Some interview Questions..

Interview Questions that are brain teasers that I encountered:

(read wip = what is probability)

1. x1,x2,x3 iid from roll of dice. wip that sum(w^xi)=0 where w is
complex cube root of 1.

2. Regression-x,y are financial time series. x=b1y+c1; y=b2x+c2. What
is b1/b2 approx.

3. Price of call when: sigma = 30% vs. E(sigma)=30%. which is
greater-(Ans: depends on curvature)

4. Early excercise of American call? whats better than early excercise
(Sell the call/stock)

5. Derive E(coin tosses for 1 head| Pr(success)=p). Do the same for 2
consecutive heads.

6. integrate (-inf to inf)(exp(-x^2) cos x) (Ans: Helps to know complex numbers)

7. What are eigen values of exp(A) where A is a matrix.

8. 10 Pirates with a ranks 1 to 10 have a loot of 10 coins. Assume
infinite intelligence and rational. Highest rank proposes division. If
accepted (>=50% including highest rank) they divide otherwise kill
highest rank. Whats the division. (Ans: start with two pirates and
work your way up).

9. 2D BM. (1,1) is starting point. WIP that you hit +ve x-axis before
-ve. (Ans: use symmetry of y-axis. final answer =3/4). Extend it to
arbitrary points.

10. An infinite matrix filled with +ve integers s.t. each integer
occurs atmost 10 times. Prove there exist i,j s.t aij>ij for some i,j
(Ans:Prove by contradiction. Use entries that are constrained to be
less than n i.e. ij10n for large n).

11. what is i^i where i^2=-1

12. If you can only go up or right from origin how many ways can you
reach (5,5).

13. Draw a graph of (1/(2-x))*exp(1/x).

14. You randomly cut a 1m long rod at 2 places. WIP that you can make
a triangle

15. A,B play a game. A picks a number between [1,10]. Gives it to B. B
pics [1,10] and adds it to the existing sum and gives back to A. The
person to first get 60 loses. Who wins.

16. C++: how do you allocate memory to an object.

Here are some slightly higher quality questions:

1. If the price of an European binary call is x what is the price of American binary.

2. If X and Y follow a Geometric Brownian motion show that X/Y also follows a GBM.

3. How do you create an arbitrage when the Put price is not a convex function of the strike.

4. Does short selling restriction change the put call prices in absence of trasaction costs.

5. WIP that on a infinite sequence of die throws you get 2 evens in a row before you get 3 or 5.

6. Can you create an event with a probability of m/n (a rational) with a fair coin.

7. If you have a fair die how would you estimate pi.

8. how would you replicate a digital option using vanillas.

9. Solve the OU process(mean reverting weiner process).

Interview Questions on Brownian Motion: I got them at a final round BB bank most of these are conceptual..

i. KL transformation on BM

ii. Explain local time.. Give the SDE satisfied by |B| or Tanakas formula

iii. Derive Focker Plank Equation.. Backward kolmogorov etc..

iv. Some thing to do with method of iterated images I didnt fully understand it, but it apparently gives the probablity that a BM will not hit two boundaries in a given time had no idea how to do it.

v. Questions on HJM model and on some mathematics behind it. Something about how the drift term represents correlation between two quantities.. I will post answers on it soon..