Here are some of my favorites
1. Lets consider a IR swap: Notional=$10million, 10 years duration, half yearly payments. Suppose the swap rate for the same today is 4% and you decide to pay floating. Tomorrow the yield curve goes up (assume parallel shift) by 10 bps, what's your PnL?
2. I randomly pick 2 stocks out of the SnP 500. What do you think is the historical 3m correlation between them. Justify your answer.
3. I will give you $10million today. Where would you invest it ? (the right answer to this question, is a question)
4. What is the expected value of W^6 W being brownian motion.
5. What is the value of an ATM European call with time to maturity as infinity? what's the value of american ATM with time to maturity infinity?
6. (I really like this puzzle -- neat one). the three trader dilema. 3 traders have initial indications A, B,C for their bonus this year. You need to design a set of arguments so that each one of them knows the mean (A+B+C)/3, but no one is able to back trace the individual indications for the other 2.
7. consider the following option: you have an option to buy (maximum) M units of gas at a fixed price K for over any N days (you can exercise the option on at max N days) over the next T days. However if you choose to exercise on day i, you cannot buy more than C units. (obviously M <= C*N) how would you value this option. (I still don't know the answer to this one

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8. Your view is that tomorrow the 6 months skew (defined as the implied vol difference btw 25 delta put and 25 delta call) will reduce by 25% tomorrow. What positions would you take? What if your view is that it will reduce by 50% over the next 2 months. What positions would you take?
9) You have to fill numbers on the faces of 3 die (18 numbers in all). The property they should have is that no matter what die I choose you should be able to pick a die from the remaining 2 such that you always win in expectation. That is if B beats A, C should be such that C beats B but C loses to A (in expectation)
10) There are 60 blue ribbons in a box such that all 120 ends are hanging out and you cannot see which ends belong to which ribbons. You randomly join all 120 ends together into pretty bows and dump out the box. Depending on chance you will form anywhere from 1 to 60 loops. How many loops would you expect?
11) Which has a higher ATM implied vol (a portfolio of puts on single stocks) or (a put option on an index with the same portfolio weights as the single stocks portfolio).
12) Continuing q11, if the argument of diversification is true, why has been ATM for SP500 been consistently higher than Dow (Dow has 30 names whereas SP500, 500 names. For e.g. recently 3mATM put on Dow traded at 22%, whereas for SP500 it traded at 24%)
13) consider a one period binomial tree with r=0, u=10 d = 5. Write the equations you would write to get a risk neutral world. calculate the price of the option. what's wrong with the price you get.
14) An American call option is about to pay a large dividend tomorrow. Give me an algorithm to decide whether or not it is optimal to exercise the option today.
15) I will sell my house in an year from now. I want to buy a put option where the payoff would be (K-S(1yr)+ where S(1yr) is the price I receive from selling my house 1 yr from now. Assume that house prices are log normally distributed. Is the Black Scholes formula valid for this situation? How would price such an option ? How would hedge this option ?
16) Assume you are Toyota. Lists all the potential risks you are exposed to, and how do you plan to hedge yourself against them.
17) there are 100 statements on a wall. Statement 0 says that atmost 0 statements can be true. Statement 1 says that atmost 1 statement can be true. Statement i says atmost i statements can be true. How many true statements are there and which ones.
18) How does d(delta)/dvol vary with the spot?
19) Consider an infinite big chess board floor with individual squares of side 5 each. A coin of radius 2cm is tossed with its center initially at one of the confluence points of 4 small squares. What is the probability that the coin lands exactly inside one of the squares.
20) Draw me the basic efficient portfolio line. Where do you think your house is on this graph. Where do you think is your own investment on this graph?