Re: [作業] Ito's lemma

看板Economics (經濟學)作者pitching (p幣輸光光)時間17年前 (2008/10/17 08:05)推噓4(4推 0噓 3→)

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※ 引述《sw0079 (極限操作)》之銘言： : 學校:SFU(溫哥華西岸的學校) : 教師:R.Jones : 科目:Futures, options and other deriatives : 題目:Ito's lemma : 1. Ito's Lemma: : Let the price s(t) of a security follow the Ito process : ds =α s dt +δ s dz : (a) Use Ito's Lemma to determine the process followed by y(t)= ln s(t). S(t) follows the Geometric Brownian motion Since y(t)is a function of S,by Ito's Lemma We can derive the stochastic process followed by y dy = [αS*(1/S)+0+1/2*(δS)^2*(-1/S^2)]dt + δdz dy = (α-1/2*δ^2)dt+δdz----(a) that is,in discrete case ㏑S_t - ㏑S_0 ~ N( (α-1/2*δ^2)T,δ^2*T ) ㏑S_t ~ N( ㏑S_0+(α-1/2*δ^2)T , δ^2*T ) y(3) = ㏑S_3 ~ N( ㏑S_0 + 3*(α-1/2*δ^2) , 3*δ^2 )----(b) (c) Given S_0 ㏑S_t is normally distributed with mean ㏑S_0 +(α-1/2*δ^2)*T and variance δ^2*T X is a random draw from it and has been standardlized,that is ㏑S_t - E(㏑S_t) ㏑S_T - [㏑S_0 + (α-1/2*δ^2)*T] X = ------------------- = ----------------------------------- ~ N(0,1) √Var(㏑S_t) δ√T ㏑S_T = (δ*√T)*X + ㏑S_0 +(α-1/2*δ^2)T S_T = exp[ Xδ√T + ㏑S_0 + (α-1/2*δ^2)T] S_3 = exp[√3Xδ + ㏑S_0 + 3*(α-1/2*δ^2)] 好像是這樣已經忘的差不多了= = 有錯請指正 : (b) What is the probability distribution of y(3) in terms of y(0), and : (i.e., what is the type of distribution, its mean and its variance) : (c) If you were given s0 = s(0) and a random draw X from the type of : distribution in (b),but it was standardized to have mean 0 and variance 1, : how would you convert it into a random draw of s(3)? : (i.e., of the security price at time 3) : 翻譯: : 讓證卷價錢依照伊藤過程 : a.) 用伊藤過程決定算出y(t)=ln s(t) : b.) y(3)的可能分配是什麼? : (i.e., 怎樣的分配? 平均數跟變異數又是什麼?) : c.) 如果知道s0 = s(0) (s0的0是標在下面的) 而且現在從(b.)的分配裡面隨機抽取X, : 可是X有個mean = 0 and variance = 1, 那當s(3)的時候你的X會變成怎樣? : 我的想法: : 這題其實老師有在黑板寫可是我還是看不太懂 : 我自己用小畫家畫了一下解答 : http://0rz.tw/344Sx : yss=y的開兩次deriative : 請問這就是解答了嗎? 可是總覺得(b)還沒回答完,但是又不知道要怎麼代入 : 另外(c)老師是這樣寫的 : y=ln s --> s= e^y (^是開平方,所以e^2=e的二次方 etc) : convert to S(T) : EXP( ln S(0) + z) --> S(0)*EXP(z) : 這個我也看不懂伊藤過程好難啊這些東西到底要怎麼帶入才會算出答案呢?? -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 218.168.205.37 ※ 編輯: pitching 來自: 218.168.205.37 (10/17 08:13)