Quantitative Finance Interview Questions v 0.1 iaquant.org Quantitative Finance Interview Questions v 0.1 iaquant.org This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License Contents 1 Just a warm-up ....................................... 1 2 Probability and stochastic calculus ............................ 4 3 Basic Finance and Black-Scholes ............................. 7 4 Equity and FX Derivatives ................................ 10 5 Interest Rate Derivatives ................................. 13 6 Commodities ........................................ 16 7 Credit Derivatives ..................................... 18 8 Credit Exposure and CVA ................................. 20 9 Numerical Methods .................................... 21 10 Quantitative Risk Management .............................. 23 CONTENTS i THIS IS A DRAFT, PLEASE DO NOT DISTRIBUTE! 1 Just a warm-up 1.1 Estimate Give an estimate for the sum 1 1 1000 3 + ··· + 8000 3 : 1.2 A Zen question A Buddhist monk needs to meditate for exactly 45 minutes. He has no watch; instead, he has two incense sticks, and he is told that each of those sticks would completely burn in 1 hour. The sticks are not identical and they burn with unknown, possibly different, rates. How can he use these sticks to measure exactly 45 minutes of meditation? 1.3 Weighting Coins You have 12 identical-looking coins but you know that one of them is false. It is not known whether the false coin is heavier or lighter than the rest of the coins. How can you find the false coin by three weightings on a simple scale? 1.4 Strategy I Consider a game with two players. There is a stack of n coins on the table and players can take in turn between 1 and k coins from the stack where k < n. The winner is the one who takes the last coin from the stack. Is there a winning strategy in this game and, if so, what is it? 1.5 Unfair Coin A coin yields heads with probability p, an unknown constant. Can you use this coin to construct an algorithm with two final states each resulting with probability 1/2? In other words, can you simulate a fair coin using one that may or may not be fair? 1.6 Tower of bricks You have an infinite number of identical rectangular bricks, with dimensions 30 × 10 × 10cm. Can you construct a tower that balances, has on brick at each level and its projection on the horizontal plane is at least 100m? 1.7 Gossip village In a village far, far away there are ten people and each of them knows of one different piece of gossip. Every time two of them talk on the phone, the exchange all the gossip they know. What is the least number of calls needed so that all of these people learn all of the gossip? Just a warm-up 2 1.8 Throw all the coins Player A has one more coin than player B. Both players throw all of their coins simultaneously and observe the number that come up heads. Assuming all the coins are fair, what is the probability that A obtains more heads than B? 1.9 A fair die What is the expected number of times a fair die must be thrown until all scores appear at least once? 1.10 Consecutive tosses A fair coin is tossed n times. What is the probability that no two consecutive heads appear? 1.11 The highest score Suppose n fair 6-sided dice are rolled simultaneously. What is the expected value of the score on the highest valued die? 1.12 Consecutive tosses II A fair coin is tossed repeatedly until n consecutive heads occur. What is the expected number of times the coin is tossed? 1.13 Consecutive tosses III A fair coin is tossed n times and the outcome of each toss is recorded. Find the probability that in the resulting sequence of tosses a head immediately follows a head exactly h times and a tail immediately follows a tail exactly t times. (For example, for the sequence HHHTTHTHH, we have n = 9; h = 3 and t = 1.) 1.14 The famous Monty Hall problem Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice? 1.15 Strategy II Consider the following game with two players. Each player in turn picks on of the numbers 1; 2; 3;:::; 9 and the winner is the first one whose chosen numbers sum to 15. Each number can only be chosen once. Is there a winning strategy in this game? Just a warm-up 3 1.16 Balls in an urn An urn contains a number of coloured balls, with equal numbers of each colour. Adding 20 balls of a new colour to the urn would not change the probability of drawing (without replacement) two balls of the same colour. How many balls are in the urn initially? 1.17 Covering the table On a rectangular table there are n identical coins placed in such a way that there is no space for an additional coin that does not overlap with any of them. (A coin is considered to be on the table when its centre is on the table). Can you show that 4n overlapping coins are enough to cover the table completely? 1.18 Match boxes An absent-minded quant buys two boxes of matches and puts them in his pocket. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes. One day the quant opens a matchbox and finds that it is empty. (He must have absent-mindedly put the empty box back in his pocket when he took the last match from it.) If each box originally contained n matches, what is the probability that the other box currently contains k matches? (Where 0 ≤ k ≤ n.) 1.19 Limited space In a busy customs office they have to keep overnight for checking 100 packages a day, randomly chosen from all the packages that arrive that day. The problem is that they do not know in advance how many packages will arrive during each day and, also, they cannot warehouse more than one hundred packages at any one time. What could they do? 1.20 Random Number Generator A random number generator generates integers in the range 1; : : : ; n, where n is a parameter passed into the generator. The output from the generator is repeatedly passed back in as the input. If the initial input parameter is one googol, 10100, find to the nearest integer, the expected value of the number of iterations by which the generator first outputs the number 1. *** 2 Probability and stochastic calculus 2.1 Normal distribution I RConsider the pdf p(x) of a Normal distribution with mean µ and variance σ2. Calculate +1 −∞ p(x) dx. 2.2 Normal distribution II Calculate the mean and variance of a Normal distribution. 2.3 Binomial distribution Define the Binomial distribution. Calculate its mean and variance. 2.4 Poisson distribution Define the Poisson distribution. Calculate its mean and variance. What does the cdf of a Poisson random variable look like? 2.5 Log-normal distribution Calculate the mean and variance of a log-normal distribution. 2.6 Basic Theorems Central Limit theorem, Law of Large Numbers and Chebychev’s Inequality. Can you think of an application of each of these in the context of financial mathematics? 2.7 Brownian Motion I Give a definition of the Brownian Motion. How can it be obtained as a limit of random walk? What are its basic properties? 2.8 Brownian Motion II: Brownian Bridge What is the Brownian Bridge and how can we use it in mathematical finance? 2.9 Brownian Motion III: Reflections What is the reflection principle for Brownian Motion and how can we use it in mathematical finance? Probability and stochastic calculus 5 2.10 Itō Statements and explanation of Itō’s Lemma, Itō’s formula and Itō’s Isometry. 2.11 Martingales I What is a filtration, what is a martingale and how are these concepts used in financial mathematics? What does the martingale convergence theorem state? 2.12 Martingales II 2 − Can you prove that the Brownian Motion Bt is a martingale? Is the process Bt t also a martingale and why? 2.13 Martingales III What is a stopping time τ? What is a stopped martingale and how is this concept related to the doubling bet strategy? 2.14 Martingales IV Girsanov’s theorem and the Radon-Nikodym derivative. What are they and how are they applied to derivative pricing? 2.15 St. Petersburg A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 1 dollar and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus, the player wins 2k−1 dollars, where k heads are tossed before the first tail appears. What would be a fair price to pay the casino for entering the game? 2.16 SDE I: Stochastic Exponential Can you find a process U that satisfies dU(t) = U(t)dBt with U(0) = 1? How is this process relevant to mathematical finance? 2.17 SDE II How would you solve the following Stochastic Differential Equation? dxt = axtdWt + σxtdt Probability and stochastic calculus 6 2.18 SDE III: O-U Process Can you solve the following Stochastic Differential Equation? dxt = θ(µ − xt)dt + σdWt 2.19 Feynman-Kac What is the Feynman-Kac formula and how can we apply it in pricing financial derivatives? *** 3 Basic Finance and Black-Scholes 3.1 Futures and Forwards I What are Futures and what are Forwards? What is the price of a Forward and why? 3.2 Futures and Forwards II Price and Delta of a Forward compared to a Future.