{"id":7510,"date":"2025-10-31T23:14:26","date_gmt":"2025-10-31T19:14:26","guid":{"rendered":"https:\/\/eoninvestment.com\/?p=7510"},"modified":"2026-04-11T09:16:08","modified_gmt":"2026-04-11T05:16:08","slug":"itos-lemma-wiener-process","status":"publish","type":"post","link":"https:\/\/eoninvestment.com\/ka\/2025\/10\/31\/itos-lemma-wiener-process\/","title":{"rendered":"\u10d8\u10e2\u10dd\u10e1 \u10da\u10d4\u10db\u10d0 \u10d3\u10d0 \u10d5\u10d8\u10dc\u10d4\u10e0\u10d8\u10e1 \u10de\u10e0\u10dd\u10ea\u10d4\u10e1\u10d8<\/span>"},"content":{"rendered":"
\n\n\n\n

Without the discoveries of the Japanese mathematician Kiyoshi It\u00f4 (\u4f0a\u85e4 \u6e05, 1915\u20132008)<\/strong>, the Black\u2013Scholes\u2013Merton model<\/strong> \u2014 and therefore the modern derivatives market<\/strong> \u2014 could not exist.<\/p>\n\n\n\n

Before moving on to It\u00f4\u2019s discovery, it\u2019s helpful to understand the generalized Wiener process<\/strong>, and even before that, to get clear on the Markov<\/strong> and Wiener<\/strong> processes (you can review these using the provided links).<\/p>\n\n\n\n

\n\n\n\n

Generalized Wiener Process<\/h3>\n\n\n\n

A generalized Wiener process<\/em> is essentially a Wiener process with an added trend<\/strong>.
In an ordinary Wiener process, the variable\u2019s expected change<\/strong> over time is zero, and its variance<\/strong> increases linearly with time (equal to 1 per unit of time).
Also, the changes are not autocorrelated<\/strong> \u2014 they don\u2019t depend on past values.<\/p>\n\n\n\n

In the generalized Wiener process, however, a trend (drift)<\/strong> appears.
In a stochastic process, the expected rate of change of the mean is called the drift rate<\/strong>, and the rate of change of the variance is called the variance rate<\/strong>.<\/p>\n\n\n\n

Graphically, it looks like this:<\/p>\n\n\n\n

The generalized Wiener process is described by the formula:
dx = a*dt + b*dz<\/strong><\/p>\n\n\n\n

This expression describes the infinitesimal change of a variable over infinitely small time intervals. (Recall that d is the same as Delta, only representing an infinitesimally small amount.)<\/p>\n\n\n\n

In discrete form<\/strong>, for finite time intervals, the same process can be written as: \u0394x=a*\u0394t+b*\u03b5*\u00a0\u221a\u0394t<\/strong>
where \u03b5<\/strong> is a random variable drawn from a standard normal distribution<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Let\u2019s return to the continuous version:<\/p>\n\n\n\n

dx = a*dt + b*dz<\/strong><\/p>\n\n\n\n

The right-hand side has two parts<\/strong> \u2014 the trend<\/strong> and the Wiener process<\/strong>.
If there were no random component (no Wiener process), the movement of x would be deterministic<\/strong>, meaning the expected rate of change of x would simply be a:<\/p>\n\n\n\n

dx = a*dt<\/strong>, – Drift Rate, – a = dx\/dt.<\/strong><\/p>\n\n\n\n

Integrating over a small time interval gives: x =x0\u200b<\/sub>+at<\/strong><\/p>\n\n\n\n

The second component, b*dz,<\/strong> adds noise\/variability<\/strong> to the deterministic trend.
Its magnitude is determined by b<\/strong> times the Wiener process.<\/p>\n\n\n\n

Since the variance rate of the Wiener process per unit time is 1, the variance rate of b*dz<\/strong> is b2<\/sup><\/strong>.
<\/p>\n\n\n\n

On the graph, you can see that as the curve rises, the magnitude of fluctuations increases as well.<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n
\n\n\n\n

It\u00f4 Process<\/h3>\n\n\n\n

An It\u00f4 process<\/strong> is a type of generalized Wiener process in which ( a ) and ( b ) are functions of both time and the underlying variable<\/strong>:<\/p>\n\n\n\n

dx = a (x, t) dt + b (x, t) dz<\/strong><\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n


<\/p>\n\n\n\n

\n\n\n\n

It\u00f4\u2019s Lemma<\/h3>\n\n\n\n

It\u00f4 proved that if x<\/strong> (e.g., a stock price) follows an It\u00f4 process, and G<\/strong> is a function of ( x ) and ( t ) (e.g., an option), then G <\/strong>also follows an It\u00f4s process \u2014 with its drift rate<\/strong> and variance rate<\/strong> determined by a specific formula:<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n


and where the ( dz ) term is the same Wiener process<\/strong> as in the original equation.<\/p>\n\n\n\n

\n\n\n\n

In short, this not-so-intuitive and mathematically challenging formula gave finance professionals the ability to formally link the value of a stock and the value of its derivative (option)<\/strong>.<\/p>\n\n\n\n

To make It\u00f4\u2019s Lemma clearer, one needs to first understand how a stock price evolves over time<\/strong> \u2014 a topic that will be discussed in the next section.<\/p>\n\n\n\n

\u10ec\u10e7\u10d0\u10e0\u10dd:<\/strong> Options, Futures & Other Derivatives<\/em>, John C. Hull<\/p>\n\n\n\n

","protected":false},"excerpt":{"rendered":"

\u10d8\u10d0\u10de\u10dd\u10dc\u10d4\u10da\u10d8 \u10db\u10d0\u10d7\u10d4\u10db\u10d0\u10e2\u10d8\u10d9\u10dd\u10e1\u10d8\u10e1 Kiyoshi It\u00f4-\u10e1 (\u4f0a\u85e4 \u6e05) (1915\u20132008), \u10db\u10d8\u10d2\u10dc\u10d4\u10d1\u10d8\u10e1 \u10d2\u10d0\u10e0\u10d4\u10e8\u10d4 \u10d5\u10d4\u10e0 \u10d8\u10d0\u10e0\u10e1\u10d4\u10d1\u10d4\u10d1\u10d3\u10d0 Black-Scholes-Merton -\u10d8\u10e1 \u10db\u10dd\u10d3\u10d4\u10da\u10d8 \u10d3\u10d0 \u10e8\u10d4\u10e1\u10d0\u10d1\u10d0\u10db\u10d8\u10e1\u10d0\u10d3 \u10d3\u10d4\u10e0\u10d8\u10d5\u10d0\u10e2\u10d8\u10d5\u10d4\u10d1\u10d8\u10e1 \u10d1\u10d0\u10d6\u10d0\u10e0\u10d8… \u10e1\u10d0\u10dc\u10d0\u10db \u10d8\u10e2\u10dd\u10e1 \u10db\u10d8\u10d2\u10dc\u10d4\u10d1\u10d0\u10d6\u10d4 \u10d2\u10d0\u10d3\u10d0\u10d5\u10d8\u10d3\u10dd\u10d3\u10d4\u10d7, \u10db\u10d0\u10dc\u10d0\u10db\u10d3\u10d4 \u10d9\u10d0\u10e0\u10d2\u10d8\u10d0 \u10d2\u10d0\u10d5\u10d4\u10e0\u10d9\u10d5\u10d4\u10d7 \u10d2\u10d0\u10dc\u10d6\u10dd\u10d2\u10d0\u10d3\u10d4\u10d1\u10e3\u10da \u10d5\u10d8\u10dc\u10d4\u10e0\u10d8\u10e1 \u10de\u10e0\u10dd\u10ea\u10d4\u10e1\u10e8\u10d8, \u10db\u10d0\u10d2\u10e0\u10d0\u10db \u10d9\u10d8\u10d3\u10d4\u10d5 \u10db\u10d0\u10dc\u10d0\u10db\u10d3\u10d4 \u10db\u10d0\u10e0\u10d9\u10dd\u10d5\u10d8\u10e1\u10d0 [\u2026]<\/p>","protected":false},"author":12998525,"featured_media":0,"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":[784445728],"eon_content_type":[],"eon_source":[784445725],"eon_idea":[784445705],"eon_framework":[],"class_list":["post-7510","post","type-post","status-publish","format-standard","hentry","category-options-futures-other-derivatives-john-c-hull-ka","category-derivatives","eon_level-advanced","eon_source-options-futures-other-derivatives-john-c-hull","eon_idea-derivatives"],"yoast_head":"\nIto's Lemma & Wiener Process - 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\/31\/itos-lemma-wiener-process\/\" \/>\n<meta property=\"og:locale\" content=\"ka_GE\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Ito's Lemma & Wiener Process - Eon Investment\" \/>\n<meta property=\"og:description\" content=\"\u10d8\u10d0\u10de\u10dd\u10dc\u10d4\u10da\u10d8 \u10db\u10d0\u10d7\u10d4\u10db\u10d0\u10e2\u10d8\u10d9\u10dd\u10e1\u10d8\u10e1 Kiyoshi It\u00f4-\u10e1 (\u4f0a\u85e4 \u6e05) (1915\u20132008), \u10db\u10d8\u10d2\u10dc\u10d4\u10d1\u10d8\u10e1 \u10d2\u10d0\u10e0\u10d4\u10e8\u10d4 \u10d5\u10d4\u10e0 \u10d8\u10d0\u10e0\u10e1\u10d4\u10d1\u10d4\u10d1\u10d3\u10d0 Black-Scholes-Merton -\u10d8\u10e1 \u10db\u10dd\u10d3\u10d4\u10da\u10d8 \u10d3\u10d0 \u10e8\u10d4\u10e1\u10d0\u10d1\u10d0\u10db\u10d8\u10e1\u10d0\u10d3 \u10d3\u10d4\u10e0\u10d8\u10d5\u10d0\u10e2\u10d8\u10d5\u10d4\u10d1\u10d8\u10e1 \u10d1\u10d0\u10d6\u10d0\u10e0\u10d8... \u10e1\u10d0\u10dc\u10d0\u10db \u10d8\u10e2\u10dd\u10e1 \u10db\u10d8\u10d2\u10dc\u10d4\u10d1\u10d0\u10d6\u10d4 \u10d2\u10d0\u10d3\u10d0\u10d5\u10d8\u10d3\u10dd\u10d3\u10d4\u10d7, \u10db\u10d0\u10dc\u10d0\u10db\u10d3\u10d4 \u10d9\u10d0\u10e0\u10d2\u10d8\u10d0 \u10d2\u10d0\u10d5\u10d4\u10e0\u10d9\u10d5\u10d4\u10d7 \u10d2\u10d0\u10dc\u10d6\u10dd\u10d2\u10d0\u10d3\u10d4\u10d1\u10e3\u10da \u10d5\u10d8\u10dc\u10d4\u10e0\u10d8\u10e1 \u10de\u10e0\u10dd\u10ea\u10d4\u10e1\u10e8\u10d8, \u10db\u10d0\u10d2\u10e0\u10d0\u10db \u10d9\u10d8\u10d3\u10d4\u10d5 \u10db\u10d0\u10dc\u10d0\u10db\u10d3\u10d4 \u10db\u10d0\u10e0\u10d9\u10dd\u10d5\u10d8\u10e1\u10d0 [\u2026]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/eoninvestment.com\/ka\/2025\/10\/31\/itos-lemma-wiener-process\/\" \/>\n<meta property=\"og:site_name\" content=\"Eon Investment\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/facebook.com\/eoninvestment\" \/>\n<meta property=\"article:published_time\" content=\"2025-10-31T19:14:26+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-04-11T05:16:08+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/eoninvestment.com\/wp-content\/uploads\/2025\/10\/image-17.png?w=859\" \/>\n<meta name=\"author\" content=\"neoinsight\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"neoinsight\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 \u10ec\u10e3\u10d7\u10d8\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/eoninvestment.com\\\/ka\\\/wp-json\\\/wp\\\/v2\\\/posts\\\/7510\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/eoninvestment.com\\\/ka\\\/#website\"},\"author\":{\"name\":\"neoinsight\",\"@id\":\"https:\\\/\\\/eoninvestment.com\\\/ka\\\/#\\\/schema\\\/person\\\/9bb6a82e15a5231535d10233395afa89\"},\"headline\":\"\u10d8\u10e2\u10dd\u10e1 \u10da\u10d4\u10db\u10d0 \u10d3\u10d0 \u10d5\u10d8\u10dc\u10d4\u10e0\u10d8\u10e1 \u10de\u10e0\u10dd\u10ea\u10d4\u10e1\u10d8\",\"datePublished\":\"2025-10-31T19:14:26+00:00\",\"dateModified\":\"2026-04-11T05:16:08+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/eoninvestment.com\\\/2025\\\/10\\\/31\\\/itos-lemma-wiener-process\\\/\"},\"wordCount\":534,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/eoninvestment.com\\\/ka\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/eoninvestment.com\\\/2025\\\/10\\\/31\\\/itos-lemma-wiener-process\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/eoninvestment.com\\\/wp-content\\\/uploads\\\/2025\\\/10\\\/image-17.png?w=859\",\"articleSection\":[\"Options, Futures & Other Derivatives, - 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