i***0
发帖数: 55 | 1 【 以下文字转载自 Physics 讨论区 】
发信人: iq200 (iq 200), 信区: Physics
标 题: 求解蒙特卡洛方法的题目 1
发信站: BBS 未名空间站 (Tue Dec 25 04:54:19 2007)
1. Show that if a discrete time Markov chain satis es detailed balance and
C(t) is the time t lag auto-covariance function in equilibrium, then C(2t)
>=for all t. |
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n*******r
发帖数: 60 | 2 My boss told me to calculate the correlation between Y and X
with cov(X,Y)/(std(X)*std(Y)),
but the trick is all the covariance and stds are calculated with zero mean.
Could anybody tell me what is the inisght behind it. Thanks! Bow<< |
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n*w
发帖数: 41 | 3 Thanks.
There are two rounds phone interviews:
(1) c++ questions:
How to implement java interface in c++?
How to write pure-virtual function?
Difference between Java and c++
Heap and stack
How to implement SmartPointer
Explain polymorphism
How can c invoke c++ code? Why?
(2) Java questions:
what is jvm?
explain GWT
what is ajax? (non-java)
How to invoke c/c++ in java?
Explain SWing
What is java reflection?
(3) Math
chi-square distribution
covariance
some statistical test methods
(4) Finance
What |
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i***6
发帖数: 442 | 4 maybe you could incorporate GARCH or EWMA into Monte Carlo simulation.
However, the most widely used method for computing VaR in London is the
simple variance/covariance method, according to Jorion. |
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i*****r
发帖数: 1302 | 5 很奇怪,我用matlab写的,算上各种variance,covariance的假设,未知量少算有10来多个.
分别给初始值,然后最大化logL应该就完了.但是问题那个logL是经过一系列矩阵运算再
相加得出来的. 然后就卡在那一直算了,状态一直是busy,也不知道要算多久,10几个
variable其实也不算多,像optimization这种都要几十个了.怎么会算这么久呢? 怎么
Eviews算得这么快呢? |
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b***k
发帖数: 2673 | 7 ☆─────────────────────────────────────☆
Jadeson (Jadeson) 于 (Tue Nov 4 17:42:57 2008) 提到:
一直没有搞明白：
equilibrium returns=risk aversion coeffficient * Matrix of covariances *
Equilibrium market capitalization weights for each asset
这最后两相的乘积到底如何理解呢？
market capitalization weights是不是市场上每个asset所占的比重呢？ 但这个和
asset return怎么联系在一起的呢？
谢谢。
☆─────────────────────────────────────☆
ryou (zzz) 于 (Tue Nov 4 20:17:50 2008) 提到:
Not my area. but here are my 2 cents.
Do singular value decomposition on COV |
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M****n
发帖数: 84 | 8 请大伙指教。简单情况：06年Business Phd. 近了一个buyside institutional fund做
了几年。主要是equity quant, 不涉及derivatives。日常工作主要是和matlab, sas,
sql, python，S+, excel vba 这些东西打交道，不会java c++. 工作内容是building
factors, portfolio optimization, covariance matrix construction, performance
attribution. 现在在一个地方呆就了想换个环境多学点东西。当然工资涨些就更好了
。一般的数据库，软件之类的当然都是多少会些。还是想换到一个buyside mutual
fund或者institutional fund里面去。要去interview了请问，大概应该给人家说多少
钱呢？请给个指教。 |
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n*********3
发帖数: 21 | 9 谢谢详细解释。
我用的是principle component analysis.
我的做法是用princomp method in MATLAB to derive eigenvalues and
eigenvectors. Then find the most important eigenvalues which represents
the covariance matrix. Hence, r = 2.
Yes, with 2 known eigenvectors (corresponding to the selected
eigenvalues)
Then, what is the next? to derive the parameters given r = 2?
.,
independent |
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a*******h
发帖数: 123 | 10 看我的回帖阿。
min(t,s)是B_s 和 B_t 的 covariance. |
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b*****t
发帖数: 10 | 11 被问到过。
如果 driven by 同一个 wiener process.两个的确不独立.
请问 如何 求这两个随机变量的 covariance?
有人教我用过 双重二元随机积分 然后 出现了 一个 delta 函数
dW_t dW_s=delta(s-t)dt
明白了 就是 不记得 那本书上 有教过？
大牛们 给指点指点！ |
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w****j
发帖数: 6262 | 12 你是说计算两个wiener process的covariance?
Assume t>s
E(W_sW_t)=E(W_s(W_s+W_t-s))=E(W_s^2)+E(W_sW_t-s)=s |
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S*****y
发帖数: 567 | 13 比如说算出100只股票的covariance matrix作为risk，minimize it.
再算出它们在未来一段时间内的expected returns，设定这个式子大于某个值。
然后再设定所有portfolio的算数和是1，000，000；投入在每只股票上的在[-100000，
100000]
有没有办法控制所有的负的portfolio的和是一个固定的值，正的portfolio的和是固定
值呢？
先谢谢大牛们拉。。。 |
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y******g
发帖数: 41 | 14 Thanks for your input. Covariance = 0 means that E(XY) = E(X)E(Y), is this
what you mean by orthogonal first moments?
yup, they are independent only when they are normals, I get that part.
Thanks :) |
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t*******8
发帖数: 170 | 15 听说这里有很多大牛，所以来碰碰运气哈。
遇到一个难题了：
上次没有说清楚，表达能力太差，这次再说一遍：
其实是有１２０条序列，每条序列有１００个值，每个值的取值是２０种可能性的一种（如：Ａ，Ｂ，。。。,O）。问题是：如何确定序列的第３个位置和第６个位置的之间的关系？我想过用covariance,不过那把这些Ａ，Ｂ，。。都量化了，本身它们只是不同而已，就是ｃａｔｅｒｇｏｒｙ，没有量化关系，所以结果不对。有什么方法能够确定第３和第６位置的两列数据的关系呢？
position 1 2 3 4 5 6 7 8
obs
1 A D G A E F H A
2 F C G N L N H O
3 D D I J K F M A
.
.
.
120
这些取值都是２０种可能性的一种，这和它们的distribution的关系，我还没有想清楚。
谢谢先！ |
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z****g
发帖数: 1978 | 16 I can only come to conditional distribution of the 6th column conditional
on the 3th column. However, it seems not enough data is presented.
种（如：Ａ，
Ｂ，。。。,O）。问题是：如何确定序列的第３个位置和第６个位置的之间的关系？我
想过用
covariance,不过那把这些Ａ，Ｂ，。。都量化了，本身它们只是不同而已，就是ｃａ
ｔｅｒｇｏｒ
ｙ，没有量化关系，所以结果不对。有什么方法能够确定第３和第６位置的两列数据的
关系呢？ |
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c*********g
发帖数: 154 | 17 看看Hidden Markov Model吧。
种（如：Ａ，Ｂ，。。。,O）。问题是：如何确定序列的第３个位置和第６个位置的之
间的关系？我想过用covariance,不过那把这些Ａ，Ｂ，。。都量化了，本身它们只是
不同而已，就是ｃａｔｅｒｇｏｒｙ，没有量化关系，所以结果不对。有什么方法能够
确定第３和第６位置的两列数据的关系呢？ |
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y**y
发帖数: 25 | 18 X=int_0^t Bs ds
Y=int_0^t s dBs
Here Bs is standard brownian motion
what is cov(X,Y)? (in terms of t) |
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b***k
发帖数: 2673 | 20 我可以做出这个题，答案是t^3/6
但过程非常繁琐，一步一步推导，
要两次分部积分，ito isometry，fubini theorem等
想问一下有没有更加简单直观的方法直接给出答案的？
其实这个题很有意思，这里X和Y都具有如下性质：
E(X)=E(Y)=0
D(X)=D(Y)=t^3/3
且都是normal distribution random varible，对不对？
Old.版内查。 |
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x******a
发帖数: 6336 | 21 如果知道D(X)和D(Y),可以考虑
X+Y=tB_t,
t^3=E((X+Y)^2)=2E(XY)+2t^3/3,
E(XY)=t^3/6.
不过D(X)怎么才能更简单直观?
我可以做出这个题，答案是t^3/6
但过程非常繁琐，一步一步推导，
要两次分部积分，ito isometry，fubini theorem等
想问一下有没有更加简单直观的方法直接给出答案的？
其实这个题很有意思，这里X和Y都具有如下性质：
E(X)=E(Y)=0
D(X)=D(Y)=t^3/3
且都是normal distribution random varible，对不对？
Old.版内查。 |
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w**********y
发帖数: 1691 | 22 (d BsS)=Bs ds + s dBs, 两边同时积分,so tBt=X+Y, so X=tBt-Y
cov(X,Y)=cov(tBt-Y,Y)=t*cov(\int dBs,Y)-cov(Y,Y)
here:
cov(\int dBs,Y)=cov(\int dBs,\int sdBs)=\int s ds=t^2/2
cov(Y,Y)=\int s^2 ds=t^3/3
so cov(X,Y)=t*t^2/2-t^3/3=t^3/6
actually, cov(\int f(s)dBs, \int g(s)dBs)=\int fg ds (?should be right..) |
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b***k
发帖数: 2673 | 23 3x，你这个方法推导E(XY)倒是容易很多。
D(X)和D(Y)我也不能直观得到，
但无外乎用定义加ito isometry就可以推出。
然后用多了，结论就很容易记住了。 |
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s*********8
发帖数: 107 | 24 LZ你这个什么职位啊，怎么狠多面试都问这种calculus的题，谁还记得书上的东西啊 |
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k***p
发帖数: 115 | 25 关于D(X),如果不用分布积分，可以这样考虑
EX^2= E (\int_0^t B_s \,ds \int_0^t B_r \,dr)
=\int_{[0,t]\times[0,t]} min{r,s} \,dr \,ds
=2 \int_0^t\int_0^s r \,dr \,ds
=2 t^3/6 =t^3/3.
不过这样没有把问题变得简单。 |
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b***k
发帖数: 2673 | 26 我知道这两个是基于isometry，而且是在 E(X)=E(Y)=0 的前提下
你这个结论是基于什么理论呢？ |
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r*****r
发帖数: 630 | 27 due to isometry:
E(int_0^t f dB_s int_0^t g dB_s)= E (\int_0^t fg ds) |
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b***k
发帖数: 2673 | 28 55555555
ms是我isometry理解不够，可是我看到的都是带平方的情况，
你这个式子好像更加general的情况，
请问有reference吗？
我在Oksendal的书上没有看到啊。 |
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x******a
发帖数: 6336 | 29 consider
E[(X+Y)^2],
where X=\int_0^t f dB_s, Y=\int_0^t g dB_s. |
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b***k
发帖数: 2673 | 30 got you, cool!
thank you very much. |
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p*****k
发帖数: 318 | 32 jpmx, as already pointed out, the answer is 0.
the rule of thumb is:
if there are odd number of normal r.v.s, the expectation of
their product (central moment) vanishes;
[of course it also applies to r.v.s with symmetric p.d.f.]
if the # is even, then use the tree diagram to reduce to
products of the elements in the covariance matrix. |
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s******a
发帖数: 11 | 33 一家华尔街的Hedge Fund。位置是High Frequency Trading Quant
1. 如果X是一个covariance matrix, 问exp(X)是什么。-- 这题我不会，我估计是用泰
勒级数展
开，但是，X^2， X^3，...是什么，我也不太清楚。我觉着exp(X)没有实在的物理意义。
2. 设计一个算法，找到n个数当中第k大的数。问最好的算法的复杂度是多少。 |
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t*****j
发帖数: 1105 | 34 1. covariance matrix的定义能详细给出么？俺只修过概率论基础课。
2. quick sorting，复杂度O(nlogn)。
义。 |
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t********a
发帖数: 810 | 35
义。
1. the hint is to think in terms of pca. exp(X) have the same eigenvectors
as X, and its eigenvalues are the exp of X's eigenvalues. Therefore exp(X)
represent the covariance matrix of data consists of the same principle
components, but the variance of the projections on each principle component
is exp of the value before. |
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z****g
发帖数: 1978 | 36 1. covariance matrix is always real symmetric, which is easy to calculate
by Taylor expansion. Actually, in math, exp(x) is DEFINED by Taylor
expansion.
2. Find the max is O(logN), so, depends on K, the worst is O(N).
义。 |
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J**********g
发帖数: 213 | 37 Just done the interview with FID at MS. Messed it up. I thought he would ask
about the techinical problems, and it turned out that brainteaser and
statistics problem. I guess he had database to pick up problems from.
first question.
1. a stock has a price of 100 and a bond is 85. Tomorrow the stock price
would be 110 or 90, and bond price would be 85 and 90 correspondingly. price
a call option with strike price of 100.
Using replicating portfolio. Answer is 6. Attention: no risk-neutral involved... 阅读全帖 |
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c******r
发帖数: 300 | 38 En, I was treating the covariance to be 1/2 instead of 1/sqrt(2), but the
idea applies similarly, let
X = 1/\sqrt{2}(U-V)
Y = U
P(X>0|Y<0)=P(U>V,U<0)/P(U<0)= (1/8)/(1/2)=1/4
I think in general for rho > 0, the result will be
\frac{1}{pi}arctan(rho/\sqrt(1-rho^2)) |
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J**********g
发帖数: 213 | 39 You are absolutely right. This is the first company that I got interview
from, and I did not prepare for brainteaser at all, and I knew only basic
statistics.
For the second one, I would guess zero, and this is definitely not the answer he was looking for. but I wanted to give an
vigorous mathematical explanation anyway (he definitely gonna ask me how I got it
after I told him the result, and I cannot say it's my guess).
For the third one, I tried two ways:
1. use chosky decomposition to transfo... 阅读全帖 |
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i**w
发帖数: 71 | 40 背景:fresh physics PhD
还没offer,但年前基本上不会再折腾了。
有重复。基本上都是很标准的题。
简单的题如果人家想问倒你也是很容易的。
面试书recruiter推荐
1) Mark Joshi: "Quant Job Interviews: Questions and Answers". I have
heard very good things about this book.
2) Xinfeng Zhou, "A Practical Guide to Quantitative Finance Interviews"
个人觉得非常有用， 大部分问题都在这两本上。
算法，C++, stochastic calculus 就看比较标准的几本。
- sqrt(i)=?
- You and me roll a dice,first one gets a six wins. You roll first. what
is the probability of you winning?
- A stair of n steps. Each time you st... 阅读全帖 |
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w**********y
发帖数: 1691 | 41 多谢分享.大概做了做..欢迎补充和指正.
- sqrt(i)=?
e^{\pi/4 i} or - e^{\pi/4 i}
- You and me roll a dice,first one gets a six wins. You roll first. what
is the probability of you winning?
P(I win) = P(Y !win and I win) = 6/11
- A stair of n steps. Each time you step up 1 or 2 steps. How many
different ways are there to reach the top? what is the asymptotic limit?
Fibonacci sequence ..limF(n)/F(n-1)==x for n>2, solve x, and F(n) ~ x^{n-1}
- Moment generating function of standard model.
statistic book…
- Write a si... 阅读全帖 |
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t*******y
发帖数: 637 | 42 第二题应该是6/11吧
能讲讲这个吗? - X1 and X2 are independent random variable with pdf f and g.
what is what is the pdf of X=X1+X2
Jacobian matrix for X1+X2 and X1-X2..
多谢分享.大概做了做..欢迎补充和指正.
- sqrt(i)=?
e^{\pi/4 i} or - e^{\pi/4 i}
- You and me roll a dice,first one gets a six wins. You roll first. what
is the probability of you winning?
P(I win) = P(Y !win and I win) = 5/6*1/6
- A stair of n steps. Each time you step up 1 or 2 steps. How many
different ways are there to reach the top? what is the asymptotic... 阅读全帖 |
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i****k
发帖数: 39 | 43 From historical data, we can calculate covariance matrix among a, b, c. then
. Then minimize the variance of the portfolio. |
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w**********y
发帖数: 1691 | 44 自信之前也得double check一下吧..
你知道correlation的定义么?covariance呢? 任何一个random variable跟自身的
correlation都是1这是个常识吧,你觉得呢?
你再去check一下martingale的定义..然后你再check一下2*W_{t/4}是不是martingale;
假设你定义那个 X_t = tW_t + \sqrt{1-t^2}B_t..
你算算 E(X_{1/2} | F__{1/4})是什么..是X_{1/4}么? |
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t***l
发帖数: 3644 | 45 correlation确实我定义搞错了，我一直想的是covariance。
但是你说的那个 2*W_{t/4}是不是martingale呢？可以说是，也可以说不是，这得看你
取得F_t是什么了，具体你看我之前回avidswimmer的一个证明。
至于你说X_t是不是martingale，这还比较显然，两个martingale的linear
combination一定是martingale的。
martingale; |
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a*********r
发帖数: 139 | 46
Quant is a technical job anyway. Basic rigorous is necessary in any
technical job. Actually I train quants. Which program you were from? If you
were our quant program. you would have already failed.
Another piece of advice, don't just focus on technical ability, improve your
personality.
When something is well-know, it's not obvious. People may think it's cross-
variation, may think it's covariance function, may think...etc. Your
ignorance makes this obvious to you, but not to others.
Confidenc... 阅读全帖 |
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r**a
发帖数: 536 | 47
不对吧。X(T) is a mean zero Gaussian process with covariance E(X(T)X(S))=\
int_0^{S\wedge T}g(t)^2dt. This can be shown via the definition of Ito
integral. If I am wrong, please correct me. Thanks. |
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EM
发帖数: 715 | 48 Is the result to Q1 (sqrt(a)-sqrt(b))/(2sqrt(a)+sqrt(b))*(2X+Y)
My solution is to find a linear combination of X and Y such that its
covariance with 2X+Y is zero, so independent of 2X+Y
An example is sqrt(b)X-sqrt(a)Y
then write X-Y in terms of 2X+Y and sqrt(b)X-sqrt(a)Y
in |
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k*******d
发帖数: 1340 | 50 En... Got some idea.
请问有什么教材或者是资料详细介绍这些的吗？
are
for
APT
missing |
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