{"id":15130,"date":"2021-05-10T11:42:06","date_gmt":"2021-05-10T11:42:06","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=15130"},"modified":"2026-03-11T07:26:26","modified_gmt":"2026-03-11T07:26:26","slug":"delta-hedging","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/delta-hedging\/","title":{"rendered":"Delta Hedging"},"content":{"rendered":"<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\": \"An investor owns a portfolio with 10,000 shares of Contagia Inc stock trading at $30 per share. The investor wants to delta hedge the portfolio using call options with a delta of 0.5. What strategy creates a delta-neutral hedge?\",\r\n    \"text\": \"An investor owns a portfolio with 10,000 shares of Contagia Inc common stock currently trading at $30 per share. The investor wants to delta hedge the portfolio using call options. A call option on the Contagia shares with a strike price of $30 has a delta of 0.5.\\n\\nThe strategy to create a delta-neutral hedge most likely involves:\\n\\nA. Selling 10,000 call options.\\n\\nB. Buying 20,000 call options.\\n\\nC. Selling 20,000 call options.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is C.\\n\\nThe portfolio delta from holding 10,000 shares is +10,000 because stock has a delta of 1 per share. To achieve a delta-neutral position, the investor must offset this exposure using call options with a delta of 0.5.\\n\\nThe hedge ratio is calculated as: N_H = -Portfolio Delta \/ Delta_H.\\n\\nN_H = -10,000 \/ 0.5 = -20,000.\\n\\nThis means the investor must sell 20,000 call options to offset the positive delta from the stock position and achieve delta neutrality.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\r\n<p>&nbsp;<\/p>\r\n\r\n<p>Delta hedging involves adding up the deltas of the individual assets and options making up a portfolio. A <em><strong>delta hedged portfolio<\/strong><\/em> is one for which the weighted sums of deltas of individual assets is zero. A position with a zero delta is referred to as a delta-neutral position.<\/p>\r\n<p>Denote the delta of a hedging instrument by \\(Delta_{H}\\)<\/p>\r\n<p>The optimal number of hedging units, \\(N_{H}=-\\frac{\\text{Portfolio Delta}}{\\text{Delta}_{H}}\\)<\/p>\r\n<p>To achieve a delta hedged portfolio, short the hedging instrument if \\(N_{H}\\) is negative and long the hedging instrument if\u00a0 \\(N_{H}\\) is positive.<\/p>\r\n<p>A delta-neutral portfolio is one that does not change in value for small changes in the underlying price. Delta neutral implies that:<\/p>\r\n<p>$$\\text{Portfolio delta} +N_{H}Delta_{H}=0$$<\/p>\r\n<p>The portfolio should be rebalanced regularly to ensure that the sum of deltas remains close to zero. <em><strong>Static<\/strong><\/em> delta hedging involves constructing an initial portfolio with a sum of deltas of zero, at time 0, and never adjusting it. On the other hand, <em><strong>dynamic <\/strong><\/em>delta hedging involves continuously rebalancing the portfolio to maintain a constant total portfolio delta of zero.<\/p>\r\n<div style=\"text-align: center; margin: 20px 0;\"><a style=\"display: inline-block; padding: 10px 18px; border: 2px solid #1e5bd8; color: #1e5bd8; border-radius: 9999px; text-decoration: none; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Practice delta hedging and option risk questions for CFA Level II <\/a><\/div>\r\n<h3>Example 1<\/h3>\r\n<p>Consider a portfolio composed of 1,500 shares. Call options with a delta of +0.50 are used to hedge this portfolio. A delta hedge could be implemented by selling enough calls to make the portfolio delta neutral.<\/p>\r\n<p>The optimal number of hedging units is determined as follows:<\/p>\r\n<p>$$N_{H}=-\\frac{\\text{Portfolio delta}}{\\text{Delta}_{H}}$$<\/p>\r\n<p>Portfolio delta= 1,500<\/p>\r\n<p>\\(\\text{Delta}_{\\text{H}} = +0.50\\)<\/p>\r\n<p>Thus, \\(\\text{N}_{H}=-\\frac{1,500}{0.50}=-3,000\\)<\/p>\r\n<p>This means that we must sell 3,000 calls to achieve delta neutrality.<\/p>\r\n<h3>Example 2<\/h3>\r\n<p>Given the following information:<\/p>\r\n<p>\\(S-{0}=60\\)<\/p>\r\n<p>\\(K=50\\)<\/p>\r\n<p>\\(r=2\\%\\)<\/p>\r\n<p>\\(T=1\\)<\/p>\r\n<p>\\(\\sigma=20\\%\\)<\/p>\r\n<p>\\(Delta_{c}=0.537\\)<\/p>\r\n<p>\\(Delta_{p}=-0.463\\)<\/p>\r\n<p>Assume that the underlying asset does not pay a dividend.<\/p>\r\n<p>Consider a <em><strong>short position<\/strong><\/em> of 5,000 shares of stock.<\/p>\r\n<h6>Hedging using call options<\/h6>\r\n<p>The optimal number of hedging units,<\/p>\r\n<p>$$N_{H}=-\\frac{\\text{Portfolio delta}}{\\text{Delta}_{H}}$$<\/p>\r\n<p>Where: \\(\\text{Portfolio Delta}=-5,000\\)<\/p>\r\n<p>\\(\\text{Delta}_{H}=0.537\\)<\/p>\r\n<p>$$N_{H}=-\\frac{-(-5,000)}{0.537}=9,311$$<\/p>\r\n<p>This means that we must buy 9,311 calls to make the portfolio delta neutral.<\/p>\r\n<h6>Hedging using put options<\/h6>\r\n<p>We have portfolio delta \\(=-5,000\\)<\/p>\r\n<p>\\(\\text{Delta}_{H}=-0.463\\)<\/p>\r\n<p>$$N_{H}=-\\frac{-5,000}{-0.463}=-10,799$$<\/p>\r\n<p>This means that we must sell 10,799 put options.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>An investor owns a portfolio with 10,000 shares of Contagia Inc. common stock currently trading at $30 per share. The investor wants to delta hedge the portfolio using call options. A call option on the Contagia shares with a strike price of $30 has a delta of 0.5.<\/p>\r\n<p>The strategy to create a delta-neutral hedge <em>most likely<\/em> involves:<\/p>\r\n<ol style=\"list-style-type: upper-alpha;\">\r\n\t<li>Selling 10,000 call options<\/li>\r\n\t<li>Buying 20,000 call options<\/li>\r\n\t<li>Selling 20,000 call options<\/li>\r\n<\/ol>\r\n<h3>Solution<\/h3>\r\n<p><strong>The correct answer is C:<\/strong><\/p>\r\n<p>Portfolio delta \\(= 10,000\\)<\/p>\r\n<p>\\(\\text{Delta}_{H}=0.5\\)<\/p>\r\n<p>The optimal number of call options required to hedge against movements in the stock price is determined as:<\/p>\r\n<p>$$N_{H}=-\\frac{\\text{Portfolio delta}}{\\text{Delta}_{H}}$$<\/p>\r\n<p>$$N_{H}=\\frac{-10,000}{0.5}=-20,000$$<\/p>\r\n<p>This means that the investor must sell 20,000 calls to achieve delta neutrality.<\/p>\r\n<\/blockquote>\r\n<p><em>Reading 38: Valuation of Contingent Claims<\/em><\/p>\r\n<p><em>LOS 38 (i) describe how a delta hedge is executed<\/em><\/p>\r\n\r\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-block; padding: 12px 24px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a>\r\n<p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Strengthen your CFA Level II derivatives skills with exam-style practice on delta hedging, option sensitivities, and dynamic hedging strategies.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>&nbsp; Delta hedging involves adding up the deltas of the individual assets and options making up a portfolio. A delta hedged portfolio is one for which the weighted sums of deltas of individual assets is zero. A position with a&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,302],"tags":[216,338,304,336],"class_list":["post-15130","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-cfa-level-2","tag-delta-hedging","tag-derivatives","tag-reading-38","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Delta Hedging - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Delta hedging involves adding up the deltas of the individual assets and options making up a portfolio.\" \/>\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\/delta-hedging\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Delta Hedging - 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