{"id":18625,"date":"2021-07-28T13:50:37","date_gmt":"2021-07-28T13:50:37","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=18625"},"modified":"2024-04-07T05:24:23","modified_gmt":"2024-04-07T05:24:23","slug":"identify-an-arbitrage-opportunity-involving-options-and-describe-the-related-arbitrage","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/identify-an-arbitrage-opportunity-involving-options-and-describe-the-related-arbitrage\/","title":{"rendered":"Arbitrage Opportunities Involving Options"},"content":{"rendered":"\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/jmbJLCdBEgs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<h2>Call Option<\/h2>\r\n<p>A hedging portfolio can be created by going long \\(\\phi\\) units of the underlying asset and going short the call option such that the portfolio has an initial value of:<\/p>\r\n<p>$$ V_0=\\phi S_0-c_0 $$<\/p>\r\n<p>Where:<\/p>\r\n<p>\\(S_0\\) = The current stock price<\/p>\r\n<p>\\(c_0\\) = Current call value<\/p>\r\n<p>After a one-time-period, this portfolio will be worth:<\/p>\r\n<p>\\(V_1 =\\phi S_ou-c_u\\) if the asset price jumps up<\/p>\r\n<p>or<\/p>\r\n<p>\\(V_1 =\\phi S_od-c_d\\) if the asset price jumps down<\/p>\r\n<p>If we equate the values of the up and down portfolios, the number of units of the underlying asset can be obtained as:<\/p>\r\n<p>$$ \\phi=\\frac{c_u-c_d}{S_0u-S_0d} $$<\/p>\r\n<p>\\({\\phi}\\) is referred to as the <em><strong>hedge ratio<\/strong><\/em>. It is the ratio that makes a trader indifferent to the movement of the underlying asset price. An arbitrageur creates a hedged portfolio to eliminate price risk. This way, they satisfy <strong>Rule 2: \u201cDo not take any price risk.\u201d<\/strong><\/p>\r\n<p>Suppose at time step 0, a trader borrows the present value of:<\/p>\r\n<p>$$ -\\phi S_od+c_d $$<\/p>\r\n<p>Assuming there is no-arbitrage, we shall have:<\/p>\r\n<p>$$ c_0-\\phi S_0 = PV(\u2013\\phi S_0d + c_d ) $$<\/p>\r\n<p>Since \\(\u2013\\phi S_0d + c_d= \u2013\\phi S_0u+ c_u\\).<\/p>\r\n<p>This is can also be expressed as:<\/p>\r\n<p>$$ c_0\u2013\\phi S_0 = PV\u2013 \\phi S_0u + c_u $$<\/p>\r\n<p>The no-arbitrage single-period valuation approach leads to the following single-step call option valuation equation for call options:<\/p>\r\n<p>$$ \\begin{align*} c_0 &amp;=\\phi S_0 + PV(\u2013\\phi S_0d + c_d) \\\\ c_0 &amp; =\\phi S_0 + PV(\u2013\\phi S_0u + c_u) \\end{align*} $$<\/p>\r\n<p>Therefore, a call option is similar to owning \\(\\phi\\) units of the underlying asset and borrowing \\(PV(-\\phi S_od+c_d)\\). This makes the transaction completely arbitrage-free, hence satisfying <strong>Rule 1: \u201cDo not use own money.\u201d<\/strong> Moreover, we can view a call option as a leveraged position in the underlying asset.<\/p>\r\n<p>We can use the idea that a hedged portfolio returns the risk-free rate to determine the initial value of a call or a put option.<\/p>\r\n<h4>Example: Value of a Call Option<\/h4>\r\n<p>Consider a one-period binomial model of a non-dividend-paying stock whose current price is $20. Assume that:<\/p>\r\n<ul>\r\n\t<li>Over the single period under consideration, the stock price can either jump up to $25 or down to $16.<\/li>\r\n\t<li>The continuously compounded risk-free rate of return is 4% per period.<\/li>\r\n<\/ul>\r\n<p>The current value of a one-period European call option that has an exercise price of $20 is <em>closest to<\/em>:<\/p>\r\n<h4>Solution<\/h4>\r\n<p>The binomial tree in respect of the stock price is as follows:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26443\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Stock-Price.jpg\" alt=\"Example - Stock Price\" width=\"1549\" height=\"1131\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Stock-Price.jpg 1549w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Stock-Price-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Stock-Price-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Stock-Price-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Stock-Price-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Stock-Price-400x292.jpg 400w\" sizes=\"auto, (max-width: 1549px) 100vw, 1549px\" \/>Similarly, consider a corresponding binomial tree with respect to the payoff provided by the call option at time 1, i.e., the profit paid at exercise:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26444\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Call.jpg\" alt=\"Example - Payoffs (Call)\" width=\"1549\" height=\"1131\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Call.jpg 1549w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Call-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Call-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Call-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Call-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Call-400x292.jpg 400w\" sizes=\"auto, (max-width: 1549px) 100vw, 1549px\" \/>We can determine \\(c_0\\) by using the single-period call option valuation equation as follows:<\/p>\r\n<p>$$ \\begin{align*} c_0 &amp;=\\phi S_0 + PV(\u2013 \\phi S_d + c_d) \\\\ \\phi &amp;=\\frac{c_u-c_d}{S_0u-S_0d} \\\\ \\phi &amp;=\\frac{5-0}{25-16}=0.56 \\end{align*} $$<\/p>\r\n<p>Therefore,<\/p>\r\n<p>$$ c_0=0.56\\times20+e^{-0.04}\\left[-0.56\\times16+0\\right]=$2.59 $$<\/p>\r\n<p>This implies that buying a call option for $2.59 is equivalent to buying 0.56 units of the underlying stock for $11.20 and lending $8.61 such that the effective payment is $2.59.<\/p>\r\n<h2>Put Options<\/h2>\r\n<p>The no-arbitrage single period valuation equation for put options is expressed as:<\/p>\r\n<p>$$ p=\\phi S_0+PV\\left(-\\phi S_0d+p_d\\right) $$<\/p>\r\n<p>Equivalently,<\/p>\r\n<p>$$ p=\\phi S_0+PV\\left(-\\phi S_0u+p_u\\right) $$<\/p>\r\n<p>Where the hedge ratio, \\(\\phi\\), is given as:<\/p>\r\n<p>$$ \\phi=\\frac{p_u-p_d}{S_0u-S_0d} \\le 0 $$<\/p>\r\n<p>Note that the hedge ratio, in this case, will be negative as \\(p_u\\) is less than \\(p_d\\). Therefore, the arbitrageur should short-sell the underlying and lend a portion of the proceeds to replicate a long-put position.<\/p>\r\n<p>Therefore, a put option can be viewed as equivalent to shorting the underlying asset and lending \\(PV\u2013 \\phi S_0 u + p_u\\).<\/p>\r\n<h4>Example: Value of a Put Option<\/h4>\r\n<p>Consider a one-period binomial model of a non-dividend-paying stock whose current price is $20. Assume that:<\/p>\r\n<ul>\r\n\t<li>Over the single period under consideration, the stock price can either jump up to $25 or down to $16.<\/li>\r\n\t<li>The continuously compounded risk-free rate of return is 4% per period.<\/li>\r\n<\/ul>\r\n<p>The current value of a one-period European put option that has an exercise price of $20 is <em>closest to<\/em>:<\/p>\r\n<h4>Solution<\/h4>\r\n<p>The payoff provided by the put option at time one is represented in the following binomial tree:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26445\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Put.jpg\" alt=\"Example - Payoffs (Put)\" width=\"1549\" height=\"1131\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Put.jpg 1549w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Put-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Put-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Put-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Put-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-Payoffs-Put-400x292.jpg 400w\" sizes=\"auto, (max-width: 1549px) 100vw, 1549px\" \/>$$ p=\\phi S_0+PV\u2013 \\phi S_0 u + p_u $$<\/p>\r\n<p>Where:<\/p>\r\n<p>$$ \\begin{align*} \\phi &amp;=\\frac{p_u-p_d}{S_0u-S_0d}\\le0 \\\\ \\phi &amp;=\\frac{0-4}{25-16}=-0.44 \\\\ p &amp; =-0.44\\times20+e^{-0.04}\\left(&#8211;0.44\\times25+0\\right)=$1.77 \\end{align*} $$<\/p>\r\n<p>Notice that buying a put option for $1.77 is equivalent to short selling 0.44 units of the underlying stock for $8.80 and lending $10.57.<\/p>\r\n<h2>Exploiting Arbitrage Opportunities<\/h2>\r\n<p>Consider a one-period binomial model of a non-dividend-paying stock whose current price is $20. Imagine that:<\/p>\r\n<ul>\r\n\t<li>Over the single period under consideration, the stock price can either jump up to $25 or down to $16.<\/li>\r\n\t<li>The continuously compounded risk-free rate of return is 4% per period.<\/li>\r\n<\/ul>\r\n<p>In the previous section, we determined the current value of this call option as $2.59, given a strike price of $20.<\/p>\r\n<p>Now, assume that the call option has a market price of $4.50. Assuming that we trade 1,000 call options, we can illustrate how this opportunity can be exploited to earn an arbitrage profit.<\/p>\r\n<p>Since the call option is overpriced, we will sell 1,000 call options and buy several shares of the underlying determined by the hedge ratio.<\/p>\r\n<p>$$ \\begin{align*} \\phi &amp; =\\frac{c_u-c_d}{S_0u-S_0d} \\\\ \\phi &amp;=\\frac{$5-$0}{$25-$16}=\\frac{5}{9} \\text{ shares per option} \\end{align*} $$<\/p>\r\n<p>Therefore, we will purchase \\(1,000\\times\\frac{5}{9}=555.5555 \\text{ shares}\\).<\/p>\r\n<p>The net cost of a portfolio with 555.55 shares of the stock held long at $20 per share and 1,000 calls held short at $4.50 is:<\/p>\r\n<p>$$ \\text{Net cost of the portfolio}=\\left(555.55\\times$20\\right)-\\left(1,000\\times$4.50\\right)\\approx$6,611 $$<\/p>\r\n<ul>\r\n\t<li>Assume that we begin with $0.<\/li>\r\n\t<li>Then, we borrow $6,611 at 4%.<\/li>\r\n\t<li>At the end of the one-time period, we repay the loan of \\($6,611\\times1.04\\approx$6,875\\).<\/li>\r\n<\/ul>\r\n<p>The portfolio value will be the same at maturity regardless of whether the stock price moves up to $25 or down to $16.<\/p>\r\n<p>The value of the portfolio after stock price moves up is:<\/p>\r\n<p>$$ V_u = \\left(555.55 \\times $25\\right) \u2013 1,000 \\times $5 \\approx $8,889 $$<\/p>\r\n<p>The value of the portfolio after stock price moves down is:<\/p>\r\n<p>$$ V_d= \\left(555.55 \\times $16\\right)\u2013 1,000 \u00d7 $0\\approx $8,889 $$<\/p>\r\n<p>The arbitrage profit on this portfolio at the end of one year if the price moves up or down after repayment of the loan is $8,889 \u2013 $6,875 = $2,014.<\/p>\r\n<p>The discounted value of the arbitrage profit is therefore:<\/p>\r\n<p>$$ PV \\left(\\text{Arbitrage profit} \\right)=\\frac{$2,014}{1.04}=$1,936.54 $$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Consider a non-dividend-paying stock with a current price of $50 and an exercise price of $50. The stock price can be modeled by assuming that it will either increase by 12% or decrease by 10% each year, independent of the price movement in other years. A trader constructs a portfolio consisting of 100 call options. If a call option is overpriced, which portfolio <em>most likely <\/em>leads to an arbitrage profit?<\/p>\r\n<ol type=\"A\">\r\n\t<li>Buy 100 call options, short 54.55 shares.<\/li>\r\n\t<li>Buy 100 call options, short 45.45 shares.<\/li>\r\n\t<li>Sell 100 call options, buy 54.55 shares.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is C.<\/strong><\/p>\r\n<p>\\(u=1.12\\)<\/p>\r\n<p>\\(d=0.90\\)<\/p>\r\n<p>We can represent the above information in the following binomial tree:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26447\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-2-Stock-Price.jpg\" alt=\"Example 2 - Stock Price\" width=\"1549\" height=\"1131\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-2-Stock-Price.jpg 1549w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-2-Stock-Price-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-2-Stock-Price-1024x748.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-2-Stock-Price-768x561.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-2-Stock-Price-1536x1122.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Example-2-Stock-Price-400x292.jpg 400w\" sizes=\"auto, (max-width: 1549px) 100vw, 1549px\" \/>The call payoffs are as follows:<\/p>\r\n<p>$$ \\begin{align*} c_u &amp;=max{\\left($56-$50,0\\right)}=$6 \\\\ c_d &amp;=max{\\left($45-50,0\\right)}=$0 \\end{align*} $$<\/p>\r\n<p>Since the option is overpriced, the trader will sell 100 call options and purchase several shares determined by the hedge ratio:<\/p>\r\n<p>$$ \\begin{align*} \\phi &amp;=\\frac{c_u-c_d}{S_0u-S_0d} \\\\ \\phi &amp;=\\frac{$6-$0}{$56-$45}=0.5455 \\text{ shares per option} \\end{align*} $$<\/p>\r\n<p>$$ \\text{Total number of shares to purchase} = 100 \\times 0.5455 = 54.55 $$<\/p>\r\n<p>Buying 54.55 shares of stock will produce a riskless hedge. The payoff at expiry will return more than the risk-free rate on the hedge portfolio\u2019s net cost. Borrowing to finance the hedge portfolio and earning a higher rate than the borrowing rate produces riskless profits.<\/p>\r\n<\/blockquote>\r\n<p>Reading 34: Valuation of Contingent Claims<\/p>\r\n<p><em>LOS 34 (c) Identify an arbitrage opportunity involving options and describe the related arbitrage.<\/em><\/p>\r\n\r\n","protected":false},"excerpt":{"rendered":"<p>Call Option A hedging portfolio can be created by going long \\(\\phi\\) units of the underlying asset and going short the call option such that the portfolio has an initial value of: $$ V_0=\\phi S_0-c_0 $$ Where: \\(S_0\\) = The&#8230;<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,302],"tags":[216,304],"class_list":["post-18625","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-cfa-level-2","tag-derivatives","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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