{"id":7447,"date":"2025-10-18T11:49:49","date_gmt":"2025-10-18T07:49:49","guid":{"rendered":"https:\/\/eoninvestment.com\/?p=7447"},"modified":"2026-04-11T09:32:25","modified_gmt":"2026-04-11T05:32:25","slug":"how-randomness-shapes-markets","status":"publish","type":"post","link":"https:\/\/eoninvestment.com\/ka\/2025\/10\/18\/how-randomness-shapes-markets\/","title":{"rendered":"<span class=\"eon-recovered-georgian-title\" data-no-translation=\"\">\ud83c\udfb2 \u10e8\u10d4\u10db\u10d7\u10ee\u10d5\u10d4\u10d5\u10d8\u10d7\u10dd\u10d1\u10d4\u10d1\u10d8\u10e1 \u10d2\u10d0\u10d5\u10da\u10d4\u10dc\u10d0 \u10d1\u10d0\u10d6\u10e0\u10d4\u10d1\u10d6\u10d4<\/span>"},"content":{"rendered":"<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\ud83c\udfb2 <\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Have you ever noticed how life, business, and markets share one thing in common \u2014 <em>uncertainty<\/em>?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Finance doesn\u2019t ignore that; it embraces it.<br \/>John Hull\u2019s classic work on derivatives explains how we can actually <strong>model uncertainty<\/strong> \u2014 and that\u2019s what this post is about. Let\u2019s walk through it in plain words \ud83d\udc47<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83c\udf27\ufe0f 1. Stochastic Process \u2013 Life as a Sequence of Uncertainties<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Imagine you\u2019re walking home, and at every intersection, you toss a coin to decide whether to go left or right.<br \/>You don\u2019t know exactly where you\u2019ll end up, but you can describe the <em>probability<\/em> of reaching different places.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That\u2019s what a <strong>stochastic process<\/strong> is \u2014 it describes how something (like a stock price, temperature, or your location) <em>evolves through time with uncertainty<\/em>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd01 2. Markov Process \u2013 \u201cMemoryless\u201d Future<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A <strong>Markov process<\/strong> says: <em>the future only cares about the present, not the past.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Think of <strong>Google Maps\u2019 live traffic<\/strong> \u2014 when predicting how long your trip will take, it uses <em>current<\/em> traffic, not the route you took to get there.<br \/>The road behind you doesn\u2019t matter \u2014 only the situation right now.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Stock prices behave a bit like that: today\u2019s price already reflects everything known from the past. What happens next depends only on the present state.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83c\udf43 3. Wiener Process \u2013 Pure Random Drift<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A <strong>Wiener process<\/strong> is like a <strong>drunk person walking in a park<\/strong>: each step is random, and there\u2019s no direction \u2014 sometimes left, sometimes right, with small unpredictable changes.<br \/>On average, they stay where they started (no drift), but over time, their position becomes more uncertain.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So if you track them for longer, the range of where they might be spreads out \u2014 that\u2019s the growing variance.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83c\udf0a 4. Generalized Wiener Process \u2013 Random Drift with a Trend<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Now imagine our drunk walker is on a <strong>gentle slope<\/strong>.<br \/>They still stumble randomly, but gravity pulls them slightly downhill \u2014 that\u2019s the <strong>drift<\/strong>.<br \/>And maybe some days they\u2019re more unstable than others \u2014 that\u2019s the <strong>variance rate<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is closer to how stock prices move: they wander unpredictably, but with an overall upward drift (the average return).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd04 5. Ito Process \u2013 When Conditions Change Over Time<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A <strong>Wiener process<\/strong> assumes drift and volatility are constant.<br \/>An <strong>Ito process<\/strong> says they can change \u2014 depending on time or on where you are.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Imagine a <strong>boat drifting on the sea<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>When near the shore, the waves are calmer (less volatility).<\/li>\n\n\n\n<li>In the open ocean, the waves get rougher.<\/li>\n\n\n\n<li>The current may change direction with the wind (time-dependent drift).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">So the randomness is still there, but its intensity and direction vary with circumstances.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83c\udfb2 6. Monte Carlo Simulation \u2013 Playing Out Many Possible Futures<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">You can imagine the future price of a stock as millions of possible paths, each depending on random daily movements.<br \/>To understand what\u2019s <em>likely<\/em>, we simulate these paths many times \u2014 like playing out <strong>a thousand versions of tomorrow<\/strong> to see the range of outcomes.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It\u2019s like testing how many ways your weekend trip could go wrong or right depending on the weather.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83e\uddee 7. Ito\u2019s Lemma \u2013 How Uncertainty Transfers Between Variables<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Ito\u2019s Lemma helps us understand how <strong>randomness in one variable<\/strong> (like a stock price) affects <strong>another variable that depends on it<\/strong> (like the value of a stock option).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Think of it like <strong>wind and a kite<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The wind (stock price movement) is random.<\/li>\n\n\n\n<li>The kite (option value) reacts to it in a more complex way \u2014 rising, falling, curving.<br \/>Ito\u2019s Lemma tells us exactly how that randomness transfers from wind to kite.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udcc8 8. Geometric Brownian Motion \u2013 How Stock Prices \u201cGrow Randomly\u201d<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Stock prices follow what\u2019s called <strong>geometric Brownian motion<\/strong> \u2014 randomness combined with a tendency to grow.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Think of a <strong>tree in the wind<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It grows taller over time (the drift \u2014 average return).<\/li>\n\n\n\n<li>But it also sways unpredictably (the random shocks).<br \/>After a while, no two trees (or stock price paths) look the same, but all have grown <em>on average<\/em>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This idea is the foundation of the <strong>Black-Scholes model<\/strong>, which prices options by assuming stock prices evolve with this kind of \u201crandom growth.\u201d<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">Adapted from: Options, Futures &#038; Other Derivatives, John C. Hull<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>","protected":false},"excerpt":{"rendered":"<p>\u10d1\u10d8\u10d6\u10dc\u10d4\u10e1\u10e1\u10d0 \u10d3\u10d0 \u10e4\u10d8\u10dc\u10d0\u10dc\u10e1\u10e3\u10e0 \u10d1\u10d0\u10d6\u10e0\u10d4\u10d1\u10e1, \u10d8\u10e1\u10d4\u10d5\u10d4 \u10e0\u10dd\u10d2\u10dd\u10e0\u10ea \u10d6\u10dd\u10d2\u10d0\u10d3\u10d0\u10d3 \u10ea\u10ee\u10dd\u10d5\u10e0\u10d4\u10d1\u10d0\u10e1 \u10d4\u10e0\u10d7\u10d8 \u10e4\u10e3\u10dc\u10d3\u10d0\u10db\u10d4\u10dc\u10e2\u10e3\u10e0\u10d8 \u10e0\u10d0\u10db \u10d0\u10d4\u10e0\u10d7\u10d8\u10d0\u10dc\u10d4\u10d1\u10d7 \u2014 \u10d2\u10d0\u10e3\u10e0\u10d9\u10d5\u10d4\u10d5\u10da\u10dd\u10d1\u10d0. \u10d7\u10d0\u10dc\u10d0\u10db\u10d4\u10d3\u10e0\u10dd\u10d5\u10d4 \u10e4\u10d8\u10dc\u10d0\u10dc\u10e1\u10e3\u10e0\u10d8 \u10db\u10d4\u10ea\u10dc\u10d8\u10d4\u10e0\u10d4\u10d1\u10d0 \u10ea\u10d3\u10d8\u10da\u10dd\u10d1\u10e1 \u10e0\u10dd\u10db \u10d4\u10e1 \u10d2\u10d0\u10e3\u10e0\u10d9\u10d5\u10d4\u10d5\u10da\u10dd\u10d1\u10d0 \u10e0\u10d0\u10e6\u10d0\u10ea \u10e9\u10d0\u10e0\u10e9\u10dd\u10d4\u10d1\u10e8\u10d8 \u10db\u10dd\u10d0\u10e5\u10ea\u10d8\u10dd\u10e1, \u10e0\u10dd\u10db \u10db\u10dd\u10db\u10d0\u10d5\u10d0\u10da\u10d8 \u10e3\u10e4\u10e0\u10dd \u10ec\u10d8\u10dc\u10d0\u10e1\u10ec\u10d0\u10e0\u10db\u10d4\u10e2\u10e7\u10d5\u10d4\u10da\u10d4\u10d1\u10d0\u10d3\u10d8 \u10d3\u10d0 \u10d2\u10d0\u10d3\u10d0\u10ec\u10e7\u10d5\u10d4\u10e2\u10d8\u10da\u10d4\u10d1\u10d4\u10d1\u10d8 \u10e3\u10e4\u10e0\u10dd \u10e0\u10d0\u10ea\u10d8\u10dd\u10dc\u10d0\u10da\u10e3\u10e0\u10d8 [\u2026]<\/p>","protected":false},"author":12998525,"featured_media":1556,"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":[4269,1868],"eon_topic":[],"eon_learning_path":[],"eon_collection":[],"eon_level":[],"eon_content_type":[],"eon_source":[],"eon_idea":[784445705],"eon_framework":[],"class_list":["post-7447","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-options-futures-other-derivatives-john-c-hull-ka","category-derivatives","tag-finance","tag-philosophy","eon_idea-derivatives"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - 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