{"id":43907,"date":"2023-01-02T07:29:28","date_gmt":"2023-01-02T07:29:28","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=43907"},"modified":"2026-02-17T18:44:18","modified_gmt":"2026-02-17T18:44:18","slug":"put-call-forward-parity","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/put-call-forward-parity\/","title":{"rendered":"Put-Call Forward 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 CFA\u00ae Level I Derivatives lesson (Module 9) explains option replication using put\u2013call parity. The video covers the put\u2013call parity formula, the construction of protective puts and fiduciary calls, and the creation of synthetic calls and puts using no-arbitrage relationships. It also extends parity to put\u2013call forward parity and connects option theory to firm value by explaining equity as a call option and credit risk through the put premium framework. The lesson focuses on exam-relevant intuition, replication logic, and arbitrage-free pricing relationships.\",\n  \"uploadDate\": \"2024-04-05\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/Dvq2OTfQLSM\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=Dvq2OTfQLSM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/Dvq2OTfQLSM\",\n  \"duration\": \"PT38M46S\"\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\": \"Which of the following best describes the replication of a risk-free bond under the put-call parity?\",\n    \"text\": \"Which of the following best describes the replication of a risk-free bond under the put-call parity?\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. A risk-free bond can be replicated by holding a long position in the underlying asset, a short position in a call option, and a long position in a put option. Under put-call parity, the relationship implies that this combination reproduces the payoff of a risk-free bond.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/Dvq2OTfQLSM?si=5L-0UI1qIaBukpQW\" 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>The put-call forward parity extends the put-call parity to include the forward contracts. To get the put-call forward parity, we substitute the present value of the forward price, \\(F_0(T)\\), for the underlying price:<br>$$F_0(T)\\left(1+r\\right)^{-T}+p_0=c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Deriving Put-Call Forward Parity<\/h2>\n\n\n\n<p>Consider an investor whose main objective is to benefit from an increase in underlying value and hedge against a decrease in underlying value. Consider the following portfolios:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Portfolio A<\/h4>\n\n\n\n<p>At time \\(t=0\\), an investor buys a forward contract and a risk-free bond whose face value is equal to the forward price \\(F_0\\left(T\\right)\\). The investor then puts an option on the underlying at a price of \\(p_0\\) whose exercise price is \\(X\\) at \\(t=T\\). The cost of this strategy is:<\/p>\n\n\n\n<p>$$F_0(T)\\left(1+r\\right)^{-T}+p_0$$<\/p>\n\n\n\n<div style=\"margin:18px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" \n     target=\"_blank\" \n     rel=\"noopener noreferrer\"\n     style=\"\n        display:block;\n        text-align:center;\n        padding:14px 18px;\n        border:2px solid #2F5BFF;\n        border-radius:18px;\n        color:#2F5BFF;\n        font-weight:600;\n        font-size:16px;\n        text-decoration:none;\n        background-color:#f9faff;\n     \">\n     Practice CFA questions on put-call forward parity.\n  <\/a>\n<\/div>\n\n\n\n<h4>Portfolio B<\/h4>\n<p>At time \\(t=0\\), an investor buys a call option at a price of \\(c_0\\) on the same underlying exercise price of \\(X\\) and a risk-free bond redeemable at a price of \\(X\\) at time \\(t=T\\). The cost of this transaction is:<\/p>\n<p>$$c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n<p>Portfolio A is called <strong>synthetic protective put<\/strong><em>. <\/em>Compared to a synthetic put, a synthetic protective put replaces the underlying cash position with a synthetic position using forward purchase and a risk-free bond.<\/p>\n<p>Portfolio B is the same fiduciary call as in the put-call parity seen previously.<\/p>\n<p>Cash flows at time \\(t=T\\) for the synthetic protective put and the fiduciary call are shown in the following table:<\/p>\n<p>$$<br \/>\\begin{array}{l|c|c|c}<br \/>\u00a0\\textbf { Portfolio Position } &amp; \\begin{array}{l}<br \/>\\textbf { Put exercised } \\\\<br \/>\\boldsymbol{S}_{\\boldsymbol{T}}&lt;\\boldsymbol{X}<br \/>\\end{array} &amp; \\begin{array}{l}<br \/>\\textbf { No Exercise } \\\\<br \/>\\boldsymbol{S}_{\\boldsymbol{T}}=\\boldsymbol{X}<br \/>\\end{array} &amp; \\begin{array}{l}<br \/>\\textbf { Call Exercised } \\\\<br \/>\\boldsymbol{S}_{\\boldsymbol{T}}&gt;\\boldsymbol{X}<br \/>\\end{array} \\\\<br \/>\\hline \\textbf { Fiduciary Call: } \\\\<br \/>\\hline \\begin{array}{l}<br \/>\\text { Purchased Call } \\\\<br \/>\\text { Option }<br \/>\\end{array} &amp; 0 &amp; 0 &amp; S_T-X \\\\<br \/>\\hline \\text { Risk-free Asset } &amp; X &amp; X &amp; X \\\\<br \/>\\hline \\text { Total: } &amp; X &amp; X\\left(=S_T\\right) &amp; S_T \\\\<br \/>\\hline \\textbf { Synthetic Protective Put: } \\\\<br \/>\\hline \\text { Purchased Put at } p_0 &amp; X-S_T &amp; 0 &amp; 0 \\\\<br \/>\\hline \\begin{array}{l}<br \/>\\text { Purchased Forward } \\\\<br \/>\\text { Contract }<br \/>\\end{array} &amp; S_T-F_0(T) &amp; S_T-F_0(T) &amp; S_T-F_0(T) \\\\<br \/>\\hline \\begin{array}{l}<br \/>\\text { Risk-free bond are } \\\\<br \/>\\text { currently priced as } \\\\<br \/>F_0(T)(1+r)^{-T}<br \/>\\end{array} &amp; F_0(T) &amp; F_0(T) &amp; F_0(T) \\\\<br \/>\\hline \\text { Total: } &amp; X &amp; S_T(=X) &amp; S_T \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>Since portfolios A and B have identical payoffs at time \\(t=T\\), the costs of these portfolios must be identical at time \\(t=0\\). Therefore, based on no-arbitrage conditions, the put-call forward parity is given by:<\/p>\n<p>$$F_0(T)\\left(1+r\\right)^{-T}+p_0=c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n<h4><strong>Example: Put-Call Forward Parity<\/strong><\/h4>\n<p>Capital Investments would like to buy a 6-month put option on a company\u2019s shares, whose current price is $195 per share. The exercise price of the put options is $190.00 per share.<\/p>\n<p>The 6-month call option on the same shares trades at $64 per share with the same exercise price of $190.00. Using the put-call forward parity and assuming a 1.5% risk-free rate, the price of the put option is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Using the put-call forward parity<\/p>\n<p>$$F_0(T)\\left(1+r\\right)^{-T}+p_0=c_0+X\\left(1+r\\right)^{-T}$$<\/p>\n<p>Making \\(p_0\\) the subject of the formula, we get:<\/p>\n<p>$$p_0=c_0+\\left(X-F_0\\left(T\\right)\\right)^{-T}$$<\/p>\n<p>We need to solve for \\(F_0\\left(T\\right)\\), which, if you recall, is given by:<\/p>\n<p>$$\\begin{align}F_0\\left(T\\right)&amp;=S_0\\left(1+r\\right)^T\\\\&amp;=195\\left(1.015\\right)^{0.5}\\\\&amp;=\\$196.4571\\end{align}$$<\/p>\n<p>As such,<\/p>\n<p>$$\\begin{align}p_0&amp;=c_0+{\\left(X-F_0\\left(T\\right)\\right)(1.015)}^{-T}\\\\&amp;=64+{\\left(190-196.4571\\right)(1.015)}^{-0.5}\\\\&amp;=\\$57.59\\end{align}$$<\/p>\n<h2>Option Put-Call Parity Applications: Firm Value<\/h2>\n<p>The put-call parity relationship can be used to define a firm&#8217;s value based on equity holders&#8217; and debt holders&#8217; interests.<\/p>\n<p>As a rule of thumb, at time \\(t=0\\), a company&#8217;s market value, \\(V_0\\), is equivalent to the present value of its outstanding debt obligations, \\(PV(D)\\), and equity, \\(E_0\\), where the borrowed funds are in zero-coupon debt with a face value of \\(D\\).<\/p>\n<p>In an equation, we can express this relationship as:<\/p>\n<p>$$V_0=E_0+PV(D)$$<\/p>\n<p>In the event of debt maturity at \\(t=T\\), the assets and debts of the company will be split between debtholders and shareholders, with two possible outcomes based on the company&#8217;s value at that given time:<\/p>\n<h4><strong>Solvency (\\(V_T&gt;D\\))<br \/><\/strong><\/h4>\n<p>Recall that solvency refers to a company&#8217;s ability to meet its financial obligations and long-term debt. If at time \\(T\\), a firm\u2019s value (\\(V_T\\)) is greater than the face value of the debt, ( \\(\\mathbf{V}_\\mathbf{T}&gt;\\mathbf{D})\\) the firm is solvent, and thus able to return capital to both the shareholders and the debtholders.<\/p>\n<p>Debt holders come first when distributing capital returns. As such, they receive the debt repayments (D) in full. On the other hand, the shareholders receive what remains. That is \\(E_T=V_T-D\\).<\/p>\n<p>In summary, we&#8217;ve established that shareholders benefit if a company can meet its debt obligation and maintain solvency. On the other hand, debt holders benefit when a company is solvent and hence meets its debt obligations.<\/p>\n<h4><strong>Ins<\/strong><strong style=\"color: initial;\">olvency (\\(V_T&lt;D\\))<\/strong><\/h4>\n<p>Insolvency refers to a company&#8217;s inability to meet its financial obligations and long-term debts. This occurs if, at debt maturity (\\(T\\)), a company&#8217;s value is less than the debt&#8217;s face value, \\(V_T&lt;D\\).<\/p>\n<p>When a firm is insolvent, the shareholders receive the residual, which is equal to zero \\(\\left(E_T=0\\right)\\), and the debtholders are owed more than the firm\u2019s total assets. As such, the debtholders receive \\(V_T&lt;D\\) to cover the debt of \\(D\\) at time \\(T\\).<strong style=\"color: initial;\"><br \/><\/strong><\/p>\n<h3>Payoff in Terms of Options<\/h3>\n<p>Note that shareholders retain the unlimited upside potential in solvency and limited downside potential in insolvency. On the other hand, the debtholders are limited to receiving debt repayment in the case of solvency and principal and interest in the case of insolvency.<\/p>\n<p>Intuitively, the payoff profiles can be mathematically represented as follows:<\/p>\n<ul>\n<li>At time \\(t=T\\), the payoff of the shareholders can be expressed as \\(max(0,D-V_T)\\).<\/li>\n<li>On the other hand, the debt holder&#8217;s payoff is expressed as \\(min(V_T,D)\\).<\/li>\n<\/ul>\n<p>In terms of options, the payoff profiles can be expressed as follows:<\/p>\n<ul>\n<li><strong>Shareholders\u2019 payoff<\/strong><b>:<\/b> They hold a long position in the underlying firm\u2019s assets value. For instance, assume that we have bought a put option on the firm value, \\(V_{T}\\), with the exercise price of \\(D\\). The payoff is \\(max(0, D-V_T)\\).<\/li>\n<li><strong>Debtholders\u2019 payoff:<\/strong> The hold a long position in the risk-free bond, valued at \\(D\\), and have sold a put option to the shareholders on the firm value, \\(V_T\\), with an exercise price of \\(D\\).<\/li>\n<\/ul>\n<p>Remember the put-call parity relationship:<\/p>\n<p>$$ S_0+p_0=c_0+PV\\left(X\\right)$$<\/p>\n<p>If we replace the underlying asset, \\(\\left(S_0\\right)\\), for the company&#8217;s value at time 0, \\(\\left(V_0\\right)\\), and further replace the risk-free bond, \\(\\left(X\\right)\\), with debt, \\(\\left(D\\right)), the equation becomes:<\/p>\n<p>$$V_0+p_0=c_0+PV(D)$$<\/p>\n<p>We can also rearrange the formula to solve for the value of the company, \\(\\left(V_0\\right)\\):<\/p>\n<p>$$V_0=c_0+PV\\left(D\\right)-p_0$$<\/p>\n<p>From the above results, the shareholders have a payoff equivalent to that of a call option (\\(c_0\\)) on the firm\u2019s value. On the other hand, the debtholders hold a position of \\(\\left(D\\right)-p_0\\), which is the risk-free debt plus a short position in a put option.<\/p>\n<p>This put option may be seen as a <strong>credit spread<\/strong> on a company&#8217;s debt or the premium above the risk-free rate a company must pay debtholders to bear insolvency risk. The value of the put option to shareholders rises as the probability of insolvency grows.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Which of the following <em>best <\/em>describes the replication of a risk-free bond under the put-call parity?<\/p>\n<p>A. Long underlying, short call option, and long put option.<\/p>\n<p>B. Long underlying, short risk-free bond, and long put option.<\/p>\n<p>C. Short underlying, long risk-free bond, and long call option<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is<strong> A.<\/strong><\/p>\n<p>Recall that the put-call parity relationship may be expressed as:<\/p>\n<p>$$\\begin{align}c_0+X\\left(1+r\\right)^{-T}&amp;=p_0+s_0\\\\\\Rightarrow X\\left(1+r\\right)^{-T}&amp;=p_0+s_0-c_0\\end{align}$$<\/p>\n<p>The risk-free bond replicating individual positions under put call parity is a long underlying, short call option and long put option.<\/p>\n<p><strong>B is incorrect<\/strong>: Call option individual replication position equals long underlying, short risk-free bond, and long put option.<\/p>\n<p>$$c_0=p_0+s_0-X\\left(1+r\\right)^{-T}$$<\/p>\n<p><strong>C is incorrect<\/strong>: Put option position equals short underlying, long risk-free bond, and long call option.<\/p>\n<p>$$p_0=c_0+X\\left(1+r\\right)^{-T}-s_0$$<\/p>\n<\/blockquote>\n\n\n<p><\/p>\n\n\n\n<div style=\"text-align:center; margin:40px 0 20px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" \n     target=\"_blank\" \n     rel=\"noopener noreferrer\"\n     style=\"\n        display:inline-block;\n        background:#3E73D9;\n        color:#ffffff;\n        padding:16px 34px;\n        border-radius:50px;\n        font-weight:700;\n        font-size:18px;\n        text-decoration:none;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"\n        margin-top:18px;\n        font-size:16px;\n        line-height:1.6;\n        max-width:700px;\n        margin-left:auto;\n        margin-right:auto;\n        color:#333333;\n     \">\n     Strengthen your CFA Level I Derivatives preparation with exam-style questions, QBank drills, and full mock exams designed to help you master option pricing relationships and forward valuation concepts.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>The put-call forward parity extends the put-call parity to include the forward contracts. To get the put-call forward parity, we substitute the present value of the forward price, \\(F_0(T)\\), for the underlying price:$$F_0(T)\\left(1+r\\right)^{-T}+p_0=c_0+X\\left(1+r\\right)^{-T}$$ Deriving Put-Call Forward 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-43907","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 Forward Parity Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Put-call forward parity extends traditional put-call parity by incorporating forward contracts, linking options 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