Codes are shown in the HTML file.<\/p>\n\n\n\n
The Markov Property states that the expected value of the random variable \\(S_i\\) conditional upon all of the past events only depends on the previous value \\(S_{i\u22121}\\). The implication of the Markov Property is that the <\/p>\n\n\n\n
best expectation of future price, \\((S_{t+k}\\), is the current price, \\(S_t\\). The current price contains all the information until now.<\/p>\n\n\n\n
$$ \\mathbb{E}(S_{t+k}|F_t)=S_t, \\text{ for k>=0} $$<\/p>\n\n\n\n
That is called the Martingale Property.<\/p>\n\n\n\n
The Quadratic Variation is defined as,<\/p>\n\n\n\n
$$ \\sum_{i=1}^k (S_i – S_{i-1})^2 $$<\/p>\n\n\n\n
In the coin toss case, each movement must be either “1” or “-1”, so \\(S_i – S_{i-1} = \\pm 1\\). Thus, \\((S_i – S_{i-1})^2 =1\\) for all “i”. Therefore, the Quadratic Variation equals “k”.<\/p>\n\n\n\n
<\/p>\n\n\n\n
Here below is the Code Realisation of a Brownian motion – Chapter 5.6 Paul Wilmott Introduces Quantitative Finance.<\/p>\n\n\n\n
Codes are shown in the HTML file. Markov Property The Markov Property states that the expected value of the random variable \\(S_i\\) conditional upon all of the past events only depends on the previous value \\(S_{i\u22121}\\). The implication of the Markov Property is that the Martingale Property best expectation of future price, \\((S_{t+k}\\), is the … Continue reading Brownian Motion to Normal Distribution<\/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\/4554"}],"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=4554"}],"version-history":[{"count":31,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4554\/revisions"}],"predecessor-version":[{"id":4596,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4554\/revisions\/4596"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4554"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4554"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4554"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}