{"id":43675,"date":"2022-12-23T12:29:44","date_gmt":"2022-12-23T12:29:44","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=43675"},"modified":"2025-05-20T09:27:10","modified_gmt":"2025-05-20T09:27:10","slug":"arbitrage-in-contingent-claims","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/","title":{"rendered":"Arbitrage in Contingent Claims"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Pricing and Valuation of Options (2025 Level I CFA\u00ae Exam \u2013 Derivatives \u2013 Module 8)\",\n  \"description\": \"This video lesson covers option pricing and valuation concepts in the CFA Level I curriculum, including intrinsic (exercise) value, time value, and moneyness. It explains arbitrage, replication, and the six key factors affecting option values\u2014like volatility and interest rates\u2014while emphasizing their impact on calls and puts.\",\n  \"uploadDate\": \"2022-12-19T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/AR1hs3m4xDk\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=AR1hs3m4xDk\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/AR1hs3m4xDk\",\n  \"duration\": \"PT40M12S\"\n}\n<\/script>\n\n\n\n<p>[vsw id=&#8221;AR1hs3m4xDk&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;]<\/p>\n\n\n\n<p>Recall that arbitrage opportunities occur if the law of one price does not hold. The no-arbitrage conditions in options are based on the payoff at maturity.<\/p>\n\n\n\n<p>Unlike forward commitments with symmetric profiles (as presented earlier), contingent claims have asymmetric payoff profiles. That is:<\/p>\n\n\n\n<p>$$\\begin{align*}c_T&amp;=\\text{max}(0,S_T-X)\\\\ p_T&amp;=\\text{max}(0,X-S_T)\\end{align*}$$<\/p>\n\n\n\n<p>Moreover, in contrast to forward commitments with an initial value of zero at initiation, the option buyer pays the seller a <strong>premium<\/strong> \\(c_0)\\)&nbsp;for call options and \\(p_0\\) for a put options. Profits at maturity are:<\/p>\n\n\n\n<p>$$\\begin{align*}\u03c0_\\text{call}&amp;=\\text{max}(0,S_T-X)-c_0\\\\ \u03c0_\\text{put}&amp;=\\text{max}(0,X-S_T)-p_0\\end{align*}$$<\/p>\n\n\n\n<p>An option is only exercised when it is in the money. As such, this condition calls for upper and lower no-arbitrage price bounds at any time \\(t\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Upper and Lower Arbitrage Bounds<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Call Option<\/h3>\n\n\n\n<p>A call option is exercisable if the underlying price exceeds the exercise price. That is \\(S_t&gt;X\\). As such, the lower bound of a call price is the underlying price minus the present value of the exercise price or zero, whichever is greater.<\/p>\n\n\n\n<p>$$\\text{Lower bound}=\\text{max}(0,S_t-X(1+r)^{-(T-t)})$$<\/p>\n\n\n\n<p>A call buyer will not pay more than the underlying price for the right to buy the underlying. &nbsp;As such, the upper bound is the current underlying price.<\/p>\n\n\n\n<p>$$\\text{Upper bound}=\\text{S}_{\\text{t}}$$<\/p>\n\n\n\n<p>Generally, the no-arbitrage bounds of a call option&nbsp;are stated as follows:<\/p>\n\n\n\n<p>$$\\text{max}(0,S_t-X(1+r)^{-(T-t)}&lt;c_t\\leq\\text{S}_t)$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Put Options<\/h3>\n\n\n\n<p>A call option buyer exercises a put option only if \\(S_T&lt;X\\). As such, the upper bound on the put value is thus the exercise price.<\/p>\n\n\n\n<p>$$\\text{Upper bound}=X$$<\/p>\n\n\n\n<p>The lower bound is the present value of the exercise price minus the spot price or zero, whichever is greater:<\/p>\n\n\n\n<p>$$\\text{Lower bound}=\\text{max}(0,X(1+r)^{-(T-t)}-S_t)$$<\/p>\n\n\n\n<p>Generally, the no-arbitrage bounds of a put option are stated as follows:<\/p>\n\n\n\n<p>$$\\text{max}(0,X(1+r)^{-(T-t)}-S_t)&lt;p_t\\leq\\text{X}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Example: No-arbitrage Bounds of a Call Option<\/h3>\n\n\n\n<p>Consider a 3-year call option with an exercise price of USD 100 and a risk-free rate of 1.5%. If, after six months, the spot price of the underlying is USD 105, the no-arbitrage upper and lower bounds are <em>closest to<\/em>:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Solution<\/h4>\n\n\n\n<p>For a call option,<\/p>\n\n\n\n<p>$$\\begin{align*}\\text{Lower bound}&amp;=\\text{max}(0,S_t-X(1+r)^{-(T-t)})\\\\&amp;=\\text{max}(0,105-100(1.015)^{-2.5})\\\\&amp;=\\text{USD 8.65}\\\\ \\text{Upper bound}&amp;= S_t=\\text{USD 105}\\end{align*}$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Replication in Contingent Claims<\/h2>\n\n\n\n<p>Note that replication refers to a strategy in which a derivative\u2019s cash flow stream may be recreated using a combination of long or short positions in an underlying asset and borrowing or lending cash.<\/p>\n\n\n\n<p>Replication mirrors or offsets a derivative position, given that the law of one price holds and arbitrage does not exist. A trader can take opposing positions in a derivative and the underlying, creating a default risk-free hedge portfolio and replicating the payoff to a risk-free asset.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Replicating Call Options<\/h3>\n\n\n\n<p>Replication of a call option at the contract initiation involves borrowing at a risk-free rate, \\(r\\), and then utilizing the proceeds to buy the underlying asset at a price of \\(S_0\\).<\/p>\n\n\n\n<p>At the expiration date \\(t=T\\), there exist two replication outcomes:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If \\(S_T&lt;X\\),\u00a0exercise the option: Sell the underlying at \\(S_T\\)\u00a0\u00a0and use the proceeds to repay the risk-free loan.<\/li>\n\n\n\n<li>If \\(S_T&lt;X\\), no exercise: No settlement is needed.<\/li>\n<\/ul>\n\n\n\n<p>If the exercise of the option is certain, we will borrow \\(X(1+r)^{-T}\\) just like in forwards. However, the exercisability of the option is not certain. As such, a proportion of \\(X(1+r)^{-T}\\)&nbsp;is borrowed depending on the likelihood of exercise at time \\(T\\) and linked to the moneyness of an option.<\/p>\n\n\n\n<p>The <strong>non-linear nature of option payoff<\/strong> requires replicating transactions to be adjusted based on the likelihood of exercise.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Replicating Put Options<\/h3>\n\n\n\n<p>Replication of a put option at the contract initiation involves selling the underlying short at a price of \\(S_0\\)&nbsp;and lending the proceeds at the risk-free rate, \\(r\\).<\/p>\n\n\n\n<p>At the expiration date \\((t=T)\\), there exist two replication outcomes:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If \\(S_T&lt;X\\),\u00a0exercise the option: Buy the underlying at \\(S_T\\)\u00a0from the proceeds of the risk-free loan.<\/li>\n\n\n\n<li>If \\(S_T&lt;X\\),\u00a0no exercise: No settlement is needed.<\/li>\n<\/ul>\n\n\n\n<p>As with call options, a proportion of \\(X(1+r)^{-T}\\)&nbsp;is borrowed depending on the likelihood of exercise at time \\(T\\) and linked to the moneyness of an option.<\/p>\n\n\n<blockquote>\n<h2>Question<\/h2>\n<p>A 6-month put option on an underlying stock with no associated costs or benefits has an exercise price of $50. The underlying price at the contract inception is $47, and the risk-free rate is 1.5%. After three months, the underlying stock price is $45.75.<\/p>\n<p>The lower bound of the put option price is <em>closest<\/em> to:<\/p>\n<p>A. $4.06.<\/p>\n<p>B. $50.<\/p>\n<p>C. $45.75.<\/p>\n<h3>Solution<\/h3>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>The lower bound of a put option is given by:<\/p>\n<p>$$\\begin{align*}\\text{Lower bound}&amp;=\\text{max}(0,X(1+r)^{-(T-t)}-S_t)\\\\&amp;=\\text{max}(0,50(1.015)^{-(0.5-0.25)}-45.75)\\\\&amp;=\\text{max}(0,4.064)\\\\&amp;=$4.064\\end{align*}$$<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Recall that arbitrage opportunities occur if the law of one price does not hold. The no-arbitrage conditions in options are based on the payoff at maturity. Unlike forward commitments with symmetric profiles (as presented earlier), contingent claims have asymmetric payoff&#8230;<\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10],"tags":[48],"class_list":["post-43675","post","type-post","status-publish","format-standard","hentry","category-derivatives","tag-arbitrage-in-contingent-claims","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>Arbitrage in Contingent Claims | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn about arbitrage opportunities in contingent claims, including upper and lower bounds in options pricing and risk-free profit strategies.\" \/>\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\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Arbitrage in Contingent Claims | CFA Level 1\" \/>\n<meta property=\"og:description\" content=\"Learn about arbitrage opportunities in contingent claims, including upper and lower bounds in options pricing and risk-free profit strategies.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2022-12-23T12:29:44+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-05-20T09:27:10+00:00\" \/>\n<meta name=\"author\" content=\"Irene Rotich\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Irene Rotich\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/\"},\"author\":{\"name\":\"Irene Rotich\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/5b869539aaec1c35c9bd0229b3cab96a\"},\"headline\":\"Arbitrage in Contingent Claims\",\"datePublished\":\"2022-12-23T12:29:44+00:00\",\"dateModified\":\"2025-05-20T09:27:10+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/\"},\"wordCount\":896,\"keywords\":[\"Arbitrage in Contingent Claims\"],\"articleSection\":[\"Derivatives\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/\",\"name\":\"Arbitrage in Contingent Claims | CFA Level 1\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"datePublished\":\"2022-12-23T12:29:44+00:00\",\"dateModified\":\"2025-05-20T09:27:10+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/5b869539aaec1c35c9bd0229b3cab96a\"},\"description\":\"Learn about arbitrage opportunities in contingent claims, including upper and lower bounds in options pricing and risk-free profit strategies.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Arbitrage in Contingent Claims\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/5b869539aaec1c35c9bd0229b3cab96a\",\"name\":\"Irene Rotich\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/366a023b8445bac92d497d497e299b92b6cda999c48a7af5ac49daea5ddb7d0f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/366a023b8445bac92d497d497e299b92b6cda999c48a7af5ac49daea5ddb7d0f?s=96&d=mm&r=g\",\"caption\":\"Irene Rotich\"},\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/irene-rotichanalystprep-com\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Arbitrage in Contingent Claims | CFA Level 1","description":"Learn about arbitrage opportunities in contingent claims, including upper and lower bounds in options pricing and risk-free profit strategies.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/","og_locale":"en_US","og_type":"article","og_title":"Arbitrage in Contingent Claims | CFA Level 1","og_description":"Learn about arbitrage opportunities in contingent claims, including upper and lower bounds in options pricing and risk-free profit strategies.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2022-12-23T12:29:44+00:00","article_modified_time":"2025-05-20T09:27:10+00:00","author":"Irene Rotich","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Irene Rotich","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/"},"author":{"name":"Irene Rotich","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/5b869539aaec1c35c9bd0229b3cab96a"},"headline":"Arbitrage in Contingent Claims","datePublished":"2022-12-23T12:29:44+00:00","dateModified":"2025-05-20T09:27:10+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/"},"wordCount":896,"keywords":["Arbitrage in Contingent Claims"],"articleSection":["Derivatives"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/","name":"Arbitrage in Contingent Claims | CFA Level 1","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"datePublished":"2022-12-23T12:29:44+00:00","dateModified":"2025-05-20T09:27:10+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/5b869539aaec1c35c9bd0229b3cab96a"},"description":"Learn about arbitrage opportunities in contingent claims, including upper and lower bounds in options pricing and risk-free profit strategies.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/arbitrage-in-contingent-claims\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Arbitrage in Contingent Claims"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/5b869539aaec1c35c9bd0229b3cab96a","name":"Irene Rotich","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/366a023b8445bac92d497d497e299b92b6cda999c48a7af5ac49daea5ddb7d0f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/366a023b8445bac92d497d497e299b92b6cda999c48a7af5ac49daea5ddb7d0f?s=96&d=mm&r=g","caption":"Irene Rotich"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/irene-rotichanalystprep-com\/"}]}},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/43675","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=43675"}],"version-history":[{"count":13,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/43675\/revisions"}],"predecessor-version":[{"id":56166,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/43675\/revisions\/56166"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=43675"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=43675"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=43675"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}