{"id":7895,"date":"2025-12-21T13:42:41","date_gmt":"2025-12-21T09:42:41","guid":{"rendered":"https:\/\/eoninvestment.com\/?p=7895"},"modified":"2026-04-11T08:09:17","modified_gmt":"2026-04-11T04:09:17","slug":"binomial-valuation-of-futures-options","status":"publish","type":"post","link":"https:\/\/eoninvestment.com\/ka\/2025\/12\/21\/binomial-valuation-of-futures-options\/","title":{"rendered":"<span class=\"eon-recovered-georgian-title\" data-no-translation=\"\">\u10d0\u10db\u10d4\u10e0\u10d8\u10d9\u10e3\u10da\u10d8 \u10e4\u10d8\u10e3\u10e9\u10d4\u10e0\u10e1-\u10dd\u10e4\u10ea\u10d8\u10d4\u10d1\u10d8<\/span>"},"content":{"rendered":"<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Valuation of American-Style Futures Options using Binomial Trees<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">American-style futures options are valued using <strong>binomial trees<\/strong>. The difference between a futures option and a stock option is that exercising a futures option <strong>does not require an upfront payment<\/strong> to enter into the futures contract.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To illustrate with a simple one-step binomial tree, we first need to create a <strong>risk-neutral portfolio<\/strong>. This portfolio consists of a <strong>short position in one option<\/strong> and a <strong>long position of \u0394 futures contracts<\/strong>, where \u0394 (the hedge ratio) is chosen such that, <strong>regardless of whether the futures price goes up or down, the portfolio value remains the same<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example, suppose we have the following data:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"395\" height=\"141\" data-attachment-id=\"7900\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/12\/21\/binomial-valuation-of-futures-options\/image-629\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-11.png?fit=395%2C141&amp;ssl=1\" data-orig-size=\"395,141\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-11.png?fit=395%2C141&amp;ssl=1\" class=\"wp-image-7900\" style=\"width: 300px\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-11.png?resize=395%2C141&#038;ssl=1\" alt=\"\"\/><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>One-step binomial alternatives: the futures price rises to <strong>$33<\/strong> or falls to <strong>$28<\/strong> in one month.<\/li>\n\n\n\n<li>\u0394 is calculated using the standard formula:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"221\" height=\"92\" data-attachment-id=\"7903\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/12\/21\/binomial-valuation-of-futures-options\/image-630\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-12.png?fit=221%2C92&amp;ssl=1\" data-orig-size=\"221,92\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-12.png?fit=221%2C92&amp;ssl=1\" class=\"wp-image-7903\" style=\"width: 150px\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-12.png?resize=221%2C92&#038;ssl=1\" alt=\"\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In our example, \u0394 = 0.8.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"505\" height=\"232\" data-attachment-id=\"7905\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/12\/21\/binomial-valuation-of-futures-options\/image-631\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-13.png?fit=505%2C232&amp;ssl=1\" data-orig-size=\"505,232\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-13.png?fit=505%2C232&amp;ssl=1\" class=\"wp-image-7905\" style=\"width: 300px\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-13.png?resize=505%2C232&#038;ssl=1\" alt=\"\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By taking \u0394 = 0.8 and assembling the portfolio accordingly, <strong>the portfolio value remains unchanged whether the futures price rises or falls<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"752\" height=\"469\" data-attachment-id=\"7907\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/12\/21\/binomial-valuation-of-futures-options\/image-632\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-14.png?fit=752%2C469&amp;ssl=1\" data-orig-size=\"752,469\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-14.png?fit=752%2C469&amp;ssl=1\" class=\"wp-image-7907\" style=\"width: 300px\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-14.png?resize=752%2C469&#038;ssl=1\" alt=\"\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Next, if we <strong>discount the portfolio value at the risk-free interest rate<\/strong>, we get <strong>&#8211; $1.59<\/strong>, which, considering that entering a futures contract costs nothing, implies that the <strong>option price today is $1.59<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Risk-Neutral Probability<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>risk-neutral probability<\/strong> is a probability that ensures the portfolio has the <strong>same value in all future scenarios<\/strong>, as obtained by the formulas above.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To clarify, in the risk-neutral probability:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The probability of a price increase is <strong>p<\/strong><\/li>\n\n\n\n<li>The probability of a price decrease is <strong>1 \u2013 p<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">p is calculated purely algebraically, so it can also be used as a <strong>shortcut to determine the option price<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"471\" height=\"142\" data-attachment-id=\"7914\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/12\/21\/binomial-valuation-of-futures-options\/image-634\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-16.png?fit=471%2C142&amp;ssl=1\" data-orig-size=\"471,142\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-16.png?fit=471%2C142&amp;ssl=1\" class=\"wp-image-7914\" style=\"width: 300px\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-16.png?resize=471%2C142&#038;ssl=1\" alt=\"\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In our case, <strong>p = 0.4<\/strong>, which results in an option price of <strong>$1.59<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Excel<\/strong>: <strong>B<\/strong><a href=\"https:\/\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/17.5-valuing-futures-options.xlsx\"><strong>inomial Valuation of Futures Options<\/strong> model.<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Note:<\/strong> The formula for <strong>p differs from the stock option case<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"697\" height=\"118\" data-attachment-id=\"7919\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/12\/21\/binomial-valuation-of-futures-options\/image-636\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-18.png?fit=697%2C118&amp;ssl=1\" data-orig-size=\"697,118\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-18.png?fit=697%2C118&amp;ssl=1\" class=\"wp-image-7919\" style=\"width: 300px\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/12\/image-18.png?resize=697%2C118&#038;ssl=1\" alt=\"\"\/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>(Source: Options, Futures &#038; Other Derivatives, John C. Hull)<\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>","protected":false},"excerpt":{"rendered":"<p>\u10d0\u10db\u10d4\u10e0\u10d8\u10d9\u10e3\u10da\u10d8 \u10e1\u10e2\u10d8\u10da\u10d8\u10e1 \u10e4\u10d8\u10e3\u10e9\u10d4\u10e0\u10e1 \u10dd\u10e4\u10ea\u10d8\u10d4\u10d1\u10d8\u10e1 \u10e8\u10d4\u10e4\u10d0\u10e1\u10d4\u10d1\u10d0 \u10ee\u10d3\u10d4\u10d1\u10d0 \u10d1\u10d8\u10dc\u10dd\u10db\u10d8\u10d0\u10da\u10e3\u10e0\u10d8 \u10ee\u10d4\u10d4\u10d1\u10d8\u10e1 \u10db\u10d4\u10e8\u10d5\u10d4\u10dd\u10d1\u10d8\u10d7. \u10e4\u10d8\u10e3\u10e9\u10d4\u10e0\u10e1\u10d8\u10e1\u10d0 \u10d3\u10d0 \u10d0\u10e5\u10ea\u10d8\u10d8\u10e1 \u10dd\u10e4\u10ea\u10d8\u10d0\u10e1 \u10e8\u10dd\u10e0\u10d8\u10e1 \u10d2\u10d0\u10dc\u10e1\u10ee\u10d5\u10d0\u10d5\u10d4\u10d1\u10d0 \u10d8\u10db\u10d0\u10e8\u10d8\u10d0, \u10e0\u10dd\u10db \u10dd\u10e4\u10ea\u10d8\u10d8\u10e1 \u10d2\u10d0\u10dc\u10d0\u10e6\u10d3\u10d4\u10d1\u10d8\u10e1\u10d0\u10e1 \u10e4\u10d8\u10e3\u10e9\u10d4\u10e0\u10e1\u10e3\u10da \u10d2\u10d0\u10e0\u10d8\u10d2\u10d4\u10d1\u10d0\u10e8\u10d8 \u10e8\u10d4\u10e1\u10d5\u10da\u10d0 \u10d0\u10e0 \u10d8\u10d7\u10ee\u10dd\u10d5\u10e1 \u10ec\u10d8\u10dc\u10d0\u10e1\u10ec\u10d0\u10e0 \u10d0\u10dc\u10d0\u10d6\u10e6\u10d0\u10e3\u10e0\u10d4\u10d1\u10d0\u10e1. \u10db\u10d0\u10e0\u10e2\u10d8\u10d5\u10d8 \u10d1\u10d8\u10dc\u10dd\u10db\u10d8\u10d0\u10da\u10e3\u10e0\u10d8 \u10ee\u10d8\u10d7 \u10e0\u10dd\u10db \u10db\u10dd\u10d5\u10d0\u10ee\u10d3\u10d8\u10dc\u10dd\u10d7 \u10d8\u10da\u10e3\u10e1\u10e2\u10e0\u10d0\u10ea\u10d8\u10d0, [\u2026]<\/p>","protected":false},"author":12998525,"featured_media":7907,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_wpcom_ai_launchpad_first_post":false,"_eon_insights_quality":"","_eon_insights_decision":"","_eon_insights_destination":"","_eon_insights_merge_group":"","_eon_insights_required_changes":"","_eon_insights_start_order":0,"_eon_featured_insight":0,"_eon_insights_map_version":"","_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[31110612,784237235,106538530],"tags":[],"eon_topic":[],"eon_learning_path":[],"eon_collection":[],"eon_level":[],"eon_content_type":[],"eon_source":[],"eon_idea":[784445705],"eon_framework":[],"class_list":["post-7895","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-excel-models","category-options-futures-other-derivatives-john-c-hull-ka","category-derivatives","eon_idea-derivatives"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - 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