{"id":18693,"date":"2021-07-29T14:42:47","date_gmt":"2021-07-29T14:42:47","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=18693"},"modified":"2026-03-24T07:22:58","modified_gmt":"2026-03-24T07:22:58","slug":"describe-how-the-black-model-is-used-to-value-european-options-on-futures","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-how-the-black-model-is-used-to-value-european-options-on-futures\/","title":{"rendered":"Black Option Valuation Model"},"content":{"rendered":"\r\n<p><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\": \"How is the Black model used to value a European call option on a futures contract?\",\r\n    \"text\": \"The US 30 index is at $30,605. A futures contract on it trades at $30,400. The exercise price is $30,000, the continuously compounded risk-free rate is 1.75%, the time to futures and options expiration is two months, and volatility is 15%. The dividend yield is 1.8%. The following results have been determined:\\n\\nN(d1) = 0.6388\\nN(\u2212d1) = 0.3612\\nN(d2) = 0.6156\\nN(\u2212d2) = 0.3844\\nc\u2080 = 948.75\\np\u2080 = 549.91\\n\\nWhich of the following options best describes how the Black model is used to value a European call option on the futures contract?\\n\\nA. The call value is the present value of the difference between the current futures price times 0.6388 and the exercise price times 0.6156.\\n\\nB. The call value is the present value of the difference between the exercise price times 0.6156 and the current futures price times 0.6388.\\n\\nC. The call value is the present value of the difference between the current spot price times 0.6388 and the exercise price times 0.6156.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is A. Under the Black model, the value of a European call option on a futures contract is c\u2080 = e^(\u2212rT)[F\u2080(T)N(d1) \u2212 KN(d2)]. This means the option value is the present value of the difference between the futures price multiplied by N(d1) and the exercise price multiplied by N(d2). The model uses the futures price rather than the spot price in valuing options on futures.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/rnMud0L9-g0\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>The <em><strong>Black options valuation model<\/strong><\/em> is a modified version of the BSM model used for options on underlying securities that are costless to carry, including options on futures and forward contracts.<\/p>\r\n<p>Similar to the BSM model, the Black model assumes that future prices follow geometric Brownian motion. The Black option model values for European call and put options are expressed as:<\/p>\r\n<p>$$ \\begin{align*} \\text{European call}: c_0 &amp;= e^{\u2013rT} \\left[F_0(T)N(d_1) \u2013 KN(d_2) \\right] \\\\ \\text{European put}: p_0 &amp;=e^{-rT} \\left[KN\\left({-d}_2\\right)-F_0\\left(T\\right)N\\left(-d_1\\right) \\right] \\end{align*} $$<\/p>\r\n<p>Where:<\/p>\r\n<p>$$ d_1=\\frac{\\ln{\\left(\\frac{F_0\\left(T\\right)}{K}\\right)}+\\frac{\\sigma^2}{2}T}{\\sigma\\sqrt T} $$<\/p>\r\n<p>and<\/p>\r\n<p>$$ d_2=d_1-\\sigma\\sqrt T $$<\/p>\r\n<p>\\(F_0\\left(T\\right)\\) = Futures price at time 0 that expires at time T.<\/p>\r\n<p>\\(\\sigma\\) = Volatility of returns on the futures price.<\/p>\r\n<div style=\"text-align: center; margin: 25px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 10px 18px; border: 2px solid #1a73e8; border-radius: 999px; color: #1a73e8; text-decoration: none; font-weight: 500; background-color: #f5f9ff; white-space: nowrap;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Price options on futures using the Black model <\/a><\/div>\r\n<h4>Example: Black Option Valuation<\/h4>\r\n<p>The NASDAQ index currently stands at $12,900. A three-month futures contract on the index trades at $12,800. The exercise price is $12,750, the continuously compounded risk-free rate is 1%, and volatility is 15%. Finally, the index has a dividend yield of 2%.<\/p>\r\n<p>The above information has been used to obtain the following results for both call and put options on the futures contract:<\/p>\r\n<p>$$ \\begin{array}{cc|cc} \\textbf{Calls} &amp; &amp; \\textbf{Puts} &amp; \\\\ \\hline N(d_1) &amp; 0.5636 &amp; N(\u2013d_1) &amp; 0.4364 \\\\ \\hline N(d_2) &amp; 0.5339 &amp; N(\u2013d_2) &amp; 0.4661 \\\\ \\hline c_0 &amp; 405.84 &amp; p_0 &amp; 355.83 \\end{array} $$<\/p>\r\n<p>The values of a European call option and the put option on the futures contract are <em>closest<\/em> to:<\/p>\r\n<h4>Solution<\/h4>\r\n<p>$$ \\begin{align*} \\text{European call}: c_0&amp; = e^{\u2013rT} \\left[F_0(T)N(d_1) \u2013 KN(d_2) \\right] \\\\ c_0 &amp;= e^{\u20130.01\u00d70.25} \\left[12,800\u00d70.5636\u2013 12,750\u00d70.5339 \\right] \\\\ &amp;=$405.84 \\\\ \\text{European put}: p_0 &amp;=e^{-rT} \\left[KN\\left({-d}_2\\right)-F_0\\left(T\\right)N\\left(-d_1\\right) \\right] \\\\ p_0 &amp;=e^{-0.01\\times0.25}\\left[12,750\\times0.4661-12,800\\times0.4364\\right]\\\\ &amp; =$355.96 \\end{align*} $$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>The US 30 index is at $30,605. A futures contract on it trades at $30,400. The exercise price is $30,000, the continuously compounded risk-free rate is 1.75%, the time to a futures contract and options expiration is two months, and the volatility is 15%. The US 30 dividend yield is 1.8%. The following results have been determined using the above information:<\/p>\r\n<p>$$ \\begin{array}{cc|cc} \\textbf{Calls} &amp; &amp; \\textbf{Puts} &amp; \\\\ \\hline N(d_1) &amp; 0.6388 &amp; N(\u2013d_1) &amp; 0.3612 \\\\ \\hline N(d_2) &amp; 0.6156 &amp; N(\u2013d_2) &amp; 0.3844 \\\\ \\hline c_0 &amp; 948.75 &amp; p_0 &amp; 549.91 \\end{array} $$<\/p>\r\n<p>Which of the following options <em>best<\/em> describes how the black model is used to value a European call option on the futures contract?<\/p>\r\n<p>The call value is the present value of the difference between:<\/p>\r\n<ol type=\"A\">\r\n\t<li>The current futures price times 0.6388 and the exercise price times 0.6156.<\/li>\r\n\t<li>The exercise price times 0.6156, and the current futures price times 0.6388.<\/li>\r\n\t<li>The current spot price times 0.6388, and the exercise price times 0.6156.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>The value of a European call option on a futures contract is obtained using the formula:<\/p>\r\n<p>$$ c_0= e^{\u2013rT}\\left[F_0(T)N(d_1)\u2013 KN(d_2) \\right] $$<\/p>\r\n<p>The above formula implies that the valuation of a European call option based on the black model involves calculating the present value of the difference between the futures price and the exercise price.<\/p>\r\n<\/blockquote>\r\n<p>Reading 34: Valuation of Contingent Claims<\/p>\r\n<p><em>LOS 34 (i) Describe how the Black model is used to value European options on futures.<\/em><\/p>\r\n\r\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 20px; border-radius: 999px; background-color: #1a73e8; color: #ffffff; text-decoration: none; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a>\r\n<p style=\"font-size: 15px; margin-top: 12px; color: #555;\">Learn how the Black model values European options on futures using forward prices, volatility, and discounting, and how d\u2081 and d\u2082 are used to estimate option value in CFA Level II derivatives questions.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>The Black options valuation model is a modified version of the BSM model used for options on underlying securities that are costless to carry, including options on futures and forward contracts. Similar to the BSM model, the Black model assumes&#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-18693","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>Black Model for European Options on Futures | CFA II<\/title>\n<meta name=\"description\" content=\"Learn how the Black model values European options on futures, including the formula, assumptions, and how it differs from the Black\u2013Scholes model.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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