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First round was a brainteaser/statistics/probability round. First question: you have two fair dice, the sum of which is equal to the value of a security. You have a call option on this security with a strike price of 7. If you are two roll each die once to give you the security value, what are you willing to pay for this call option? Second Question: You have two coins, one fair coin and one with heads on both sides. You randomly choose one of the coins and flip it twice. The results are heads and heads. What is the probability that the coin you flipped is a fair coin. Third Question: You have a spherical balloon that is deflating at a rate such that the radius is decreasing at a constant rate of 4cm/min. At what rate is the balloon deflating when the radius is .25cm. Fourth Question: You have a circle with a radius of one. What is the probability that two randomly chosen points connected by a chord have a length greater than 1? Second Round was all fit. Seems like a pretty unique firm in terms of culture (ski trips, paintball, golf, etc.). The first two guys I interviewed with seemed pretty cool, but the guy from the third round seemed like a hardo. Third Round: They ask you to spend 10 hours reading Hull's textbook on options and futures, which is just about enough time to learn the very basics of options and futures if you have no background. I interviewed with a French guy with a very heavy accent, making it real difficult to comprehend anything coming out of his mouth. First Question: You have a revolver with six chamber slots, and you load two bullets next to each other. You fire the gun once and it's not the bullet. You have the option to either re-spin the chamber, or fire again. Which gives you the higher probability. Second Question: Call option at the money with strike of 100. What is a rough value for the price of the this option. What is the delta of this option?

10

^agree on 1 and 2. For 3) Volume = 4/3pi * r^3 so dVolume/dt (in minutes) = 4/3 pi * 3 r^2 *dr/dt (in minutes) = 4 * pi *r^2 *4 cm/min. Since r = 1/4 cm, we have volume decreasing at a rate of pi cm^3/minute. 4) 2 points on a line can be connected by a triangle with two sides of length r going through the center, and a 'base' that is a chord. We want this chord to be >1. We know the triangle is isosceles since the other sides that connect the center of the circle to the points are length r = 1. Let the angle between the two equal sides be theta. Then splitting our isosceles triangle in half we have a right triangle, and theta is related to the cord and r by: sin(theta/2) = (chord/2) / r from the relation sin = opposite/hypotenuse. r = 1, so we have sin(theta/2) = chord/2 or chord = 2*sin(theta/2). We want chord > 1 so sin(theta/2) must be >1/2. We know that theta/2 has to be >30 degrees (for theta between 0 and 180), and so theta has to be greater than 60. This is easier to visualize with a picture, but basically there is a 60 degree zone to the left and right of our first point where, if our second point is drawn, our chord won't be >1. So, we see that we don't have 120 degrees of our circle to pick a point from if we want a chord of greater than 1 => picking a theta at random gives us a 2/3 chance (360 degrees without 120 available) to get our chord. This question is easy to draw, harder to explain in words. 2nd round: 1) Don't spin. 4 spots to put the pair of bullets in the remaining 5 chambers, but only 1 spot kills you (75% survival). If you spin its just a simple 66% chance since there are 6 spots for the pair to be loaded and 2 result in killing you. 2) Any at the money call/put has a delta of 0.5. Rough price is really subjective, standard pricing models require a volatility as input to determine the premium over parity of an option, not sure what they want. A few dollars (<5) is probably fine.

Anonymous on

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First Question: The value of the call option (assuming no discount) should be 35/36.

Anonymous on

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Agreed with the second post. 2nd question of the 2nd round: I think they were asking about trader's rule of thumb. If time to maturity is short and interest rate is low, ATM option price is close to 0.4* sigma * spot price * sqrt(T)

Anonymous on

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Question 4: This is actually an area problem. Think of the chord as infinitely small, and there's an infinite number of points along the edge of the circle that can start a chord. If you layered a large number of chords of 1cm or more on the circle one by one, you'd eventually notice that the center would show a greater density first. The smallest chord is 1cm, therefore, you can use that with Pythagorean theorem to find the the radius of the inner circle. From which you get a^2+.5^2 =1, r =sqrt(.75) = .866 Circle (outer) with r = 1cm has an area of 1pi. Inner Circle r =.866 has an area of .866pi. Therefore, if a chard must pass through that inner circle, and can't pass through the outer circle, the odds of picking two points that form a chord of at least 1cm is .866pi/1pi... .866

Jeff on

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To clarify, since every chord must be be tangent to or pass through the center circle, find the odds that a chord passes through that circle...

Jeff on

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First question: if the price you pay for the option is greater than zero, the expected profit of the trade is negative.

Aaron Soellinger on

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Second Question: 1/5 Third Question: 4cm/min, it will be empty in 1/16 of a minute Fourth Question: no 2 points in the unit disk will define a segment that has length greater than 1. so the answer is 0.

Aaron Soellinger on