{"id":2261,"date":"2023-02-18T05:23:16","date_gmt":"2023-02-18T05:23:16","guid":{"rendered":"https:\/\/www.goodacademic.com\/blog\/questions\/geometric-brownian-motion-question\/"},"modified":"2023-02-18T05:23:16","modified_gmt":"2023-02-18T05:23:16","slug":"geometric-brownian-motion-question","status":"publish","type":"questions","link":"https:\/\/www.goodacademic.com\/blog\/questions\/geometric-brownian-motion-question\/","title":{"rendered":"Geometric Brownian Motion question"},"content":{"rendered":"<div class=\"col-sm-12 messageContent\">\n <b>Learning Goal: <\/b>I&#8217;m working on a finance question and need guidance to help me learn.<\/p>\n<p>Problem 6Let S1(t) and S2(t) be the prices of 2 securities obeying geometric Brownian motions:dSi(t) = (mi \u00e2\u02c6\u2019 qi) Si(t)dt + \u00cf\u0192i Si(t)dBi(t), i = 1, 2where qi is the annual dividend yield rate, \u00cf\u0192i is the annual volatility, and mi \u00e2\u02c6\u2019 qi is theexpected continuously compounded rate at which the mean price of the ith securityincreases. Suppose that B1(t) and B2(t) are are independent standard Brownian motions.Let f (t, x, y) be a twice continuously differentiable (non-random) function. Establish a2-dimensional It\u00cb\u2020o formula for f (t, S1, S2).Hint: Start with Taylor\u00e2\u20ac\u2122s formula for f (t, x, y) and use the multiplication rules:dBi(t) dBj(t) = \u00ce\u00b4ijdt, dBi(t)dt = 0, (dt)a = 0 for a &gt; 1,where\u00ce\u00b4ij =\u00ef\u00a3\u00b1\u00ef\u00a3\u00b2\u00ef\u00a3\u00b31 if i = j0 if i 6= j.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Learning Goal: I&#8217;m working on a finance question and need guidance to help me learn. Problem 6Let S1(t) and S2(t) be the prices of 2 securities obeying geometric Brownian motions:dSi(t) = (mi \u00e2\u02c6\u2019 qi) Si(t)dt + \u00cf\u0192i Si(t)dBi(t), i = 1, 2where qi is the annual dividend yield rate, \u00cf\u0192i is the annual volatility, and [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"closed","template":"","meta":[],"disciplines":[676],"paper_types":[],"tagged":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/questions\/2261"}],"collection":[{"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/questions"}],"about":[{"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/types\/questions"}],"author":[{"embeddable":true,"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/comments?post=2261"}],"version-history":[{"count":0,"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/questions\/2261\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/media?parent=2261"}],"wp:term":[{"taxonomy":"disciplines","embeddable":true,"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/disciplines?post=2261"},{"taxonomy":"paper_types","embeddable":true,"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/paper_types?post=2261"},{"taxonomy":"tagged","embeddable":true,"href":"https:\/\/www.goodacademic.com\/blog\/wp-json\/wp\/v2\/tagged?post=2261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}