{"id":7401,"date":"2025-10-11T17:25:01","date_gmt":"2025-10-11T13:25:01","guid":{"rendered":"https:\/\/eoninvestment.com\/?p=7401"},"modified":"2026-04-11T09:36:34","modified_gmt":"2026-04-11T05:36:34","slug":"risk-neutral-vs-replication-in-option-pricing","status":"publish","type":"post","link":"https:\/\/eoninvestment.com\/ka\/2025\/10\/11\/risk-neutral-vs-replication-in-option-pricing\/","title":{"rendered":"<span class=\"eon-recovered-georgian-title\" data-no-translation=\"\">\u10e0\u10d8\u10e1\u10d9-\u10dc\u10d4\u10d8\u10e2\u10e0\u10d0\u10da\u10e3\u10e0\u10d8 \u10e1\u10d0\u10db\u10e7\u10d0\u10e0\u10dd\u10e1\u10d0 \u10d3\u10d0 \u10e0\u10d4\u10de\u10da\u10d8\u10d9\u10d0\u10ea\u10d8\u10d8\u10e1 \u10db\u10d4\u10d7\u10dd\u10d3\u10d4\u10d1\u10d8\u10e1 \u10e8\u10d4\u10d3\u10d0\u10e0\u10d4\u10d1\u10d0<\/span>"},"content":{"rendered":"<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Using the Risk-Neutral Probability Method in Option Valuation<\/strong> is a bit counterintuitive. Why doesn\u2019t the option price depend on the expected movement of the underlying stock \u2014 its probability of going up or down? Why does it not matter if the probability of the stock going up is 95% or 15%? Why do calculations done in a risk-neutral world work even in the real world of risk?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The point is that we are not valuing the option in absolute terms; we are valuing it <strong>as a function of the current price of the underlying stock<\/strong>. The stock price itself already incorporates risk and expectations (Hull, <em>Options, Futures &#038; Other Derivatives<\/em>, Chapter 12, Binomial Trees).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Substantially, I understand it this way: In a simple model (<a href=\"https:\/\/eoninvestment.com\/ka\/2024\/12\/21\/risk-free-world\/\">Risk-Free World<\/a>), we have a stock price that can either <strong>increase by X% or decrease by Y%<\/strong>. Accordingly, a call option has two possible outcomes: positive payoff if the stock increases, or zero if the stock decreases. To calculate the PV of an investment in such an option, we need a <strong>discount rate<\/strong>, which is unknown, even though we know the range of volatility. We know X and Y, but we don\u2019t know the real probability that would help weight them.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">However, we know that <strong>discounting risky cash flows at the appropriate risky rate<\/strong> and <strong>discounting riskless cash flows at the risk-free rate<\/strong> will give the same PV. Therefore, we can <strong>adjust the probabilities<\/strong> so that, when weighting X% and Y%, we use the <strong>risk-free rate<\/strong>. In other words, we know what the \u201cup\u201d and \u201cdown\u201d probabilities would be under <strong>risk-neutral cash flows<\/strong>. Then, discounting the expected option payoff using these probabilities at the <strong>risk-free rate<\/strong> gives the fair value of the option.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Interestingly, the fair value obtained using this method <strong>matches the fair value obtained from the asset replication method<\/strong> (\u0394 Stock + Bond, <a href=\"https:\/\/eoninvestment.com\/ka\/2024\/12\/21\/portfolio-replication-method\/\">Portfolio Replication Method<\/a>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">P.S.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As ChatGPT showed me, these two methods (often referred to as separate methods in academic literature) are in fact <strong>algebraically connected<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-10.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"971\" height=\"343\" data-attachment-id=\"7433\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/10\/11\/risk-neutral-vs-replication-in-option-pricing\/image-562\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-10.png?fit=971%2C343&amp;ssl=1\" data-orig-size=\"971,343\" 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\/10\/image-10.png?fit=971%2C343&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-10.png?resize=971%2C343&#038;ssl=1\" alt=\"\" class=\"wp-image-7433\"\/><\/a><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-11.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"983\" height=\"337\" data-attachment-id=\"7435\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/10\/11\/risk-neutral-vs-replication-in-option-pricing\/image-563\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-11.png?fit=983%2C337&amp;ssl=1\" data-orig-size=\"983,337\" 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\/10\/image-11.png?fit=983%2C337&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-11.png?resize=983%2C337&#038;ssl=1\" alt=\"\" class=\"wp-image-7435\"\/><\/a><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-12.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"980\" height=\"341\" data-attachment-id=\"7436\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/10\/11\/risk-neutral-vs-replication-in-option-pricing\/image-564\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-12.png?fit=980%2C341&amp;ssl=1\" data-orig-size=\"980,341\" 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\/10\/image-12.png?fit=980%2C341&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-12.png?resize=980%2C341&#038;ssl=1\" alt=\"\" class=\"wp-image-7436\"\/><\/a><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-13.png?ssl=1\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"968\" height=\"498\" data-attachment-id=\"7437\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2025\/10\/11\/risk-neutral-vs-replication-in-option-pricing\/image-565\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-13.png?fit=968%2C498&amp;ssl=1\" data-orig-size=\"968,498\" 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\/10\/image-13.png?fit=968%2C498&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-13.png?resize=968%2C498&#038;ssl=1\" alt=\"\" class=\"wp-image-7437\"\/><\/a><\/figure>","protected":false},"excerpt":{"rendered":"<p>Risk Neutral-Probability &#8211; \u10db\u10d4\u10d7\u10dd\u10d3\u10d8\u10e1 \u10d2\u10d0\u10db\u10dd\u10e7\u10d4\u10dc\u10d4\u10d1\u10d0 \u10dd\u10e4\u10ea\u10d8\u10d4\u10d1\u10d8\u10e1 \u10e8\u10d4\u10e4\u10d0\u10e1\u10d4\u10d1\u10d0\u10e8\u10d8, \u10ea\u10dd\u10e2\u10d0 \u10d0\u10e0\u10d0\u10d8\u10dc\u10e2\u10e3\u10d8\u10ea\u10d8\u10e3\u10e0\u10d8\u10d0. \u10e0\u10d0\u10e2\u10dd\u10db \u10d0\u10e0 \u10d0\u10e0\u10d8\u10e1 \u10dd\u10e4\u10ea\u10d8\u10d8\u10e1 \u10e4\u10d0\u10e1\u10d8 \u10d3\u10d0\u10db\u10dd\u10d9\u10d8\u10d3\u10d4\u10d1\u10e3\u10da\u10d8 \u10e5\u10d5\u10d4\u10db\u10d3\u10d4\u10d1\u10d0\u10e0\u10d4 \u10d0\u10e5\u10ea\u10d8\u10d8\u10e1 \u10e4\u10d0\u10e1\u10d8\u10e1 \u10db\u10dd\u10da\u10dd\u10d3\u10d8\u10dc\u10d6\u10d4? &#8211; \u10db\u10d8\u10e1\u10d8 \u10d6\u10e0\u10d3\u10d8\u10e1\u10d0 \u10d3\u10d0 \u10e8\u10d4\u10db\u10ea\u10d8\u10e0\u10d4\u10d1\u10d8\u10e1 \u10d0\u10da\u10d1\u10d0\u10d7\u10dd\u10d1\u10d0\u10d6\u10d4? \u10e0\u10d0\u10e2\u10dd\u10db \u10d0\u10e0 \u10d0\u10e5\u10d5\u10e1 \u10db\u10dc\u10d8\u10e8\u10d5\u10dc\u10d4\u10da\u10dd\u10d1\u10d0 \u10d0\u10e5\u10ea\u10d8\u10d8\u10e1 \u10e4\u10d0\u10e1\u10d8\u10e1 \u10d6\u10e0\u10d3\u10d8\u10e1 [\u2026]<\/p>","protected":false},"author":12998525,"featured_media":7437,"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":[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-7401","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","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 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Risk-Neutral vs Replication in Option Pricing - Eon Investment<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/eoninvestment.com\/ka\/2025\/10\/11\/risk-neutral-vs-replication-in-option-pricing\/\" \/>\n<meta property=\"og:locale\" content=\"ka_GE\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Risk-Neutral vs Replication in Option Pricing - Eon Investment\" \/>\n<meta property=\"og:description\" content=\"Risk Neutral-Probability - \u10db\u10d4\u10d7\u10dd\u10d3\u10d8\u10e1 \u10d2\u10d0\u10db\u10dd\u10e7\u10d4\u10dc\u10d4\u10d1\u10d0 \u10dd\u10e4\u10ea\u10d8\u10d4\u10d1\u10d8\u10e1 \u10e8\u10d4\u10e4\u10d0\u10e1\u10d4\u10d1\u10d0\u10e8\u10d8, \u10ea\u10dd\u10e2\u10d0 \u10d0\u10e0\u10d0\u10d8\u10dc\u10e2\u10e3\u10d8\u10ea\u10d8\u10e3\u10e0\u10d8\u10d0. \u10e0\u10d0\u10e2\u10dd\u10db \u10d0\u10e0 \u10d0\u10e0\u10d8\u10e1 \u10dd\u10e4\u10ea\u10d8\u10d8\u10e1 \u10e4\u10d0\u10e1\u10d8 \u10d3\u10d0\u10db\u10dd\u10d9\u10d8\u10d3\u10d4\u10d1\u10e3\u10da\u10d8 \u10e5\u10d5\u10d4\u10db\u10d3\u10d4\u10d1\u10d0\u10e0\u10d4 \u10d0\u10e5\u10ea\u10d8\u10d8\u10e1 \u10e4\u10d0\u10e1\u10d8\u10e1 \u10db\u10dd\u10da\u10dd\u10d3\u10d8\u10dc\u10d6\u10d4? - \u10db\u10d8\u10e1\u10d8 \u10d6\u10e0\u10d3\u10d8\u10e1\u10d0 \u10d3\u10d0 \u10e8\u10d4\u10db\u10ea\u10d8\u10e0\u10d4\u10d1\u10d8\u10e1 \u10d0\u10da\u10d1\u10d0\u10d7\u10dd\u10d1\u10d0\u10d6\u10d4? \u10e0\u10d0\u10e2\u10dd\u10db \u10d0\u10e0 \u10d0\u10e5\u10d5\u10e1 \u10db\u10dc\u10d8\u10e8\u10d5\u10dc\u10d4\u10da\u10dd\u10d1\u10d0 \u10d0\u10e5\u10ea\u10d8\u10d8\u10e1 \u10e4\u10d0\u10e1\u10d8\u10e1 \u10d6\u10e0\u10d3\u10d8\u10e1 [\u2026]\" \/>\n<meta property=\"og:url\" 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