------------------------------------------------ Partial list of Quant interview questions:
1. You have two glass balls and a 100-story building. You would like to determine the lowest floor from which a glass ball will break when dropped. What is the strategy that will minimize the worst case scenario for number of drops? What is your strategy if you have 100 glass balls?
2. A rabbit sits at the bottom of a staircase with n stairs. The rabbit can hop up only one or two stairs at a time. How many different ways are there for
the rabbit to ascend to the top of the stairs?
3. A drunken man is on a 100 meter long bridge at the 17 meter position. He has a 50% chance of staggering forward and backward one meter per step. What is the probability he will make it to the end of the bridge before the beginning?
4. Assume zero interest rate and a stock with current price $1. When the price
hits level $H for the first time you can exercise the option and will receive $1. What is this option worth to you today?
5. A family has two children. If one of them is a boy, what is the probability the other child is a boy?
6. You start with one amoeba. After one minute the amoeba can die, stay the same, split into two or split into three with equal probability. What is the probability the amoeba population will ever die out?
7. Suppose you have a coin with probability p of tossing heads. What is the expected number of coin tosses to get two heads in a row?
8. A banker gets off work at some random time between 6:00 pm and 7:00 pm. He walks to the subway and takes the first of two trains that arrives. One of these travels downtown to his mother, the other travels uptown to his girlfriend. His mother complains that she never sees him since she has only seen him twice in the last 20 working days. Explain this.
9. Two people arrive at the center of town at some random time between 5:00 am
and 6:00 am. They stay exactly five minutes and then leave. What is the probability they will meet on a given day?
10. A baker sells on average 20 cakes in one day. What is the probability he will sell an even number of cakes on any given day?
11. Airborne spores produce tiny mold colonies on gelatin plates in a laboratory. The many plates average 3 colonies per plate. What fraction of plates will have exactly 3 colonies?
12. You walk out of an airport, and the first bus you see has number 26 on it.
Assume buses with all numbers up to some number N exist. Estimate N.
13. Coupons in cereal boxes are numbered 1 to 5, and a set of one each is required for a prize. With one coupon per box, how many boxes on average are required to make a complete set?
14. Explain linear regression. What is the model? How do you estimate the parameters? Prove this.
15. Solve the following ordinary differential equations. y'' + y' + y = 1 y'' + 2y' + y = 1
16. Graph vega for a digital option.
17. Prove it is never optimal to exercise an American call on a non-dividend paying stock.
18. What is the martingale measure of the forward price for a non-dividend- paying stock?
19. Describe a function for which using antithetic variables in a Monte Carlo simulation produces no advantage.
20. Suppose you have two random variables, one uniformly distributed on inter-
val [3; 3] and the other standard normal. Is it possible for them to have correlation 1 or -1? What if the first variable is T-distributed?
21. What is the price of a swap? Choose the swap rate so the initial price is zero. Let V_t be the price of the swap. You would like to model V_t using Brownian motion. Would this work? Why or why not?
22. Derive Newton's method for finding the zero of a function.
23. Define Brownian Motion. Suppose W~N(0; 1) and t measures time. Is the random variable Sqrt(t)*W Brownian Motion?
24. Use a binomial tree to price a call option with strike K. Assume zero interest rate. Use parameters S0 = 18 Su = 22 Sd = 16 K = 19.
25. Design a C++ routine that will efficiently compute exp(x) to n terms.
26. Design a class or classes in C++ that will model a deck of cards.
27. Which pricing measure is most relevant for pricing credit derivatives, risk- neutral or physical? Why?
28. Give an example where Value at Risk is not subadditive.
29. Describe a situation where Value at Risk is always subadditive.
30. Suppose you have a string that is one unit long, and you cut it in two places uniformly distributed. What is the probability you can make a triangle out of the three pieces? If you instead make the second cut on the longest piece, what is then the probability you can make a triangle out of the pieces?
31. Define standard Brownian motion. Compute P(W1 > 0;W2 < 0), where W1 and W2 are standard Brownian motion at times 1 and 2.
32. Prove that standard Brownian motion hits a level with probability 1. Is this also true in higher dimensions?
33. What is the distribution of the minimum of a Brownian motion? How about a Brownian motion plus a Poisson process?
34. How would you price a swaption today when the swap begins today?
35. How would you price a swaption today when the swap begins in one month?
36. What is a Eurodollar future?
37. What is a swaption?
38. What is smile and skew? How do you model them?
39. What is the advantage of HJM over regular affine models? Can you fit the initial term structure for any affine models?
40. Draw on one graph the Black Scholes price of a call option as a function of the stock price for two option, one with maturity 1 and the other with maturity 2.
