More Questions !

Discuss all those difficult questions here.

More Questions !

Postby f0X_in_s0X » Thu Nov 20, 2008 8:08 pm

here are some questions I was asked on a test, for a quantitative analyst position

1) Let a discrete distribution have probability function pk , and let a second discrete distribution have probability function qk . Then the relative entropy of p with respect to q is defined by ,
d = sum_k (pk log_2(pk/qk))

Prove that d >=0 and d=0 iff pk=qk

2) A question based on the knapsack algorithm

3) A and B play a game. A wins a game with prob p and B with q. A player needs to win 2 consecutive games to register a win
Calc the prob that A wins in less than or equal to 100 games
............that play continues beyond N games,

4) A rather difficult problem askin to prove that the rank of a given matrix was always atmost two. I cant seem to recall the structure of the matrix.

5) Xi, i=1 to n iid Normal variables
Calc P (sum_i Xi <= n)

and 2 others...
f0X_in_s0X
 
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Re: More Questions !

Postby salientxu » Mon Jun 21, 2010 3:47 pm

1. Suppose the prob space has only finite (say n) states. Here is one solution.
View d as a function of $(q_1,\cdots, q_{n-1})$ with $0<=q_i, q_1+...+q_{n-1}<=0$. The extremal values can only be obtained at boundary points or the interior points with $\partial d/\partial q_i=0$. Compute $\partial d/\partial q_i=-p_i/q_i+p_n/q_n$. Hence if an extremal point is interior then $p_1/q_1=p_2/q_2=...p_n/q_n$. Then $p_i=q_i$. Now consider boundary points. These points must have at least 1 $q_i$ to be 0. Hence $d=+\infty$ at boundary. Thus the minimal point of $d$ is 0 when $p_i=q_i$.
This method does not directly apply when we have infinite states.
2. do not know what a knapsack algorithm is
3.This problem may be rephrased as the problem of tossing a biased coin with two consecutive heads: suppose that the probability of obtaining a head is p; what is the probability of two consecutive heads before time 100. Consider the inverse problem. Let the probability of not getting two consecutive heads before time $n$ be $P_n$. Then we obtain a recursive formula for $P_n$:
P_n=(1-p)P_{n-1}+p(1-p)P_{n-2}
with P_{1}=1, P_2=1-p^2.
Solve $P_{100}$ and the required probability will be 1-P_{100}.
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Re: More Questions !

Postby bigperm » Tue Dec 21, 2010 12:18 am

1) This is the Kullback-Leibler divergence written [; D_{KL}(p||q) ;]. The most common proof involves Jensen's inequality. We have
[; \sum_k p_k \log (p_k/q_k) = -\sum_k p_k \log(q_k/p_k) >= -\log(\sum_k p_k q_k/p_k) = -\log (\sum_k q_k) = -\log 1 = 0 ;].
The inequality is due to Jensen and by the strict convexity of -log the inequality is equality iff [; p_k = q_k ;] for all k. This proof works if the sum is replaced by an integral.

5) There is no general method that works for all random variables (asymptotic methods e.g. Cramers theorem do apply however). Since these rv's are normal, we know that their sum is also normal. It has mean zero and variance n. This defines the distribution of the sum completely, and the probability is given by the appropriate integral.
--BTW, special [; formatting ;] for mathematics is meant to be read with http://thewe.net/tex/
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Re: More Questions !

Postby nablaQuadrat » Fri Feb 25, 2011 8:52 am

I have a couple of them :

Firm 1 :

1. Explain, what LIA is. Derive the LIA rate and explain, why it does or does not depend on the volatility.
2. If the YEN is getting stronger against USD, is the PRDC worth more ? What will ultimately happen ?
3. What angle do the short and the long hand of the clock close at 15:15 ?
4. Derive the HJM drift condition. Why is it important ? Why is the HJM paper considered so revolutionary ?
5. Integrate int_{0}^{T}W_s ds
6. Elaborate on ADI.
7. You work with a 2Y->10Y CMS product. To what do you calibrate it ? Why ?
8. YOu have a swap : at T_1 you pay X1 USD, at T2 you receive T2 USD. You work in the EUR world. (1) What is the expected value of T(0) of this swap ? (2) To which volatility is this swap more sensitive ? (3) How would you price an option to enter this swap, in the period 0 < t < T1 ?
9. Tell me a method to calculate the reserves for a PRDC.
10. If you have a 3 factor model for a PRDC then (a) what is the big danger of this model ? (2) what are the three factors ? (3) How would you cricumvent the "big danger"? (4) What are the disadvantages of your workaround ?
11. Where would you use Hull White and where would you use BGM and why ?

Firm 2:
1. The interest rate is 5%, vol is 0, Spot is 100, strike is 100, maturity is 1Y. Price this call option quickly.
2. Is (W_t)^{\pi} a martingale ? ( W_t is the Brownian )
3. You fall asleep on the beach and you wake up in some time. What is the expected value of the minute hand of the watch and what is its variance ?
4. Tell me about feynman Kac.
5. Tell me, how do you price a swaption with Monte carlo in a LIBOR Market BGM setting ?
6. If you price a CMS spread option in BGM, what are the problems which may occur ?
7. What is an inline function ?

The moral is :
1. The stuff is realtively simple. If you exercise and keep track of the literature ( and have at least SOME talent and feeling ) it if a.s., that you get a position.
2. The interviewers are helpful, mostly. They also want you to answer them and if you take their hints ( and think together with them ) they appreciate it a lot.
3. DO get worried, if HR is involved. Then your soft skills will be inspected, not just your hard skills. And I experienced, that if you perform rerasonably well on the quant interviews, then you could, by your mindset, easily fail with a blonde HR lady.
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Re: More Questions !

Postby singlau714 » Fri Feb 01, 2013 8:32 am

nablaQuadrat, please tell me that "they are not interviews for an entry level quant position" ?? And they are just for the fixed-income or related division, right?
They seem very hardcore... at least to me...
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