{"id":18647,"date":"2021-07-28T15:28:21","date_gmt":"2021-07-28T15:28:21","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=18647"},"modified":"2024-04-07T05:30:15","modified_gmt":"2024-04-07T05:30:15","slug":"describe-how-the-value-of-a-european-option-can-be-analyzed-as-the-present-value-of-the-options-expected-payoff-at-expiration","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-how-the-value-of-a-european-option-can-be-analyzed-as-the-present-value-of-the-options-expected-payoff-at-expiration\/","title":{"rendered":"Expectations Valuation Approach"},"content":{"rendered":"\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/jmbJLCdBEgs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<h2>One-step Binomial Tree<\/h2>\r\n<p>Since a hedged portfolio returns the risk-free rate, it can determine the initial value of a call or put. The expectations approach calculates the values of the option by taking the present value of the expected terminal option payoffs. This approach utilizes risk-neutral probabilities instead of true probabilities.<\/p>\r\n<p>Therefore, the initial value of a call and put respectively are determined using the following formulas:<\/p>\r\n<p>$$ c_0=\\frac{qc_u+\\left(1-q\\right)c_d}{1+r} $$<\/p>\r\n<p>And<\/p>\r\n<p>$$ p_0=\\frac{qp_u+\\left(1-q\\right)p_d}{1+r} $$<\/p>\r\n<p>Where:<\/p>\r\n<p>$$ q=\\frac{\\left(1+r\\right)-d}{u-d} $$<\/p>\r\n<p>Where<\/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, while \\((1-q)\\) gives the probability of a downward move.<\/p>\r\n<h4>Example: Expectations Approach for One-step Binomial Tree<\/h4>\r\n<p>Consider a stock that is currently trading at $50. Assume that the up jump and down jump factors for the stock price are <em>u<\/em> = 1.20 and <em>d<\/em> = 0.80. The risk-free rate compounded periodically is 4%. Given a strike price of $50, we can use a single period binomial model to price European call and put options.<\/p>\r\n<h4>Price of a European Call Option<\/h4>\r\n<p>Note that:<\/p>\r\n<p>$$ \\begin{align*} c_T &amp;=max{\\left(S_T-K,0\\right)} \\\\ S_0u &amp; =50\\times1.20=$60 \\\\ S_0d &amp;=50\\times0.80=$40 \\\\ c_u &amp;=max\\left($60-$50,0\\right)=$10 \\\\ c_d &amp;=max\\left($40-$50,0\\right)=$0 \\end{align*} $$<\/p>\r\n<p>The value of the call option can then be determined using the formula:<\/p>\r\n<p>$$ c_0=\\frac{qc_u+\\left(1-q\\right)c_d}{1+r} $$<\/p>\r\n<p>Where:<\/p>\r\n<p>$$ \\begin{align*} q &amp;=\\frac{\\left(1+r\\right)-d}{u-d} \\\\ q &amp;=\\frac{\\left(1.04\\right)-0.8}{1.20-0.80}=0.6 \\\\ c_0 &amp;=\\frac{0.6\\times$10+\\left(1-0.6\\right)\\times0}{1.04}=$5.77 \\end{align*} $$<\/p>\r\n<h4>Price of a European Put Option<\/h4>\r\n<p>$$ \\begin{align*} p_T &amp;=max{\\left(K-S_T,0\\right)} \\\\ p_u &amp;=max\\left($50-$60,0\\right)=$0 \\\\ p_d &amp;=max\\left($50-$40,0\\right)=$10 \\\\ p_0 &amp;=\\frac{qp_u+\\left(1-q\\right)p_d}{1+r} \\\\ p_0 &amp;=\\frac{0.60\\times0+\\left(1-0.60\\right)\\times$10}{1.04} \\\\ p_0 &amp;=$3.85 \\end{align*} $$<\/p>\r\n<h2>Two-step Binomial Tree<\/h2>\r\n<p>The expectations approach can also be applied to the two-step binomial model to determine the value of options.<\/p>\r\n<p>Let \\(q\\) be the risk-neutral probability of an up move. In this instance, the price of a European call option can be determined using the two-step binomial model:<\/p>\r\n<p>$$ c_o=\\frac{q^2c_{uu} + 2q\\left(1-q\\right)c_{ud} +\\left(1 &#8211; q\\right)^2c_{dd}}{\\left(1 + r\\right)^2} $$<\/p>\r\n<p>The two-period European put value is given as:<\/p>\r\n<p>$$ p_o=\\frac{q^2p_{uu} + 2q\\left(1-q\\right)p_{ud} +\\left(1- q\\right)^2p_{dd}}{\\left(1 + r\\right)^2} $$<\/p>\r\n<h4>Example: Expectations Approach for two-Step Binomial Model<\/h4>\r\n<p>Assume that you have a stock that is currently trading at $60. A two-year European call option on the stock is available with a strike price of $60. The risk-free rate is 2% per annum. Given that the up-move factor is 1.10 and the down-move factor is 0.90, the value of the call option using a two-period binomial model is <em>closest<\/em> to:<\/p>\r\n<h4>Solution<\/h4>\r\n<p>The risk-neutral probability of an up-move is given by:<\/p>\r\n<p>$$ \\begin{align*} q &amp;=\\frac{\\left(1+r\\right)-d}{u-d} \\\\ q &amp;=\\frac{1.02-0.90}{1.1-0.9}=0.6 \\end{align*} $$<\/p>\r\n<p>$$ \\text{The probability of down move } (1-q) = 1-0.6 = 0.4 $$<\/p>\r\n<p>The two-period binomial tree is shown below:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26453\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example.jpg\" alt=\"Two-step Binomial Model - Example\" width=\"1590\" height=\"1161\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example-400x292.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>The two-period binomial value of the call option:<\/p>\r\n<p>$$ \\begin{align*} c_o &amp;=\\frac{q^2c_{uu} + 2q\\left(1-q\\right)c_{ud} +\\left(1 &#8211; q\\right)^2c_{dd}}{\\left(1 + r\\right)^2} \\\\ c_0 &amp;=\\frac{{0.6}^2\\times$12.60+2\\times0.6\\times0.4\\times0+{0.4}^2\\times0}{\\left(1.02\\right)^2} \\\\ c_0 &amp; =$4.36 \\end{align*} $$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Nabi Gudka, CFA, applies the expectations approach to value a European call option on the common shares of Wipro Inc. The expectation approach <em>most likely<\/em> utilizes:<\/p>\r\n<ol type=\"A\">\r\n\t<li>A risk premium for discounting.<\/li>\r\n\t<li>Risk-neutral probabilities.<\/li>\r\n\t<li>Actual probabilities.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>Under the expectations approach, the expected future payoff is calculated using risk-neutral probabilities, and the expected payoff is discounted at the risk-free rate.\u2003<\/p>\r\n<\/blockquote>\r\n<p>Reading 34: Valuation of Contingent Claims<\/p>\r\n<p><em>LOS 34 (e) Describe how the value of a European option can be analyzed as the present value of the option\u2019s expected payoff at expiration.<\/em><\/p>\r\n\r\n","protected":false},"excerpt":{"rendered":"<p>One-step Binomial Tree Since a hedged portfolio returns the risk-free rate, it can determine the initial value of a call or put. The expectations approach calculates the values of the option by taking the present value of the expected terminal&#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-18647","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 v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Expectations Valuation Approach - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"The expectations approach can also be applied to the two-step binomial model to determine the value of options.\" \/>\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\/describe-how-the-value-of-a-european-option-can-be-analyzed-as-the-present-value-of-the-options-expected-payoff-at-expiration\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Expectations Valuation Approach - 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