{"id":4597,"date":"2022-11-04T15:58:42","date_gmt":"2022-11-04T07:58:42","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=4597"},"modified":"2022-11-21T13:18:13","modified_gmt":"2022-11-21T05:18:13","slug":"a-bit-about-stochastic-calculus","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=4597","title":{"rendered":"A bit Stochastic Calculus"},"content":{"rendered":"\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-b3413ae6-1865-4022-897b-773097ce7b3f\" href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/11\/A-bit-about-Stochastic-Calculus-3.html\">A-bit-about-Stochastic-Calculus<\/a><a href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2022\/11\/A-bit-about-Stochastic-Calculus-3.html\" class=\"wp-block-file__button\" download aria-describedby=\"wp-block-file--media-b3413ae6-1865-4022-897b-773097ce7b3f\">Download<\/a><\/div>\n\n\n\n<p>See the HTML file for full detals.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Key Takeaways<\/h4>\n\n\n\n<ul><li>Property of <span class=\"katex math inline\">\\{W_t\\}<\/span>:<\/li><\/ul>\n\n\n\n<ol><li>$ W_t &#8211; W_s \\sim N(0, t-s) $<\/li><li>$(W_t &#8211; W_{t-1})$ and $W_{t-i} &#8211; W_{t-i+1}$ are uncorrelated. So, $\\int_0^t dW_u = \\sum^t dW_u = W_t$ <\/li><\/ol>\n\n\n\n<ul><li>For,<\/li><\/ul>\n\n\n\n<p>$$ dS_t = \\mu S_t \\ dt +\\sigma S_t \\ dW_t $$<\/p>\n\n\n\n<p>Why the Geometric Brownian Motion of <span class=\"katex math inline\">\\{S_t\\}<\/span> is designed in that form?<\/p>\n\n\n\n<p>The answer might be,<\/p>\n\n\n\n<p>$$ dS_t \/S_t = \\mu  \\ dt +\\sigma \\ dW_t $$<\/p>\n\n\n\n<p>$$\\int_0^t dS_t \/S_t = \\int_0^t \\mu  \\ dt + \\int_0^t \\sigma \\ dW_t $$<\/p>\n\n\n\n<p>$$ log(S_t) = \\mu \\  t +\\sigma \\ W_t $$<\/p>\n\n\n\n<p>Taking the first difference is similar to differentiation. (<span class=\"katex math inline\">d(log(S_T)) = log(S_t\/S_t-1) = log(1+r_t) \\ approx r_t<\/span>).<\/p>\n\n\n\n<p>$$ r_t = \\mu + \\sigma \\ \\Delta W_t $$<\/p>\n\n\n\n<p>The return, <span class=\"katex math inline\">\\{r_t\\}<\/span>, is equal to a mean, <span class=\"katex math inline\">\\mu<\/span>, plus a stochastic term. That is a random walk.<\/p>\n\n\n\n<ul><li>We apply a transformation, if <span class=\"katex math inline\">S_t<\/span> follows a Geometric Brownian Motion, then <span class=\"katex math inline\">f(S_t)<\/span> follows another, <\/li><\/ul>\n\n\n\n<p>In calculating <span class=\"katex math inline\">d f(S_t)<\/span>, we would get, (by Taylor Expansion)<\/p>\n\n\n\n<p>$$ df(S_t) = \\bigg( \\frac{\\partial f}{\\partial t} +  \\frac{\\partial f}{\\partial S_t}\\mu S_t +\\frac{1}{2}\\frac{\\partial^2 f}{\\partial S_t^2}\\sigma^2 S_t^2  \\bigg)dt + \\frac{\\partial f}{\\partial S_t}\\sigma S_t \\ dW_t  $$<\/p>\n\n\n\n<ul><li>A Special Form of <span class=\"katex math inline\">f(\\cdot)<\/span> is <span class=\"katex math inline\">f(S) = log(S)<\/span>,<\/li><\/ul>\n\n\n\n<p>$$ d\\ log(S_t) = \\bigg( \\mu &#8211; \\frac{1}{2}\\sigma^2 \\bigg)dt + \\sigma \\ dW_t $$<\/p>\n\n\n\n<p>Integrate the above equation from time 0 to time t, then we would get,<\/p>\n\n\n\n<p>$$ log(S_t) = log(S_0) + (\\mu &#8211; \\frac{1}{2}\\sigma^2)t + \\sigma W_t $$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>See the HTML file for full detals. Key Takeaways Property of \\{W_t\\}: $ W_t &#8211; W_s \\sim N(0, t-s) $ $(W_t &#8211; W_{t-1})$ and $W_{t-i} &#8211; W_{t-i+1}$ are uncorrelated. So, $\\int_0^t dW_u = \\sum^t dW_u = W_t$ For, $$ dS_t = \\mu S_t \\ dt +\\sigma S_t \\ dW_t $$ Why the Geometric Brownian Motion &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=4597\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">A bit Stochastic Calculus<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,18,26],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4597"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4597"}],"version-history":[{"count":37,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4597\/revisions"}],"predecessor-version":[{"id":4812,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4597\/revisions\/4812"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4597"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4597"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}