{"id":43902,"date":"2023-01-02T02:17:19","date_gmt":"2023-01-02T02:17:19","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=43902"},"modified":"2026-03-27T08:40:53","modified_gmt":"2026-03-27T08:40:53","slug":"put-call-parity","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/put-call-parity\/","title":{"rendered":"Put-Call Parity"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Option Replication Using Put-Call Parity (2025 Level I CFA\u00ae Exam \u2013 Derivatives \u2013 Module 9)\",\n  \"description\": \"This lesson explains option replication using put-call parity for the 2025 CFA\u00ae Level I Derivatives curriculum. It introduces the put-call parity relationship, compares protective puts and fiduciary calls, and shows how to construct synthetic calls and puts using no-arbitrage principles. The video also extends the framework to put-call forward parity and connects option parity to firm value concepts, including equity as a call option and credit risk reflected through put option premiums.\",\n  \"uploadDate\": \"2024-04-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/Dvq2OTfQLSM\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/Dvq2OTfQLSM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/Dvq2OTfQLSM\",\n  \"duration\": \"PT38M14S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e.png\",\n  \"caption\": \"Illustration related to CFA Level I quantitative analysis\",\n  \"width\": 974,\n  \"height\": 641,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\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\": \"European put and call options have an exercise price of $50 and expire in four months. The underlying asset is priced at $52 and makes no cash payments during the option\u2019s life. The risk-free rate is 4.5%, and the put is selling for $3.80. According to the put-call parity, the price of the call option should be closest to:\",\n    \"text\": \"European put and call options have an exercise price of $50 and expire in four months. The underlying asset is priced at $52 and makes no cash payments during the option\u2019s life. The risk-free rate is 4.5%, and the put is selling for $3.80. According to the put-call parity, the price of the call option should be closest to:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. Using put-call parity for a non-dividend-paying asset: S0 + p0 = c0 + X(1 + r)^(-T). Solving for the call price gives c0 = S0 + p0 - X(1 + r)^(-T) = 52 + 3.80 - 50(1.045)^(-0.25) \u2248 6.35.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/Dvq2OTfQLSM?si=BhPbkcNOxOn_xCDE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n<p>Put-call parity is a no-arbitrage concept. It involves a combination of cash and derivative instruments in a portfolio. Put-call parity allows pricing and valuation of these positions without directly modeling them using non-arbitrage conditions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Deriving Put-Call Parity<\/h2>\n\n\n\n<p>Consider an investor whose main objective is to benefit from an increase in the underlying value and hedge an investment against a decrease in underlying value.<\/p>\n\n\n\n<p>Consider the following portfolios:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Portfolio A<\/strong><\/h4>\n\n\n\n<p>At time \\(t=0\\), an investor buys a call option at a price of \\(c_0\\) on an underlying with an exercise price of \\(X\\) and a risk-free bond that is redeemable at \\(X\\) at time \\(t=T\\). Intuitively, assuming the call option expires at time t=T, the cost of this strategy is<\/p>\n\n\n\n<p>$$c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n\n\n\n<p>In this portfolio, the investor buys a call option with a positive payoff if the underlying price exceeds the exercise price \\((S_T&gt;X)\\) and invests cash in a risk-free bond. This strategy is called the <strong>fiduciary call<\/strong>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Portfolio B<\/strong><\/h4>\n\n\n\n<p>At time \\(t=0\\), an investor buys an underlying at a price of \\(S_0\\) and a put option on the underlying price of \\(p_0\\) whose exercise price is \\(X\\) at time \\(t=T\\). Intuitively, the cost of this strategy is<\/p>\n\n\n\n<p>$$p_0+S_0$$<\/p>\n\n\n\n<div style=\"text-align:center; margin: 25px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" 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;\">\n    Apply put-call parity with a free trial\n  <\/a>\n<\/div>\n\n\n<p>The strategy applied in portfolio B is called <strong>protective put<\/strong>. Protective put involves holding an asset and buying a put on the same asset.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/57e.png\" alt=\"protective-puts-and-fiduciary-calls\" \/><\/p>\n<p>Both portfolios allow the investor to benefit from the rise in underlying price without exposure to a decrease below the exercise price. Moreover, portfolios A and B have identical profiles. Based on the no-arbitrage condition, assets with similar future payoff profiles must trade at the same price, ignoring associated transaction costs.<\/p>\n<p>Consider the following table:<\/p>\n<p>$$<br \/>\\begin{array}{l|c|c|c}<br \/>\\textbf { Portfolio Position } &amp; \\begin{array}{l}<br \/>\\textbf { Put exercised } \\\\<br \/>\\boldsymbol{S}_{\\boldsymbol{T}}&lt;\\boldsymbol{X}<br \/>\\end{array} &amp; \\begin{array}{l}<br \/>\\textbf { No Exercise } \\\\<br \/>\\boldsymbol{S}_{\\boldsymbol{T}}=\\boldsymbol{X}<br \/>\\end{array} &amp; \\begin{array}{l}<br \/>\\textbf { Call Exercised } \\\\<br \/>\\boldsymbol{S}_{\\boldsymbol{T}}&gt;\\boldsymbol{X}<br \/>\\end{array} \\\\<br \/>\\hline \\textbf { Fiduciary Call: } &amp; \\\\<br \/>\\hline \\text { Call Option } &amp; 0 &amp; 0 &amp; S_T-X \\\\<br \/>\\hline \\text { Risk-free Asset } &amp; X &amp; X &amp; X \\\\<br \/>\\hline \\text { Total: } &amp; \\boldsymbol{X} &amp; \\boldsymbol{X}\\left(=\\boldsymbol{S}_{\\boldsymbol{T}}\\right) &amp; \\boldsymbol{S}_{\\boldsymbol{T}} \\\\<br \/>\\hline \\textbf { Protective Put: } \\\\<br \/>\\hline \\text { Underlying Asset } &amp; S_T &amp; S_T &amp; S_T \\\\<br \/>\\hline \\text { Put option } &amp; X-S_T &amp; 0 &amp; 0 \\\\<br \/>\\hline \\text { Total: } &amp; \\boldsymbol{X} &amp; \\boldsymbol{S}_{\\boldsymbol{T}}(=\\boldsymbol{X}) &amp; \\boldsymbol{S}_{\\boldsymbol{T}} \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>Therefore, since portfolios A and B have identical payoffs at time \\(t=T\\), the costs of these portfolios must be similar at time \\(t=0\\). For this reason, the put-call parity equation:<\/p>\n<p>$$S_0+p_0=c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n<p>Where:<\/p>\n<p>\\(S_0 =\\) Price of the underlying asset.<\/p>\n<p>\\(p_0=\\) Put premium.<\/p>\n<p>\\(c_0=\\) Call option premium.<\/p>\n<p>\\(X=\\) Exercise price.<\/p>\n<p>\\(r=\\) Risk-free rate.<\/p>\n<p>\\(T=\\) Time to expiration.<\/p>\n<p>Put-call parity holds for European options that have similar exercise prices and expiration times. These similarities ensure a no-arbitrage relationship between the put option, call option, the underlying asset, and risk-free asset prices. Put-call parity implies that at time \\(t=0\\), the price of the long underlying asset plus the long put must be equal to the price of the long call option plus the risk-free asset.<\/p>\n<h4><strong>Example: Put-Call Parity<\/strong><\/h4>\n<p>Consider European put and call options, where both have an exercise price of $50 and expire in 3 months. The underlying asset is priced at $52 and makes no cash payments during the life of the options.<\/p>\n<p>If the put is selling for $3.80 and the risk-free rate is 4.5%, the price of the call option is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The put-call parity is given by:<\/p>\n<p>$$S_0+p_0=c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n<p>We need to rearrange the formula to make \u00a0the subject of the formula so that:<\/p>\n<p>$$\\begin{align}c_0 &amp;=S_0+p_0-X\\left(1+r\\right)^{-T}\\\\&amp;=52+3.80-50\\left(1.045\\right)^{-0.25}\\\\&amp;=$6.35\\end{align}$$<\/p>\n<h2>Option Replication Using Put-Call Parity<\/h2>\n<p>We can rearrange the put-call parity equation to solve for the put option premium, \\(p_0\\):<\/p>\n<p>$$p_0=c_0+X\\left(1+r\\right)^{-T}-S_0$$<\/p>\n<p>The right side of this equation is referred to as a <strong>synthetic put<\/strong>. It consists of a long call, a short position in the underlying, and a long position in the risk-free bond.<\/p>\n<p>We can make another re-arrangement to solve for a long call, \\(c_0\\):<\/p>\n<p>$$c_0=p_0+S_0-X\\left(1+r\\right)^{-T}$$<\/p>\n<p>The right side of this equation is equivalent to a call option and is referred to as a <strong>synthetic call<\/strong>. It consists of a long put, a long position in the underlying asset, and a short position in the risk-free bond.<\/p>\n<p>Also, we can further rearrange the put-call parity as follows:<\/p>\n<p>$$S_0-c_0=X\\left(1+r\\right)^{-T}-p_0$$<\/p>\n<p>The right-hand side of the above equation is called the <strong>covered call position<\/strong>. Intuitively, a covered call is equivalent to a long risk-free bond and short put option.<\/p>\n<p>In summary, synthetic relationships with options occur by replicating a one-part position under put-call parity. Study the following table.<\/p>\n<p>$$<br \/>\\begin{array}{l|c|c|c|c}<br \/>\\textbf { Position } &amp; \\begin{array}{c}<br \/>\\textbf { Underlying } \\\\<br \/>\\left(\\boldsymbol{S}_{\\mathbf{0}}\\right)<br \/>\\end{array} &amp; \\begin{array}{c}<br \/>\\textbf { Risk-free Bond } \\\\<br \/>\\left((\\mathbf{1}+\\boldsymbol{r})^{-T}\\right)<br \/>\\end{array} &amp; \\begin{array}{c}<br \/>\\textbf { Call Option } \\\\<br \/>\\left(\\boldsymbol{c}_{\\mathbf{0}}\\right)<br \/>\\end{array}\u00a0&amp; \\begin{array}{c}<br \/>\\textbf { Put Option } \\\\<br \/>\\left(\\boldsymbol{p}_{\\mathbf{0}}\\right)<br \/>\\end{array} \\\\<br \/>\\hline \\text { Underlying }\\left(\\boldsymbol{S}_{\\mathbf{0}}\\right) &amp; &#8211; &amp; \\text { Long } &amp; \\text { Long } &amp; \\text { Short } \\\\<br \/>\\hline \\begin{array}{c}<br \/>\\text { Risk-free bond } \\\\<br \/>\\left(\\frac{x}{(1+r)^{\\mathrm{T}}}\\right)<br \/>\\end{array} &amp; \\text { Long } &amp; &#8211; &amp; \\text { Short } &amp; \\text { Long } \\\\<br \/>\\hline \\text { Call option }\\left(\\boldsymbol{c}_{\\mathbf{0}}\\right) &amp; \\text { Long } &amp; \\text { Short } &amp; &#8211; &amp; \\text { Long } \\\\<br \/>\\hline \\text { Put Option }\\left(\\boldsymbol{p}_{\\mathbf{0}}\\right) &amp; \\text { Short } &amp; \\text { Long } &amp; \\text { Long } &amp; &#8211; \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>If the put-call parity does not hold, an arbitrage opportunity exists. The arbitrage opportunity can be exploited by selling the most expensive portfolio and purchasing the cheaper one.<\/p>\n<p><strong>Example<\/strong>:<strong> Arbitrage Opportunity<\/strong><\/p>\n<p>A European call option with a strike price of $25 sells at $7. The price of a European put option with the same strike price is also $7. If the underlying stock sells for $28, and the one-year risk-free rate is 4%, determine if there is an arbitrage opportunity.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The put-call parity equation:<\/p>\n<p>$$\\begin{align}p_0+s_0 &amp; \u225f c_0+X(1+r)^{-T}\\\\7+28&amp;\u225f7+25(1.04)^{-1}\\\\35 &amp; \\neq31.0385\\end{align}$$<\/p>\n<p>To exploit the opportunity, we need to:<\/p>\n<ul>\n<li>Sell the right side (<strong>Protective put<\/strong>) for $35.<\/li>\n<li>Buy the left side (<strong>fiduciary call<\/strong>) for $31.0385.<\/li>\n<\/ul>\n<p>We get a cash inflow of \\($35-$31.0385=$3.9615\\). Thus, the strategy provides cash inflow ($3.9615) today and no cash outflow at expiration.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>European put and call options have an exercise price of $50 and expire in four months. The underlying asset is priced at $52 and makes no cash payments during the option\u2019s life. The risk-free rate is 4.5%, and the put is selling for $3.80. According to the put-call parity, the price of the call option should be\u00a0<em>closest<\/em>\u00a0to:<\/p>\n<p>A. $5.25.<\/p>\n<p>B. $6.35.<\/p>\n<p>C. $7.12.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is<strong> B<\/strong>.<\/p>\n<p>The put-call parity is given by:<\/p>\n<p>$$S_0+p_0=c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n<p>Where:<\/p>\n<p>\\(S_0\\) =Price of the underlying asset.<\/p>\n<p>\\(p_0\\) = Put premium.<\/p>\n<p>\\(c_0\\)= Call option premium.<\/p>\n<p>\\(X\\) = Exercise price.<\/p>\n<p>\\(r\\)= Risk-free rate.<\/p>\n<p>\\(T\\)= Time to expiration.<\/p>\n<p>Making \\(c_0\\) the subject, we have:<\/p>\n<p>$$\\begin{align}c_0&amp;=S_0+p_0-X\\left(1+r\\right)^{-T}\\\\&amp;=52+3.80-50\\left(1.045\\right)^{-0.25}\\\\&amp;=6.35\\end{align}$$<\/p>\n<\/blockquote>\n\n\n<div style=\"text-align:center; margin: 40px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" 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;\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"font-size:15px; margin-top:12px; color:#555;\">\n    Practice put-call parity, synthetic positions, and arbitrage questions for CFA Level I with AnalystPrep.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Put-call parity is a no-arbitrage concept. It involves a combination of cash and derivative instruments in a portfolio. Put-call parity allows pricing and valuation of these positions without directly modeling them using non-arbitrage conditions. Deriving Put-Call Parity Consider an investor&#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-43902","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>Put-Call Parity Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Put-call parity ensures no-arbitrage pricing by linking call and put options with the underlying asset and risk-free bonds. 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