{"id":18677,"date":"2021-07-28T21:25:59","date_gmt":"2021-07-28T21:25:59","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=18677"},"modified":"2025-12-30T17:02:05","modified_gmt":"2025-12-30T17:02:05","slug":"interpret-the-components-of-the-black-scholes-merton-model-as-applied-to-call-options-in-terms-of-a-leveraged-position-in-the-underlying","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/interpret-the-components-of-the-black-scholes-merton-model-as-applied-to-call-options-in-terms-of-a-leveraged-position-in-the-underlying\/","title":{"rendered":"Components of the BSM Model"},"content":{"rendered":"<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"What is the value of the replicating portfolio for a call option under the BSM model?\",\r\n    \"text\": \"Common stock is currently trading at $50. A call option is written on it with an exercise price of $45. Further, the continuously compounded risk-free rate of interest is 4%, the interest rate volatility is 30%, and the time to the option expiry is 2 years. Using the BSM model, the following components have been calculated:\\n\\nPV(K) = $41.54\\n\\nd1 = 0.6490\\n\\nd2 = 0.2248\\n\\nN(d1) = 0.7418\\n\\nN(d2) = 0.5889\\n\\nc0 = 12.63\\n\\nThe value of the replicating portfolio is closest to:\\n\\nA. $9.57.\\nB. $11.94.\\nC. $12.63.\",\r\n    \"answerCount\": 3,\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"A. $9.57.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"B. $11.94.\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"C. $12.63.\"\r\n      }\r\n    ],\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"C. $12.63.\",\r\n      \"commentary\": \"Under Black-Scholes-Merton, the call can be replicated by buying N(d1) shares and borrowing N(d2) zero-coupon bonds (i.e., taking a short position in the bond). Using the given values: nS = N(d1) = 0.7418 and nB = \u2212N(d2) = \u22120.5889, with bond price B = PV(K) = 41.54. The replicating portfolio cost is nS\u00b7S + nB\u00b7B = 0.7418(50) + (\u22120.5889)(41.54) = 37.09 \u2212 24.46 \u2248 12.63.\",\r\n      \"url\": \"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/interpret-the-components-of-the-black-scholes-merton-model-as-applied-to-call-options-in-terms-of-a-leveraged-position-in-the-underlying\/\"\r\n    },\r\n    \"author\": {\r\n      \"@type\": \"Organization\",\r\n      \"name\": \"AnalystPrep\"\r\n    }\r\n  }\r\n}\r\n<\/script>\r\n\r\n\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/rnMud0L9-g0\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>The BSM model for pricing options on a non-dividend-paying stock is given by:<\/p>\r\n<h4><strong>European Call<\/strong><\/h4>\r\n<p>$$ c_0= S_0N(d_1) \u2013 e^{(-rT)}KN(d_2) $$<\/p>\r\n<h4><strong>European Put<\/strong><\/h4>\r\n<p>$$ p_0=e^{-rT}KN\\left({-d}_2\\right)-S_0N\\left({-d}_1\\right) $$<\/p>\r\n<p>Where:<\/p>\r\n<p>$$ \\begin{align*} d_1 &amp;=\\frac{ln{\\left(\\frac{S}{K}\\right)}+\\left(r+\\frac{1}{2}\\sigma^2\\right)T}{\\sigma\\sqrt T} \\\\ d_2 &amp;=d_1-\\sigma\\sqrt T \\end{align*} $$<\/p>\r\n<p>\\(N(x)\\) = Standard normal cumulative distribution function.<\/p>\r\n<p>\\(N(\u2013x) = 1 \u2013 N(x)\\)<\/p>\r\n<p>BSM Model has the following variables:<\/p>\r\n<p>\\(T\\) = Time to option expiration.<\/p>\r\n<p>\\(r\\) = Continuously compounded risk-free rate.<\/p>\r\n<p>\\(S_0\\) = Current share price.<\/p>\r\n<p>\\(K\\) = Exercise price.<\/p>\r\n<p>\\(\\sigma\\) = Annual volatility of asset returns.<\/p>\r\n<h2>Interpretation of the BSM Model<\/h2>\r\n<p>The BSM model can be interpreted as the present value of the expected option payoff at expiration. It can be expressed as:<\/p>\r\n<p>$$ \\begin{align*} c_0 &amp;=PV(S_0e^{rT}N\\left(d_1\\right)\u2013 KN(d_2)) \\\\ p_0 &amp;=PV(KN\\left(-d_2\\right)-S_0e^{rT}N\\left(-d_1\\right)) \\end{align*} $$<\/p>\r\n<p>Where the present value factor, in this case, is \\(e^{-rT}\\).<\/p>\r\n<p>Alternatively, the BSM model can be described as having two components, a<em> <strong>stock component<\/strong><\/em>, and a <em><strong>bond component<\/strong><\/em>.<\/p>\r\n<p>The stock component for call options is \\(S_0N(d_1)\\) while the bond component is \\(e^{\u2013rT}KN(d_2)\\). Therefore, the BSM model call value is the difference between the stock component and the bond component.<\/p>\r\n<p>The stock component is \\((S_0N(d_1))\\) and the bond component is \\(e^{rT} KN(-d_2)\\) for put options. Therefore, the BSM model put value is the bond component minus the stock component.<\/p>\r\n<p>An option can be thought of as a dynamically managed portfolio of the underlying stock and zero-coupon bonds. The initial cost of this replicating strategy is given as:<\/p>\r\n<p>$$ \\text{Replicating strategy cost} = n_SS + n_BB $$<\/p>\r\n<h3>Call Options<\/h3>\r\n<p>The equivalent number of underlying shares is \\(n_S=N\\left(d_1\\right) &gt; 0\\). \\(n_S\\) greater than 0 implies that we are buying the stock. On the other hand, the equivalent number of bonds is \\(n_B=-N\\left(d_2\\right) &lt; 0\\). \\(n_B\\) less than 0 implies that we are selling the bond. Note that selling a bond is the same as borrowing money. Therefore, a call option can be viewed as a leveraged position in the stock where \\(N(d_1)\\) units of shares are purchased using \\(e^{\u2013rT}KN(d_2)\\) of borrowed money.<\/p>\r\n<h3>Put Options<\/h3>\r\n<p>The equivalent number of underlying shares is \\(n_S=-N\\left(-d_1\\right) &lt; 0.\\) This can be interpreted as selling the shares of the underlying stock as \\(n_S &lt; 0\\).<\/p>\r\n<p>Further, the equivalent number of bonds is \\(n_B=N\\left(-d_2\\right) &gt; 0\\). The bond is being bought here since \\(n_B\\) is greater than 0. Buying a bond is similar to lending money. Therefore, a put can be viewed as buying a bond where this purchase is partially financed by short selling the underlying stock.<\/p>\r\n<h4>Example: Interpreting BSM Model Components<\/h4>\r\n<p>Consider the following information relating to call and put options on an underlying stock<\/p>\r\n<p>\\(S_0\\) = 48<\/p>\r\n<p>\\(K\\) = 40<\/p>\r\n<p>\\(r\\) = 2.5% (Continuously compounded)<\/p>\r\n<p>\\(T\\) = 2<\/p>\r\n<p>\\(\\sigma\\) = 30%<\/p>\r\n<p>The current market price of call option = 14<\/p>\r\n<p>The current market price of put option = 3<\/p>\r\n<p>The following values have been calculated using the above information:<\/p>\r\n<p>$$ PV\\left(K\\right)={40\\times e}^{-0.025\\times2} = 38.05 $$<\/p>\r\n<p>\\(d_1\\) = 0.7597<\/p>\r\n<p>\\(d_2\\) = 0.3354<\/p>\r\n<p>\\(N\\left(d_1\\right)\\) = 0.7763<\/p>\r\n<p>\\(N\\left(d_2\\right)\\) = 0.6314<\/p>\r\n<p>We can determine the replicating strategy cost and arbitrage profits on both options as follows:<\/p>\r\n<p>According to the no-arbitrage approach to replicating the call option, a trader can purchase \\(n_S = N(d_1) = 0.7763\\) shares of stock by borrowing \\(n_B = \u2013N(d_2)= -0.6314\\) shares of zero-coupon bonds priced at \\(B = Ke^{\u2013rT} = $38.05\\) per bond.<\/p>\r\n<p>$$ \\begin{align*} \\text{Replicating strategy cost} &amp; = n_SS + n_BB \\\\ \\text{Replicating strategy cost} &amp; =0.7763\\times48+\\left(-0.6314\\times\\ 38.05\\right)=$13.24 \\end{align*} $$<\/p>\r\n<p>An arbitrage profit can be realized on the call option by writing a call at the current market price of $14 and purchasing a replicating portfolio for $13.24.<\/p>\r\n<p>Therefore,<\/p>\r\n<p>$$ \\text{Arbitrage profit} = $14-$13.24 = $0.76 $$<\/p>\r\n<p>For the put option, we have:<\/p>\r\n<p>$$ \\begin{align*} N(\u2013d_1) &amp; = 1 \u2013 N(d_1)= 1 \u2013 0.7763=0.2237 \\\\ N(\u2013d_2)&amp; = 1 \u2013 0.6314= 0.3686 \\end{align*} $$<\/p>\r\n<p>The no-arbitrage approach to replicating the put option involves:<\/p>\r\n<p>Purchasing \\(n_B =N(-d_2) =0.3686\\) shares of zero-coupon bonds priced at \\(40e^{-0.025\\times 2} =$38.05\\) per bond and short-selling \\(n_S=\u2013N(-d_1)= \u20130.2237\\) shares of stock resulting in short proceeds of \\($48\\times0.2237=$10.74\\).<\/p>\r\n<p>Therefore, the replicating strategy cost for the put option is:<\/p>\r\n<p>$$ \\text{Replicating strategy cost} =-0.2237\\times$48 + 0.3686\\times$38.05=$3.29 $$<\/p>\r\n<p>A trader can exploit arbitrage profits by selling the replicating portfolio and purchasing puts for an arbitrage profit of $0.29 per put.<\/p>\r\n<p>$$ \\text{Arbitrage profit} = $3.29-$3 = $0.29 $$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Common stock is currently trading at $50. A call option is written on it with an exercise price of $45. Further, the continuously compounded risk-free rate of interest is 4%, the interest rate volatility is 30%, and the time to the option expiry is 2 Years. Using the BSM model, the following components have been calculated:<\/p>\r\n<p>\\(PV(K)\\) = $41.54<\/p>\r\n<p>\\(d_1\\) = 0.6490<\/p>\r\n<p>\\(d_2\\) = 0.2248<\/p>\r\n<p>\\(N\\left(d_1\\right)\\) = 0.7418<\/p>\r\n<p>\\(N\\left(d_2\\right)\\) = 0.5889<\/p>\r\n<p>\\(c_0\\) = 12.63<\/p>\r\n<p>The value of the replicating portfolio is <em><strong>closest<\/strong> to<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>$9.57.<\/li>\r\n\t<li>$11.94.<\/li>\r\n\t<li>$12.63.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is C.<\/strong><\/p>\r\n<p>The no-arbitrage approach to replicating the call option involves purchasing \\(n_S = N\\left(d_1\\right) = 0.7418\\) shares of stock partially financed with \\(n_B = \u2013N(d_2) = \u20130.5889\\) shares of zero-coupon bonds priced at \\(B = Ke^{\u2013rT} = $41.54\\) per bond.<\/p>\r\n<p>$$ \\begin{align*} \\text{Cost of replicating portfolio} &amp; = n_SS + n_BB \\\\ c &amp;= 0.7418 \\left(50\\right)+ (\u20130.5889)41.54 =$12.63 \\end{align*} $$<\/p>\r\n<\/blockquote>\r\n<p>Reading 34: Valuation of Contingent Claims<\/p>\r\n<p><em>LOS 34 (g) Interpret the components of the Black\u2013Scholes\u2013Merton model as applied to call options in terms of a leveraged position in the underlying.<\/em><\/p>\r\n\r\n","protected":false},"excerpt":{"rendered":"<p>The BSM model for pricing options on a non-dividend-paying stock is given by: European Call $$ c_0= S_0N(d_1) \u2013 e^{(-rT)}KN(d_2) $$ European Put $$ p_0=e^{-rT}KN\\left({-d}_2\\right)-S_0N\\left({-d}_1\\right) $$ Where: $$ \\begin{align*} d_1 &amp;=\\frac{ln{\\left(\\frac{S}{K}\\right)}+\\left(r+\\frac{1}{2}\\sigma^2\\right)T}{\\sigma\\sqrt T} \\\\ d_2 &amp;=d_1-\\sigma\\sqrt T \\end{align*} $$ \\(N(x)\\) =&#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-18677","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 v26.9 - 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