{"id":43913,"date":"2023-01-02T08:37:07","date_gmt":"2023-01-02T08:37:07","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=43913"},"modified":"2026-01-09T08:30:54","modified_gmt":"2026-01-09T08:30:54","slug":"one-period-binomial-model-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/derivatives\/one-period-binomial-model-2\/","title":{"rendered":"One-Period Binomial Model"},"content":{"rendered":"\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\/2023\/01\/Img_1-4.jpg\",\n  \"caption\": \"Image showing One-Period Binomial Model\",\n  \"width\": 1590,\n  \"height\": 1047,\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\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5.jpg\",\n  \"caption\": \"Image Showing Call Option Pricing Using a One-Period Binomial Model at t=0\",\n  \"width\": 1590,\n  \"height\": 1014,\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\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3.jpg\",\n  \"caption\": \"Image Showing Call Option Pricing Using a One-Period Binomial Model at t=1\",\n  \"width\": 1644,\n  \"height\": 1162,\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\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4-1024x641.jpg\",\n  \"caption\": \"Image Showing how to determine the Value of c0 using a Hedge Ratio\",\n  \"width\": 1024,\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\": \"Consider a one-year put option on a non-dividend paying stock with an exercise price of $50. The current stock price is $45. The stock price is expected to go up or down by 20%. Calculate the non-arbitrage price of the put option if the risk-free rate of return is 4%.\",\n    \"text\": \"Consider a one-year put option on a non-dividend paying stock with an exercise price of $50. The current stock price is $45. The stock price is expected to go up or down by 20%. Calculate the non-arbitrage price of the put option if the risk-free rate of return is 4%.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The non-arbitrage price of the put option is $5.38. Using a one-period binomial model, the stock price can rise to $54 or fall to $36. The corresponding put payoffs are $0 and $14, respectively. The hedge ratio is calculated using the difference in payoffs divided by the difference in stock prices. The replicating portfolio value at the end of the period is $42, which is discounted back at the risk-free rate to $40.38. Subtracting the stock position from the portfolio value gives a put option price of $5.38.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p>The law of arbitrage dictates that the value of any two assets (or portfolio of assets) whose payoffs are identical in all possible future scenarios at a given time must also be identical today.<\/p>\n\n\n\n<p>Unlike forward commitments that offer symmetric payoffs at a pre-determined price in the future, contingent claims offer asymmetric payoffs. For this reason, their valuation is a challenge. The binomial model can be used to model the payoffs of contingent claims.<\/p>\n\n\n\n<div style=\"margin: 0 0 20px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      border: 2px solid #1e63ff;\n      color: #1e63ff;\n      background: #ffffff;\n      padding: 10px 14px;\n      border-radius: 10px;\n      font-weight: 500;\n      line-height: 1.35;\n      text-decoration: none;\n    \"\n  >\n    Want to practice binomial tree logic and option valuation step by step? Try AnalystPrep\u2019s free trial now.\n  <\/a>\n<\/div>\n\n\n\n<h2><strong>One-Period Binomial Model<\/strong><\/h2>\n<h2><!-- \/wp:post-content --> <!-- wp:paragraph --><\/h2>\n<p><!-- \/wp:paragraph --><\/p>\n<p>The idea behind the binomial model is that at maturity, an asset&#8217;s spot price, \\(S_0\\), can either increase to \\(S_1^u\\) or decrease to \\(S_1^d\\). We do not need to know the asset&#8217;s future price in advance since it depends on a random variable&#8217;s outcome. The asset price movements, from \\(S_0\\) to either \\(S_1^u\\) or \\(S_1^d\\), can be seen as the outcome of a Bernoulli trial.<\/p>\n<p>Let us denote \\(q\\) as the probability of an increase in the asset&#8217;s price. Because of the presence of only two possibilities, i.e., an increase or a decrease in the asset&#8217;s price, we can denote the probability of a decrease in its price as \\(1-q\\), such that the two probabilities will add up to 1.<\/p>\n<p>The gross return when the asset price increases will be:<\/p>\n<p>$$R^u=\\frac{S_1^u}{S_0}&gt;1$$<\/p>\n<p>When the asset price decreases, the gross return will be:<\/p>\n<p>$$R^d=\\frac{S_1^d}{S_0}&lt;1$$<\/p>\n<p>The difference between \\(S_1^u\\) (or \\(R^uS_o)\\) and \\(S_1^d\\) (or \\(R^dS_0\\)) is the spread of possible future price outcomes.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-43999 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_1-4.jpg\" alt=\"Image showing One-Period Binomial Model\" width=\"1590\" height=\"1047\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_1-4.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_1-4-300x198.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_1-4-1024x674.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_1-4-768x506.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_1-4-1536x1011.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_1-4-400x263.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<h3><strong>Pricing a European Call Option using a One-Period Binomial Model<\/strong><\/h3>\n<p>Consider a one-year call option with an underlying price of \\(S_0\\) and an exercise price of \\(X\\). Also, assume that \\(S_1^d&lt;X&lt;S_1^u\\) and that the one-period binomial model is equivalent to the time of expiration of one year.<\/p>\n<p>The one-period binomial model gives the underlying asset values in one year, where the option value is defined as a function of the underlying value.<\/p>\n<h4><strong>At time <em>t=0<\/em>:<\/strong><\/h4>\n<p>The value of the call option is \\(c_0\\). Note that this value is unknown and needs to be calculated.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-44000 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5.jpg\" alt=\"Image Showing Call Option Pricing Using a One-Period Binomial Model at t=0\" width=\"1590\" height=\"1014\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5-300x191.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5-1024x653.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5-768x490.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5-1536x980.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_2-5-400x255.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<h4><strong>At time <em>t=1<\/em>:<\/strong><\/h4>\n<p>After one year, the option expires. At this time, the value of the option will either be \\(c_1^u=\\max(0,S_1^u-X)\\) if the underlying price rises to \\(S_1^u\\) or \\(c_1^d=\\max(0,S_1^u-X)\\) if the underlying price falls to \\(S_1^d\\).<\/p>\n<p>Intuitively, for the up movement, the call option is in the money, and for the down direction, the option is out of the money.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-44001 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3.jpg\" alt=\"Image Showing Call Option Pricing Using a One-Period Binomial Model at t=1\" width=\"1644\" height=\"1162\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3.jpg 1644w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3-300x212.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3-1024x724.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3-768x543.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3-1536x1086.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_3-3-400x283.jpg 400w\" sizes=\"auto, (max-width: 1644px) 100vw, 1644px\" \/><\/p>\n<h4><strong>Determining the Value of <em>c<sub>0<\/sub><\/em><\/strong><\/h4>\n<p>The value of \\(c_0\\) is determined by applying replication and no-arbitrage pricing. Replication implies that the value of the option and its underlying asset in any future scenario may be used to construct a risk-free portfolio.<\/p>\n<p>With that in mind, assume at time \\(t=0\\), we sell a call option for a price of \\(c_0\\) and buy \\(h\\) units of the underlying asset. Also, let the value of the portfolio be \\(V\\) so that its value at \\(t=0\\) is:<\/p>\n<p>$$V_0={\\rm hS}_0-c_0$$<\/p>\n<p>The value of the portfolio, if the underlying price moves up, is given by:<\/p>\n<p>$$V_1^u={\\rm hS}_1^u-c_1^u=h\\times R^u\\times S_0-\\max{\\left(0,S_1^u-X\\right)}$$<\/p>\n<p>And for the down movement:<\/p>\n<p>$$V_1^d={\\rm hS}_1^d-c_1^d=h\\times R^d\\times S_0-\\max{\\left(0,S_1^d-X\\right)}$$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-44002 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4.jpg\" alt=\"Image Showing how to determine the Value of c0 using a Hedge Ratio\" width=\"1590\" height=\"996\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4-300x188.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4-1024x641.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4-768x481.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4-1536x962.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_4-400x251.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<p>Assuming no-arbitrage condition, note that we have established two portfolios with identical payoff profiles at time \\(t=1\\). As such, we need to find the value of \\(h\\) such that:<\/p>\n<p>$$V_1^u=V_1^d$$<\/p>\n<p>Therefore,<\/p>\n<p>$$\\Rightarrow{hS}_1^u-c_1^u={hS}_1^d-c_1^d$$<\/p>\n<p>Making \\(h\\) the subject of the formula:<\/p>\n<p>$$h=\\frac{c_1^u-c_1^d}{S_1^u-S_1^d}$$<\/p>\n<p>The value \\(h\\) is called the hedge ratio. The hedge ratio is a proportion of the underlying that will offset the risk associated with an option.<\/p>\n<p>Since \\(V_1^u=V_1^d\\), we can draw two conclusions:<\/p>\n<ul>\n<li>We can utilize either of the two portfolios to value the option.<\/li>\n<li>The return \\(\\frac{V_1^u}{V_0}=\\frac{V_1^d}{V_0}=1+r\\).<\/li>\n<\/ul>\n<p>To avoid arbitrage, the portfolio value at \\(t=1\\), (\\(V_1={\\ V}_1^u=V_1^d\\)), must be discounted using a risk-free rate so that:<\/p>\n<p>$$V_0=V_1\\left(1+r\\right)^{-1}$$<\/p>\n<p>However, recall that \\(V_0={hS}_0-c_0\\):<\/p>\n<p>$$\\Rightarrow{\\rm hS}_0-c_0=V_1\\left(1+r\\right)^{-1}$$<\/p>\n<p>Making \\(c_0\\) the subject of the formula:<\/p>\n<p>$$c_0={\\rm hS}_0-V_1\\left(1+r\\right)^{-1}$$<\/p>\n<h4><strong>Example<\/strong>:<strong> Pricing Call Option Using One Period Binomial Model<\/strong><\/h4>\n<p>A European call option that expires in one year has an exercise price of $70 and an underlying spot price of $60. Use a one-period binomial model to estimate the call option price if the underlying&#8217;s spot price is expected to change by 25% in one year. Assume that the risk-free annual rate is 5%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>Step 1: Determine the Call Option&#8217;s Value at Maturity <\/strong><strong><em>t=1<\/em><\/strong><\/p>\n<p>At maturity, the value can either be \\(c_1^u\\) if the price of the underlying goes up or \\(c_1^d\\) if the price of the underlying goes down.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>The spot price at maturity, if the price goes up by 25%, will be:<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$S_1^u=\\frac{125}{100}\\times60=75$$<\/p>\n<p>and the gross return will be:<\/p>\n<p>$$R^u=\\frac{S_1^u}{S_o}=\\frac{75}{60}=1.25$$<\/p>\n<p>If the price goes down by 25%, the spot price at maturity will be:<\/p>\n<p>$$S_1^d=\\frac{75}{100}\\times60=45$$<\/p>\n<p>and the gross return will be:<\/p>\n<p>$$R_d=\\frac{S_1^d}{S_o}=\\frac{45}{60}=0.75$$<\/p>\n<p>If the underlying&#8217;s price goes up (the call option expires in the money):<\/p>\n<p>$$c_1^u=\\max\\ \\left(0,\\ S_1^u-X\\right)=\\max\\ \\left(0,\\ 75-70\\right)=5$$<\/p>\n<p>If the underlying&#8217;s price goes down, and the call option expires out of the money:<\/p>\n<p>$$c_1^d=\\max\\ \\left(0,\\ S_1^d-X\\right)=\\max\\ \\left(0,\\ 45-70\\right)=0$$<\/p>\n<p><strong>Step 2: Determining <em>h<\/em><\/strong>, <strong>the Hedge Ratio<\/strong><\/p>\n<p>The hedge ratio is given by<\/p>\n<p>$$\\begin{align} h&amp;=\\frac{c_1^u-c_1^d}{S_1^u-S_1^d}\\\\&amp;=\\frac{5-0}{75-45}=0.167\\end{align}$$<\/p>\n<p>The hedge ratio of 0.167 implies that we either need to buy 0.167 units of the underlying for every call option we sell or sell 6 call options for each underlying asset to equate the portfolio values at maturity (\\(t=1\\)). Therefore, the portfolio values when the price of the underlying increases and decreases respectively is:<\/p>\n<p>$$\\begin{align}V_1^u&amp;=\\left(0.167\\times75\\right)-5=7.5\\\\V_1^d&amp;=\\left(0.167\\times45\\right)-0=7.5\\end{align}$$<\/p>\n<p>The portfolio values are the same, implying that either of the portfolios can be used to value the derivative.<\/p>\n<p><strong>Step 4<\/strong>:<strong> Determining \\(\\mathbf{V_0}\\)<\/strong><\/p>\n<p>We can obtain \\(V_0\\) by discounting \\(V_1\\) at the risk-free discount rate. Remember that \\(V_1=V_1^u=V_1^d\\):<\/p>\n<p>$$V_o=7.5(1+{0.05)}^{-1}=7.14$$<\/p>\n<p><strong>Step 5<\/strong>:<strong> Determining \\(\\mathbf{c_0}\\)<\/strong><\/p>\n<p>Recall that:<\/p>\n<p>$$c_0={\\rm hS}_0-V_1\\left(1+r\\right)^{-1}=hS_o-V_0$$<\/p>\n<p>Therefore,<\/p>\n<p>$$c_o=hS_o-V_0=\\left(0.167\\times60\\right)-7.14=2.88$$<\/p>\n<p><em><strong>Note<\/strong>:<\/em> The hedging approach can be used to value many derivatives, not just call options, provided the derivative&#8217;s value depends on the underlying asset&#8217;s value at contract maturity, i.e., \\(t=1\\)<\/p>\n<h2><strong>Pricing a European Put Option One-period Binomial Model<\/strong><\/h2>\n<p>For put options, the same explanations we gave under the call option apply, albeit with a different replication strategy. Consider the following diagram:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1644\" height=\"1162\" class=\"alignnone wp-image-44003 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_5.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_5.jpg 1644w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_5-300x212.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_5-1024x724.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_5-768x543.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_5-1536x1086.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_5-400x283.jpg 400w\" sizes=\"auto, (max-width: 1644px) 100vw, 1644px\" \/><\/p>\n<p>Under the put option, the hedge ratio is given by<\/p>\n<p>$$h=\\frac{p_1^u-p_1^d}{S_1^d-S_1^u}$$<\/p>\n<p>Note that the formula remains the same as in the call option, except for the change of notations. Replication in pricing put option using a one-period binomial model involves buying the put option and \\(h\\) units of the underlying so that:<\/p>\n<p>$$V_0=p_0+hS_0$$<\/p>\n<p>Therefore,<\/p>\n<p>$$p_0=V_0-hS_0$$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Consider a one-year put option on a non-dividend paying stock with an exercise price of $50. The current stock price is $45. The stock price is expected to go up or down by 20%. Calculate the non-arbitrage price of the put option if the risk-free rate of return is 4%.<\/p>\n<p>A. 0<\/p>\n<p>B. $5.38<\/p>\n<p>C. $14.00<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is\u00a0<strong>B.<\/strong><\/p>\n<p>Consider the following diagram:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1590\" height=\"834\" class=\"alignnone wp-image-44004 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_6-1.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_6-1.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_6-1-300x157.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_6-1-1024x537.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_6-1-768x403.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_6-1-1536x806.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/01\/Img_6-1-400x210.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<p>From the above results, the hedge ratio is given by:<\/p>\n<p>$$ h=\\frac{p_1^u-p_1^d}{S_1^u-S_1^d}=\\frac{0-14}{54-36}=-0.7778 $$<\/p>\n<p>We need to calculate \\(V_1={\\ V}_1^u=V_1^d\\), which are:<\/p>\n<p>$$ \\begin{align}V_1^u&amp;={\\rm hS}_1^u+p_1^u=0.7778\\times54+0=\\$42.00\\\\V_1^d&amp;={\\rm hS}_1^d+p_1^d=0.7778\\times36+14=\\$42.00\\end{align} $$<\/p>\n<p>Next, we can either use \\(V_1^u\\) or \\(V_1^d\\) to calculate the value of \\(V_0\\):<\/p>\n<p>$$ V_0=V_1\\left(1+r\\right)^{-1}=42\\left(1.04\\right)^{-1}=\\$40.38 $$<\/p>\n<p>Therefore, the put option value is:<\/p>\n<p>$$ p_0=V_0-{\\rm hS}_0=40.38-0.7778\\times45=\\$5.38 $$<\/p>\n<\/blockquote>\n\n<!-- wp:html -->\n<div style=\"text-align: center; margin: 32px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      background-color: #1e63ff;\n      color: #ffffff;\n      padding: 12px 26px;\n      border-radius: 12px;\n      font-weight: 600;\n      font-size: 16px;\n      text-decoration: none;\n    \"\n  >\n    Start Free Trial \u2192\n  <\/a>\n\n  <div style=\"margin-top: 10px; font-size: 14px; color: #374151;\">\n    Practice binomial option pricing, risk-neutral valuation, and call\/put payoffs with full CFA\u00ae-style solutions.\n  <\/div>\n<\/div>\n\n<!-- \/wp:html -->","protected":false},"excerpt":{"rendered":"<p>The law of arbitrage dictates that the value of any two assets (or portfolio of assets) whose payoffs are identical in all possible future scenarios at a given time must also be identical today. 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