PT36M40S\"\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\": \"A European call option that expires in one year has an exercise price of GBP 70. The spot price of the underlying asset is GBP 70. Suppose that the underlying price is expected to increase or decrease by 20% within the next year, assuming a risk-free interest rate of 5%. The no-arbitrage price of a put option on the underlying asset (with similar exercise price and time to maturity) using the binomial model is closest to:\",\n    \"text\": \"A European call option that expires in one year has an exercise price of GBP 70. The spot price of the underlying asset is GBP 70. Suppose that the underlying price is expected to increase or decrease by 20% within the next year, assuming a risk-free interest rate of 5%. The no-arbitrage price of a put option on the underlying asset (with similar exercise price and time to maturity) using the binomial model is closest to:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. Using the one-period binomial model, the call option payoff is GBP 14 in the up state and GBP 0 in the down state. The hedge ratio is 0.5, and the risk-free portfolio value discounted at 5% gives a call value of approximately GBP 8.33. Applying put-call parity, the put option value is approximately GBP 5.00.\"\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\/P3O1yAGCPe4\"\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\n<p>Remember that the value of an option is not affected by the real-world probabilities of the underlying price increments or decrements but rather by the expected volatilities (\\(R^u\\) and \\(R^d\\) ), which are required to price an option.<\/p>\n\n\n\n<p>We can compute the value of a call option today by discounting its expected value at expiration at the current risk-free rate, as summarized in the equation below:<\/p>\n\n\n\n<p>$$c_0=\\frac{\\pi c_1^u+(1-\\pi)c_1^d}{(1+{r)}^T}$$<\/p>\n\n\n\n<p>Similarly, the value of the put option is given by:<\/p>\n\n\n\n<p>$$p_0=\\frac{p_i.p_1^u+(1-\\pi)p_1^d}{(1+{r)}^T}$$<\/p>\n\n\n\n<p>The <strong>risk-neutral probability<\/strong> (denoted by \\(\\pi\\)) is defined as the computed probability used in binomial option pricing that equates the discounted weighted sum of the expected values of the underlying to the option&#8217;s current price. It is calculated using the risk-free rate and the assumed up-and-down gross returns of the underlying, as follows:<\/p>\n\n\n\n<p>$$\\pi=\\frac{(1+r)-R^d}{R^u-R^d}$$<\/p>\n\n\n\n<p>Risk-neutral pricing is the process of determining the risk-neutral probability (which is used to calculate the present value of future cash flows) using only the expected volatilities, i.e., (\\(R^u\\) and \\(R^d)\\), and the risk-free rate.<\/p>\n\n\n\n<div style=\"margin: 0 0 20px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      border: 2px solid #1e63ff;\n      color: #1e63ff;\n      background: #ffffff;\n      padding: 10px 14px;\n      border-radius: 10px;\n      font-weight: 500;\n      line-height: 1.35;\n      text-decoration: none;\n    \"\n  >\n    Want to apply risk-neutral valuation to derivative pricing scenarios? Try AnalystPrep\u2019s free trial now.\n  <\/a>\n<\/div>\n\n\n<\/p>\n<h4><strong>Example<\/strong>:<strong> Risk-neutral Probability<\/strong><\/h4>\n<p>A company is considering selling a one-year call option on a non-dividend paying stock whose current price is $80. The exercise price of the call option is $85, and the risk-free interest is 4%.<\/p>\n<p>If the stock price is expected to go up or down by 30%, what is the selling price of the option? (Using risk-neutral pricing).<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Consider the following diagram:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"2145\" height=\"672\" class=\"alignnone wp-image-43927 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13.png\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13.png 2145w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13-300x94.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13-1024x321.png 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13-768x241.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13-1536x481.png 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13-2048x642.png 2048w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Binomial-13-400x125.png 400w\" sizes=\"auto, (max-width: 2145px) 100vw, 2145px\" \/><\/p>\n<p><strong>Risk neutral probability <\/strong>of an upward move is given by:<\/p>\n<p>$$\\pi=\\frac{(1+r)-R^d}{R^u-R^d}=\\frac{1.04-0.7}{1.3-0.7}=0.56667$$<\/p>\n<p>Intuitively, the risk-neutral probability of a downward move is given by:<\/p>\n<p>$$1-\\pi=1-0.56667=0.43333$$<\/p>\n<p>We need:<\/p>\n<p>$$c_0=\\frac{\\left[c_1^u+\\left(1-\\pi\\right)c_1^d\\right]}{1+r}=\\frac{0.56667\\times19+0.43333\\times0}{1.04}=$10.35$$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>A European call option that expires in one year has an exercise price of GBP 70. The spot price of the underlying asset is GBP 70. Suppose that the underlying price is expected to increase or decrease by 20% within the next year, assuming a risk-free interest rate of 5%. The no-arbitrage price of a put option on the underlying asset (with similar exercise price and time to maturity) using the binomial model is <em>closest <\/em>to:<\/p>\n<p>A. GBP 5.00.<\/p>\n<p>B. GBP 14.00.<\/p>\n<p>C. GBP 24.04.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p><strong>Step 1<\/strong>:<strong> Determine the call option&#8217;s value at maturity.<\/strong><\/p>\n<p>$$\\begin{align}S_1^u&amp;=\\frac{120}{100}\\times70=84\\\\ S_1^d&amp;=\\frac{80}{100}\\times70=56\\\\ c_1^u&amp;=\\max\\ \\left(0,\\ S_1^u-X\\right)=\\max\\ \\left(0,\\ 84-70\\right)=14 \\\\c_1^d&amp;=\\max\\ \\left(0,\\ S_1^d-X\\right)=\\max\\ \\left(0,\\ 56-70\\right)=0\\end{align}$$<\/p>\n<p><strong>Step 2<\/strong>:<strong> Determining <em>h<\/em> (the edge ratio).<\/strong><\/p>\n<p>$$h=\\frac{c_1^u-c_1^d}{S_1^u-S_1^d}=\\frac{14-0}{84-56}=0.5$$<\/p>\n<p><strong>Step 3<\/strong>: <strong>Determine the portfolio value if the price of the underlying increases or decreases.<\/strong><\/p>\n<p>$$\\begin{align}V_1^u&amp;=\\ {hS}_1^u-c_1^u=\\left(0.5\\times84\\right)-14=28 \\\\V_1^d&amp;={hS}_1^d-C_1^d=\\left(0.5\\times56\\right)-0=28\\end{align}$$<\/p>\n<p><strong>Step 4<\/strong>:<strong> Determining <em>V<sub>0<\/sub><\/em>.<\/strong><\/p>\n<p>$$V_0=\\frac{V_1}{(1+{r)}^t}=\\frac{28}{(1+{0.05)}^1}=26.67$$<\/p>\n<p><strong>Note<\/strong>: \\(V_1=V_1^d=V_1^u\\).<\/p>\n<p><strong>Step 5<\/strong>:<strong> Determining \\(p_0\\) and \\(c_0\\).<\/strong><\/p>\n<p>We need,<\/p>\n<p>$$\\begin{align} c_0 &amp;= \\frac{\\pi.c_{1}^{u} + (1 &#8211; \\pi).c_{1}^{d}}{1 + r} \\end{align}$$<\/p>\n<p>where<\/p>\n<p>$$\\begin{align}\\pi &amp;=\\frac{(1+r)-R^d}{R^u-R^d}\\\\ &amp;=\\frac{(1.05)-0.8}{1.2-0.8}\\\\ &amp;=0.625\\end{align}$$<\/p>\n<p>So,<\/p>\n<p>$$\\begin{align} c_0 &amp;= \\frac{\\pi.c_{u}^{1} + (1 &#8211; \\pi).c_{d}^{1}}{1 + r} \\\\ &amp;= \\frac{(0.625 \\times 14) + (1 &#8211; 0.625) \\times 0}{1.05} \\\\ &amp;= \\frac{8.75}{1.05} \\\\ &amp;\\approx 8.33 \\end{align}$$<\/p>\n<p>To find \\(p_0\\) we need to use put-call parity:<\/p>\n<p>$$\\begin{align}p_0&amp;=\\ c_0-S_0+X(1+{r)}^{-T}\\\\&amp;=8.33-70+70(1+{0.05)}^{-1}=4.997 \\\\&amp;\\approx. 5.00\\end{align}$$<\/p>\n<\/blockquote>\n<p><!-- \/wp:post-content --><\/p>\n<p><!-- wp:html --><\/p>\n<div style=\"text-align: center; margin: 32px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      background-color: #1e63ff;\n      color: #ffffff;\n      padding: 12px 26px;\n      border-radius: 12px;\n      font-weight: 600;\n      font-size: 16px;\n      text-decoration: none;\n    \"\n  ><br \/>\n    Start Free Trial \u2192<br \/>\n  <\/a><\/p>\n<div style=\"margin-top: 10px; font-size: 14px; color: #374151;\">\n    Practice risk-neutral pricing, expected payoffs, and derivative valuation using full CFA\u00ae-style questions and solutions.\n  <\/div>\n<\/div>\n<p><!-- \/wp:html --><\/p>","protected":false},"excerpt":{"rendered":"<p>Remember that the value of an option is not affected by the real-world probabilities of the underlying price increments or decrements but rather by the expected volatilities (\\(R^u\\) and \\(R^d\\) ), which are required to price an option. We can&#8230;<\/p>\n","protected":false},"author":15,"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-43925","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>Risk-Neutrality in Derivative Pricing | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Risk-neutral valuation involves discounting the expected value of options at the risk-free rate. 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