[問數] Probability theory題目 - 香港膠登 HKGalden

[問數] Probability theory題目

1. Let `X~N(mu, psi^2)` and `K` be a constant. Calculate

(a) `E(I{X>K})` where `I` is the indicator function.

(b) `E("max"{K-X,0})`.

Hint Use `(K-X) = (K-mu) - (X-mu)`.

(c) `E(e^{tx})`.

Leave answers where appropriate in terms of the normal cumulative distribution function
`Phi(·)`.

2. A random variable `Y` is said to be lognormally distributed with parameters `mu` and `sigma` if
`log Y ~ N(mu, sigma^2)`. We sometimes write `Y ∼"lognormal"(mu, sigma^2)`.

(a) Using your answer to Question 1(c), calculate `E(Y)` and `Var(Y)`.

(b) Suppose `mu = log f - 1/2 sigma^2` for a constant `f>0`.
Calculate `E(Y)` and show that `Var(Y) = f^2 sigma^2` (`1+{sigma^2}/2 + {sigma^4}/6 + `higher order terms).

3. A sequence of random variables `X_0, X_1, ..., X_n, ...` is defined by `X_0 = 1` and `X_n = X_{n-1}xi_{n-1}`, where `xi_i`, `i=0,1,...`, are independent and identically distributed with

\[
\xi_i =
\cases{
1+u & \text{with probability } p\cr
1+d & \text{with probability } 1 – p,
}
\]

where `u>d`.

(a) Calculate `E(X_n|X_{n-1})`.

(b) Let `Y_n = {X_n}/(1+r)^n` for a constant `r>0`.
Find in terms of `p`, `u`, and `d` the value of `r` such that `E(Y_n|Y_{n-1}) = Y_{n-1}`.
Hence find in terms of `u` and `d` the range of such possible values for `r`.

(c) Suppose `E(Y_n|Y_{n-1}) = Y_{n-1}` for all `n`.
Use any result concerning conditional expectation to prove that `E(Y_n|Y_m) = Y_m` for all `n>m>=0`.

4. Show that the first three non-zero terms of the Taylor series for the normal cumulative distribution function `Phi(x)` around zero are

`Phi(x) = 1/2 + x/sqrt{2pi} - {x^3}/{6sqrt{2pi}}`.

Thanks #adore#

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20/05/15 12:27 AM

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#120/05/15 11:09 AM

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孖芝牛堡

雖然看不懂但好像很難 #adore#

#220/05/15 11:09 AM

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Khinchin

我一看就知這些題目出自 Blyth 那本 Introduction to Quantitative Finance.....

1(a) digital option, (b) put option, (c) mgf

2. lognormal

3. Martingale....

4. saddlepoint approximation

#320/05/15 12:34 PM

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係咪玩嘢呀?

其實你有咩學術嘢係唔知?
我發現你真係所有數嘅嘢都識

#420/05/15 12:47 PM

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畀你發現到

究竟呢啲數係咪prob theory入門就會學到？我最近為咗呢幾題(因為想鑽研下QF)，專登走去睇Ross嗰本Introduction to Probability Models，睇咗Expectation嗰個章節都唔識做第一題 :~(

#520/05/15 12:52 PM

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Angry Man

有無人可以 cap screen
無 plugin 唔想裝

#620/05/15 1:15 PM

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Angry Man

Open problem [slick]

#720/05/15 1:16 PM

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#820/05/15 1:25 PM

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贊助網站

#920/05/15 5:27 PM

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還有戲唱嗎

好撚難

#1020/05/15 5:51 PM

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眺久莉雅

相關major的intro就差唔多係呢d
不過呢d係香港的程度係略為難的程度
qf聞說係香港唔係做research其實優勢不大 [sosad]

btw果位巴打連出自邊都知 [sosad]

#1120/05/15 6:07 PM

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溢出神晞

in 計數佬 we trust #adore#

#1220/05/15 6:57 PM

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二進制

我純粹係為興趣先想計下QF啲數，睇下可唔可以應用喺market度

#1320/05/15 7:44 PM

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#1420/05/15 9:10 PM

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#1521/05/15 12:13 AM

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小矶健二

好多financial math 書都要 prob theory / measure theory 底
先睇 david williams, probability with mart
再睇 shreve stochastic calculus for finance i & ii 就perfect 池 la

上面D Statistics題目有2nd year底應該可以KO九成

Undergrad d quant finance多數係旁門左道, skip晒d measure theory 同stochastic analysis
典型例子就係麥當當本derivatives markets, brownian motion個到寫得好含糊

#1621/05/15 12:14 AM

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thx望下先

#1721/05/15 12:19 AM

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小矶健二

sor9ly ,有 D 199
我意思係你打好個底先諗 Quant fin
如果唔係，就睇麥當當本書，開頭ok，後面數學一多，求其就過左

#1821/05/15 12:20 AM

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麥當當本書即係邊本？

贊助網站

#1921/05/15 1:07 AM

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重治

PURE撚表示完全唔撚明題目 [sosad]

#2021/05/15 1:10 AM

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小矶健二

http://www.math.hcmus.edu.vn/~tdhien/Financial Modelling/derivative market.pdf

#2121/05/15 1:12 AM

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數佬唔係都有學少少prob theory咩？ [sosad]

#2221/05/15 1:12 AM

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重治

我STAT剩係讀到2000-LV 仲要F撚左 #lol#

我讀其他野根本唔需要STAT #lol#

#2321/05/15 1:15 AM

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thx 望下先

#2421/05/15 1:16 AM

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二進制

屌，我記得你好似講過肥咗probi [sosad]

你有冇學過Jacobian matrix?

#2521/05/15 1:18 AM

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