【转帖】【讨论帖】一些金工面试题（没答案）

是 否 +2 论坛币

k人 参与回答

经管之家送您一份

应届毕业生专属福利!

求职就业群

赵安豆老师微信：zhaoandou666

经管之家联合CDA

送您一个全额奖学金名额~ !

立即领取

感谢您参与论坛问题回答

经管之家送您两个论坛币！

+2 论坛币

有空谁解解。
可以看看国外数学要求不低啊。http://www.markjoshi.com/books/phpBB3/viewforum.php?f=7

I recently interviewed with some banks for front office or research quant position. My background is: Computer Science BS+MS+PhD.
I had read Mark's 2 books and Crack to prepare for the interviews.
I got several problems from Crack. Below I am putting some of the problems that I hadn't seen before. At one place, I signed an NDA so those questions are not there.

1.If a is a column vector, then how many non-zero eigenvalues does the matrix aa' have? what are the eigenvalues? What are the corresponding eigenvectors? What are the eigenvectors corresponding to the zero eigen values?
2. if w is an standard brownian motion, is w^3 a martingale?
3. prove that the price of a call option is a convex function of the strike price.
4. Suppose you are throwing a dart at a circular board. What is your expected distance from the center? Make any necessary assumptions. Suppose you win a dollar if you hit 10 times inside a radius of R/2, where R is the radius of the board. You have to pay 10c for every try. If you try 100 times, how much money would you have lost/made in expectation? Does your answer change if you are a pro and your probability of hitting inside R/2 is double of hitting outside R/2?
5. Suppose you have an old machine, which does not have a capability to multiply two numbers, but does have a capability to square a number. It also has addition and bit shift operators. Implement multiplication and division (integer division only)
6. Again the previous question, now you dont even have the squaring capability, but only bit shift, and addition. Implement multiplication
7. what do you know about const.
8. What is the problem with virtualization from the point of view of optimization. What can a compiler do when a function is not virtualized?
9. How is virtuality implemented in C++
10. integrate log^n x.
11. prove, from first principles, the differential of e^cos(x).
12. given the matrix A=(5 -3;-3 5), find a matrix M, such that A=M*M. Now find a matrix M such that A=M'*M
13. Suppose x_1, x_2...x_n are IID from [0,1] uniform interval. What is the expected value of the maximum. What is the expected value (max-min).
14. Suppose I have a routine that can sort n numbers in O(n) time. Prove me wrong.
15. Suppose you have the implied vol curve for call options. What is the arb free price of a digital struck at k given this implied vol curve.
16. Pricing a barrier option with a discrete barrier.
17. Distribution of the max of a brownian motion. Use it to price digital american and european call options.
18. Explain put call parity.
19. At one interview, I was asked to explain, in great detail, whatever I knew about the current credit problem (for about 25 mins). I did well only because I was reading the WSJ.
20. Given a fair coin, what is the expected number of trials you need to go to get 2 consecutive heads. 3 consecutive heads. generalize to N.
21. What is the variance on the number of trials in the question above?

扫码加我 拉你入群

请注明：姓名-公司-职位

以便审核进群资格，未注明则拒绝

分享0 收藏3 回帖

关键词：面试题 讨论帖 Eigenvectors Differential Optimization background interviews recently research position

顶一下，金融工程的基础

Solutions to problems listed above:
1) aa' has two eigenvalues: zero and a'a.  the eigenvector of a'a is a, and the eigenvector of zero is the vectors orthogonal to vector a.
2) w^3 is not a martingale, but is a semi-martingale.
3) We note the price of a call with strike K as C(K). if we buy two calls with strike K_1 and K_2, and short two calls with strike K_3=(K_1+K_2)/2, then pay C(K_1)+C(K_2)-2*C(K_3) at first. At maturity, we will receive an non-negative payoff. So the initial value of this portfolio is non-negative. So we get the convexity.
4) I don't fully understand this question. It is easy to get the distribution of distance. And we know that the probability that dart falls inside the radius of R/2 is  p=1/4. So the expectation of gain is
     \sum_{n=0}^{n=100} C_100^n p^n (1-p)^{100-n} ([n/10]-(100-n)*0.1), where [x] means the integer part of x. And i think this expectation is negative, because we get 1 dollar for tens times,
     which means that every time we gain 10c, and we lost 10c if the dart is out of radius of R/2. So if the probability changes, so the result.
5)  Multiplication: x*y = {(x+y)^2-(x-y)^2}/4
     Division:  We just have to figure out 1/N, where N is an integer. We take the integer n satisfying 2^(n-1)<N<2^n.
                     So we have 1/N=(1/2^n)*(2^n/N), and addition can be applied to get the first number of 2^n/N, so we go on to repeat this process.
6) We take two positive integer x and y, and present x as x=2^N+x', where x' is positive and smaller than x, y as y=2^M+y', where y' is positive and smaller than y.
    So x*y = (2^N+x')*(2^M+y')=2^(N+M)+2^N*y'+2^M*x'+x'*y', and we go to to compute x'*y' until x' and y' become 0 or 1.
7) const const const, find a C++ book......
8) On my point, i will use a pure virtual class to present any function that we might to optimisation. And we can implement algorithms of optimisation. Once we want to optimise a function, we
    just present the function as a subclass of pure virtual class. If this, we don't have to rewrite algorithms of optimisation. That's what i think.
9) Find a book and check it.
10) Change variable as y=logx
11) By the proof of the derivative of compounded function. We can find it at any mathematics analyses book.
12) A is definitive, so use the Cholesky decomposition.
13) the expectation of max is N/(N+1), so the min is 1/(N+1).
14) An interesting question. I find one wich is not so convictive. Offer a better one!
15) Price of digital is the derivative of call w.r.t strike. So with imp vol, we can present it as vega* d vol / d strike
16) PDE
17) I am still working on it. And i know other method.
18)
19) Wall street Journal
20) We note the expected number of trials of N consecutive heads as T(N).
      So we has T(N)=0.5*(1+T(N))+0.5*(1+T(N-1)).
21) I have no idea to solve this problem, maybe Monte Carlo?????

Yes, this is my answer. I think it will be an easy work to find lots of grammer mistakes in my answer. So just ignore them.
And I am looking for an quantiative position for several months. But unfortunately i don't find a favorable one  even now.
Do you guys know somebody who work in this domain and could he do me a favor to recommand me?
I really need help now.......