{"id":5859,"date":"2019-02-06T10:07:37","date_gmt":"2019-02-06T10:07:37","guid":{"rendered":"https:\/\/www.wasteyourtime.co\/?p=5859"},"modified":"2020-07-07T12:21:04","modified_gmt":"2020-07-07T11:21:04","slug":"american-option-pricing-and-early-exercise","status":"publish","type":"post","link":"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/","title":{"rendered":"American option pricing and early exercise"},"content":{"rendered":"<h1><\/h1>\n<p style=\"text-align: right;\"><i><span style=\"font-weight: 400;\">3 min read<\/span><\/i><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5863 size-full\" src=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4.jpg\" alt=\"American option pricing and early exercise 3\" width=\"800\" height=\"536\" srcset=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4.jpg 800w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4-300x201.jpg 300w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4-768x515.jpg 768w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4-370x248.jpg 370w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4-213x144.jpg 213w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4-760x509.jpg 760w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-4-424x284.jpg 424w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><br \/>\nAmerican option pricing\u00a0is the binomial options pricing model that provides a generalizable numerical method for the valuation of options.<\/p>\n<p><span style=\"font-weight: 400;\">American options are contracts that may be exercised early, prior to expiry.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">These options are contrasted with European options for which exercise is only permitted at expiry. Most traded stock and futures options are &#8216;American style&#8217;, while most index options are European.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The exercise style of listed options is American by default. Except for options on equity market indexes such as the S&amp;P 500 index.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Options on futures are typically American as well.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The Black-Scholes pricing formulas are not applicable to American option pricing.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Being an algorithm, binomial option pricing models, nevertheless, can be modified to take care of the added complication in the American option.<\/span><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_74 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#Lets_see_the_differences_between_these_two_styles_European_vs_American_options\" >Let&#8217;s see the differences between these two styles: European vs American options<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#European-style\" >European-style<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#American-style\" >American-style<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#American_options_can_be_exercised_early\" >American options can be exercised early<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#By_exercising_the_option_holder_may_forego_the_time_value\" >By exercising, the option holder may forego the time value.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#The_advantage_of_the_American_style\" >The advantage of the American style<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#But_lets_make_American_option_pricing_simpler\" >But, let&#8217;s make\u00a0American option pricing simpler<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/traders-paradise.com\/magazine\/2019\/02\/american-option-pricing-and-early-exercise\/#The_bottom_line\" >The bottom line<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h3><span class=\"ez-toc-section\" id=\"Lets_see_the_differences_between_these_two_styles_European_vs_American_options\"><\/span><b>Let&#8217;s see the differences between these two styles: European vs American options<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h4><span class=\"ez-toc-section\" id=\"European-style\"><\/span><b><i>European-style<\/i><\/b><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><span style=\"font-weight: 400;\">The seller sells the (call) option to allow the buyer to buy the underlying at the price of K on the expiration date only.<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5860 size-full\" src=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise.jpg\" alt=\"American option pricing and early exercise\" width=\"800\" height=\"450\" srcset=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise.jpg 800w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-300x169.jpg 300w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-768x432.jpg 768w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-370x208.jpg 370w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-760x428.jpg 760w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-424x239.jpg 424w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/span><\/p>\n<h4><span class=\"ez-toc-section\" id=\"American-style\"><\/span><b><i>American-style<\/i><\/b><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><span style=\"font-weight: 400;\">The seller sells the (call) option to allow the buyer to buy the underlying at the price of K and another option to buy at any time no later than the expiration date.<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5861 size-full\" src=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-2.jpg\" alt=\"American option pricing and early exercise 1\" width=\"800\" height=\"450\" srcset=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-2.jpg 800w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-2-300x169.jpg 300w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-2-768x432.jpg 768w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-2-370x208.jpg 370w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-2-760x428.jpg 760w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-2-424x239.jpg 424w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400;\">In these graphs, you can see the main difference. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">The key difference between American and European options relates to when the options can be exercised: A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time. An American option, on the other hand, may be exercised at any time before the expiration date.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"American_options_can_be_exercised_early\"><\/span><b>American options can be exercised early<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Unlike a European option, the holder of an American option can exercise the option before the expiry date. Because of this additional benefit, an American option is always more expensive than a European option. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">However, is this benefit of any real use? Is there a situation where the option holder will get a better payoff by exercising the option early? <\/span><\/p>\n<p><span style=\"font-weight: 400;\">The answer is NO. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">You should never early exercise an American option, especially if it\u2019s a non-dividend paying stock. Let\u2019s look at the reasoning behind this.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The option has intrinsic value and time value. The intrinsic value of the option is always greater than 0. <\/span><br \/>\n<span style=\"font-weight: 400;\">Along with that, the cash has time value. So, you would rather delay paying the strike price by exercising it. As late as possible. <\/span><br \/>\n<span style=\"font-weight: 400;\">You could use that money to earn interest.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">So, a positive intrinsic value plus time value implies that you are better off selling the option rather than exercising it early. This is true for a non-dividend paying stock.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">However, for a dividend paying stock, the only time it may pay to exercise a call option is the day before the stock goes ex-dividend. And only if the dividend minus the cost of carry is less than the corresponding Put. <\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"By_exercising_the_option_holder_may_forego_the_time_value\"><\/span><span style=\"font-weight: 400;\">By exercising, the option holder may forego the time value. <\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">But don&#8217;t worry, it will make up from the dividend received. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">We use the word \u2018may\u2019 because the dividend may not be high enough to justify the early exercise.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Here is the model for American options pricing. Here we take into account a put as an example.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Let <\/span><br \/>\n<b>V = V (S, t) <\/b><\/p>\n<p><span style=\"font-weight: 400;\">be the option value. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">At expiry, we still have<\/span><br \/>\n<b>V (S, T)=(X \u2212 S) +<\/b><\/p>\n<p><span style=\"font-weight: 400;\">The early exercise feature gives the constraint<\/span><br \/>\n<b>V (S, t) \u2265 X \u2212 S<\/b><\/p>\n<p><span style=\"font-weight: 400;\">As before, we construct a portfolio of one long American option position and a short position in some quantity \u2206, of the underlying.<\/span><\/p>\n<p><b>\u03a0 = V \u2212 \u2206S<\/b><br \/>\n<span style=\"font-weight: 400;\">With the choice \u2206 = \u2202V\/\u2202S, the value of this portfolio changes by the amount<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5862 size-full\" src=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-3.jpg\" alt=\"American option pricing and early exercise 2\" width=\"800\" height=\"262\" srcset=\"https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-3.jpg 800w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-3-300x98.jpg 300w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-3-768x252.jpg 768w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-3-370x121.jpg 370w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-3-760x249.jpg 760w, https:\/\/traders-paradise.com\/magazine\/wp-content\/uploads\/2019\/02\/american-option-pricing-and-early-exercise-3-424x139.jpg 424w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"The_advantage_of_the_American_style\"><\/span><b>The advantage of the American style<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The principal advantage of the American style of an option contract is the flexibility it offers to its holder as it can be exercised anytime before the expiration date T. Majority derivative contracts traded in financial markets are of the American style. In mathematical modeling of American options, unlike European style options, there is the possibility of early exercising the contract at some time t* \u2208 [0, T) prior to the maturity time T.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">It is well-known that pricing an American call option on an underlying stock paying continuous dividend yield q &gt; 0 leads to a free boundary problem. In addition to a function V (t, S), we need to find the early exercise boundary function Sf (t), t \u2208 [0, T].<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The function Sf (t) has the following properties:<\/span><\/p>\n<p><b>If Sf (t) &gt; S for t \u2208 [0, T] then V (t, S) &gt; (S \u2212 E)+<\/b><br \/>\n<b>If Sf (t) \u2264 S for t \u2208 [0, T] then V (t, S) = (S \u2212 E)+<\/b><\/p>\n<p><span style=\"font-weight: 400;\">The free boundary problem for pricing an American call option consists in finding a function V (t, S) and the early exercise boundary function Sf such that V solves the Black-Scholes PDE on a time depending domain: <\/span><\/p>\n<p><b>{(t, S), 0 &lt; S &lt; Sf (t)} and V (t, Sf (t)) = Sf (t) \u2212 E, and \u2202SV (t, Sf (t)) = 1<\/b><\/p>\n<p><span style=\"font-weight: 400;\">In a stylized financial market, the price of a European style option can be computed from a solution to the well-known Black\u2013Scholes linear parabolic equation derived by Black and Scholes. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Recall that a European call option gives its owner the right but not the obligation to purchase an underlying asset at the expiration price E at the expiration time T.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"But_lets_make_American_option_pricing_simpler\"><\/span><b>But, let&#8217;s make\u00a0American option pricing simpler<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The value and pricing of stocks are fairly simple for most investors to understand. Basically, the value of a stock at any given time should reflect all known information about the company and the market. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">However, increased volatility in option value occurs when the expiration date draws close or when it is already &#8220;in-the-money.&#8221; A call option is &#8220;in-the-money&#8221; when the present price of the underlying stock is higher than the strike price. A put option is considered to be &#8220;in-the-money&#8221; when the market price is lower than the strike price. When options are &#8220;in-the-money&#8221; or close to their expiration date, their value will change at a different rate than the underlying stock. However, the Black Scholes formula is a mathematical equation that can be used to approximate the value of an option relative to its market price.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Delta (\u0394) in the Black Scholes formula is equal to the amount that the value of the option is expected to move for every 1 point of movement in the price of the underlying stock. Thus, if delta is 0.5 for stock A, then the value of the option for that stock will increase or decrease by 0.5 for every 1 point of fluctuation in the stock price. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">In addition to being affected by proximity to the expiration date and being &#8220;in&#8221; or &#8220;out&#8221; of the money, the value of delta may change due to the overall volatility of the underlying stock itself. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">There are times, however, when the Black Scholes formula fails to predict the value <\/span><span style=\"font-weight: 400;\">of the option.\u00a0<\/span><\/p>\n<h4><span class=\"ez-toc-section\" id=\"The_bottom_line\"><\/span><b>The bottom line<\/b><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><span style=\"font-weight: 400;\">The overall value of an option is actually <a href=\"https:\/\/traders-paradise.com\/magazine\/2019\/03\/simple-forex-trading-strategy\/\">determined<\/a> by six factors: strike price, the current market price of an underlying stock, dividend yield, prime interest rate, proximity to the expiration date, and the volatility of the stock prices over the course of the option.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Because these six variables combine in different ways to affect the value of an option, it is possible for the price of the underlying stock to increase while the value of the option falls. The Black Scholes formula may fail when other factors are affecting the value of the option more than the current stock price.<\/span><\/p>\n<p>American option pricing\u00a0uses a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Binomial_options_pricing_model\">&#8220;discrete-time&#8221; model<\/a> of the varying price over time of the underlying financial instrument.<\/p>\n<p><a href=\"https:\/\/traders-paradise.com\/magazine\/2018\/07\/money-risk-trading\"><b>risk disclosure<\/b><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>3 min read American option pricing\u00a0is the binomial options pricing model that provides a generalizable numerical method for the valuation of options. American options are contracts that may be exercised early, prior to expiry. These options are contrasted with European options for which exercise is only permitted at expiry. Most traded stock and futures options [&hellip;]<\/p>\n","protected":false},"author":4,"featured_media":5047,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"content-type":"","_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[1016,1012],"tags":[752,753,754,755],"class_list":["post-5859","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-how-to-master-in-trading-advanced","category-traders-secrets","tag-american-option","tag-american-option-pricing","tag-european-option","tag-european-option-pricing"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>American option pricing and how it is different from European<\/title>\n<meta name=\"description\" content=\"American option pricing\u00a0is 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