Ito's lemma - 經濟

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).

(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??





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