{"id":14968,"date":"2021-05-07T11:12:07","date_gmt":"2021-05-07T11:12:07","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=14968"},"modified":"2026-07-06T20:37:01","modified_gmt":"2026-07-06T20:37:01","slug":"valuation-of-equities-and-currencies-using-the-black-scholes-model","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/valuation-of-equities-and-currencies-using-the-black-scholes-model\/","title":{"rendered":"Valuation of Equities and Currencies using the Black-Scholes Model"},"content":{"rendered":"\r\n<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\": \"For an Australian importer buying an at-the-money spot pound call option, what underlying price, risk-free rate, and carry rate should be used in the BSM model?\",\r\n    \"text\": \"An Australian importer has to pay fixed pound amounts every three months for goods. The spot price of the currency pair is 1.78 A$\/\u00a3. If the exchange rate rises to 1.80 A$\/\u00a3, the Australian dollar will have weakened because it takes more Australian dollars to buy one pound. The importer believes that the Australian dollar will depreciate in the following months and decides to buy an at-the-money spot pound call option. The risk-free Australian rate is 3.00%, and the British risk-free rate is 2.00%. The underlying price, risk-free rate, and carry rate to use in the BSM model to value the pound call option are most likely: A. 1.78, 2.00%, 3.00%. B. 0.56, 2.00%, 3.00%. C. 1.78, 3.00%, 2.00%.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is C. The underlying price is 1.78 A$\/\u00a3 because the exchange rate is quoted as the value of the domestic currency per unit of foreign currency. Since the importer is Australian, the domestic risk-free rate is the Australian risk-free rate of 3.00%. The carry rate is the foreign risk-free rate, which is the British risk-free rate of 2.00%. Therefore, the inputs to use in the BSM model are an underlying price of 1.78, a risk-free rate of 3.00%, and a carry rate of 2.00%.\"\r\n    }\r\n  }\r\n}\r\n<\/script>\r\n\r\n<p>Some underlying instruments have carry benefits. These benefits include dividends for stock options, foreign interest rates for currency options, and coupon payments for bond options.<\/p>\r\n<p>The BSM model should be adjusted to incorporate carry benefits in the option value. Let the carry benefit be a continuous yield, . The carry adjusted BSM model is expressed as:<\/p>\r\n<p>European call:\\(c_{0}=S_{0}e^{-\\gamma T}N(d_{1})-e^{rT}KN(d_{2})\\)<\/p>\r\n<p>European put: \\(p_{0}=e^{rT}KN(-d_{2})-S_{0}e^{-\\gamma T}N(-d_{1})\\)<\/p>\r\n<p>Where:\u00a0<\/p>\r\n<p>$$d_{1}=\\frac{ln\\bigg(\\frac{S_{0}}{K}\\bigg)+\\bigg(r-\\gamma+\\frac{\\sigma^{2}}{2}\\bigg)T}{\\sigma\\sqrt{T}}$$<\/p>\r\n<p>and<\/p>\r\n<p>$$d_{2}=d_{1}-\\sigma\\sqrt{T}$$<\/p>\r\n<p>It is worth noting that carry benefits lower the expected future value of the underlying. Further, an increase in carry benefits lowers the value of a call option and raises the put option\u2019s value.<\/p>\r\n<div style=\"margin: 18px 0;\">\r\n  <a style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #2F5BFF; border-radius: 18px; color: #ffffff; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">\r\n    Master CFA Level 2 Black-Scholes valuation concepts with AnalystPrep\u2019s Free Trial.\r\n  <\/a>\r\n<\/div>\r\n\r\n<h2>BSM Valuation for Equities<\/h2>\r\n<p>Assume that the underlying equity has a continuously compounded dividend yield \\(\\gamma=\\delta\\). The BSM model can be adjusted for dividends as follows:\u00a0<\/p>\r\n<p>European call: \\(C_{0}=S_{0}e^{-\\delta T}N(d_{1})-e^{-rT}KN(d_{2})\\)<\/p>\r\n<p>European put: \\(p_{0}=e^{-rT}KN(-d_{2})-S_{0}e^{-\\delta T}N(-d_{1})\\)<\/p>\r\n<p>Where:\u00a0<\/p>\r\n<p>$$d_{1}=\\frac{ln\\bigg(\\frac{S_{0}}{K}\\bigg)+\\bigg(r-\\delta+\\frac{\\sigma^{2}}{2}\\bigg)T}{\\sigma\\sqrt{T}}$$<\/p>\r\n<p>and<\/p>\r\n<p>$$d_{2}=d_{1}-\\sigma\\sqrt{T}$$<\/p>\r\n<p>The arbitrageur of a dividend-paying stock receives dividend payments when long the stock and pays dividends when short the stock. Dividends reduce the number of shares to buy for calls and the number of shares to short-sell for puts. The higher the dividends, the lower the value of \\(d_{1}\\) and hence the lower the value of \\(N(d_{1})\\).<\/p>\r\n<h3>Example: Valuing Stock Options using the BSM Model<\/h3>\r\n<p>Consider a stock that is trading on the London Stock Exchange at \u00a350. A trader believes that the stock price will rise in the next month and decides to buy one-month call options with an exercise price of \u00a353. The risk-free annual rate of interest is 2%, and the yield on the stock is \u00a30.35%. The volatility of the stock is 20%.<\/p>\r\n<p>The BSM model inputs are as follows:<\/p>\r\n<ul>\r\n<li>The spot price of the underlying = \u00a350<\/li>\r\n<li>Exercise price = \u00a353<\/li>\r\n<li>Expiration = 1 month<\/li>\r\n<li>Risk-free rate = 2%<\/li>\r\n<li>Dividend yield = 0.35%<\/li>\r\n<li>Volatility = 0.20<\/li>\r\n<\/ul>\r\n<h2>BSM Valuation of Currencies<\/h2>\r\n<p>The BSM model can also be used to value foreign exchange options. They carry benefit for a foreign exchange option is the continuously compounded foreign risk-free interest rate.<\/p>\r\n<p>The values of\u00a0 European call and put options are determined using the following formulas:<\/p>\r\n<p>European call: \\(c_{0}=S_{0}e^{-r^{f}{T}}N(d_{1})-e^{-rT}KN(d_{2})\\)<\/p>\r\n<p>European put: \\(p_{0}=e^{-rT}KN(-d_{2})-S_{0}e^{-r^{f}{T}}N(-d_{1})\\)<\/p>\r\n<p>Where:\u00a0<\/p>\r\n<p>$$d_{1}=ln\\bigg(\\frac{S_{0}}{K}\\bigg)+\\frac{\\bigg(r-r^{f}+\\frac{\\sigma^{2}}{2}\\bigg)T}{\\sigma\\sqrt{T}}$$<\/p>\r\n<p>and<\/p>\r\n<p>$$d_{2}=d_{1}=\\sigma\\sqrt{T}$$<\/p>\r\n<p>Note that:<\/p>\r\n<p>r=domestic risk-free rate<\/p>\r\n<p>$$r^{f}=\\text{Foreign risk}-\\text{free rate}$$<\/p>\r\n<h3>Example: BSM Model Applied to Value Options on Currency<\/h3>\r\n<p>A swiss exporter will receive Euros for his watches. The exporter purchases a three-month put option with an exercise price \\(K=1.07CHF\/EUR\\) to protect against a decrease in the EUR exchange rate. The current exchange rate is \\(1.08CHF\/EUR\\)<\/p>\r\n<p>The BSM model inputs for this currency option are as follows:<\/p>\r\n<ul>\r\n<li>The underlying, \\(S_{0}\\) the value of the domestic currency per unit of the foreign currency) \\(=1.08CHF\/EUR\\)<\/li>\r\n<li>The annualized swiss risk-free rate = r<\/li>\r\n<li>The EUR rate (Carry rate)\\(=r^{f}\\)<\/li>\r\n<li>Time to expiration = 0.25 years (three months)<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h2 class=\"wp-block-heading\">Question<\/h2>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\">An Australian importer has to pay fixed pound (\u00a3) amounts every three months for goods. The spot price of the currency pair is 1.78A$ \/\u00a3. If the exchange rate rises to say 1.80 A$\/\u00a3, then the Aussie will have weakened as it will take more Aussie dollars to buy one pound. The importer believes that the Australian dollar will depreciate in the following months. Thus, he decides to buy an at-the-money spot pound call option to protect against this weakening. The risk-free Australian rate is 3.00%, and the British risk-free rate is 2.00%.<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\">The underlying price, the risk-free rate, and the carry rate to use in the BSM model to get the pound call option value is most likely:<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\">          A. 1.78, 2.00%, 3.00%.<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\">          B. 0.56, 2.00%, 3.00%.<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\">          C. 1.78, 3.00%, 2.00%.<\/p>\r\n\r\n\r\n\r\n<h3 class=\"wp-block-heading\">Solution<\/h3>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\"><strong>The correct answer is C:<\/strong><\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\">The underlying, \\(S_{0}\\) the value of the domestic currency per unit of the foreign currency) = 1.78A$ \/\u00a3. The risk-free rate is the Australian \u00a0rate, 3.00%, and the carry rate is the British rate of 2.00%<\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\"><em>Reading 38:<\/em> <em>Valuation of Contingent Claims<\/em><\/p>\r\n\r\n\r\n\r\n<p class=\"wp-block-paragraph\"><em>LOS 38 (h): Describe how the Black\u2013Scholes\u2013Merton model is used to value European options on equities and currencies;<\/em><\/p>\r\n\r\n\r\n\r\n<div style=\"text-align: center; margin: 30px 0;\">\r\n  <a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">\r\n    Start Free Trial \u2192\r\n  <\/a>\r\n  <p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">\r\n    Build confidence in CFA Level 2 derivatives with study notes, practice questions, mock exams, and video lessons covering Black-Scholes valuation, equities, currencies, and exam-style applications.\r\n  <\/p>\r\n<\/div>\r\n","protected":false},"excerpt":{"rendered":"<p>Some underlying instruments have carry benefits. These benefits include dividends for stock options, foreign interest rates for currency options, and coupon payments for bond options. The BSM model should be adjusted to incorporate carry benefits in the option value. Let&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,302],"tags":[216,304,333],"class_list":["post-14968","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-cfa-level-2","tag-derivatives","tag-valuation-of-equities-and-currencies-using-the-black-scholes-model","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Black-Scholes Valuation | CFA Level 2<\/title>\n<meta name=\"description\" content=\"Learn how the Black-Scholes model values equity and currency options, including the effects of dividends and exchange rates. 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