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

Consider∫ the pdf p(x) of a Normal distribution with mean µ and variance σ2. Calculate +∞ −∞ 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 . 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. How does the underlying asset affect pricing of these derivatives?

3.3 Black-Scholes I

What is a self-financing replicating portfolio? Can you derive the Black-Scholes equation for the pricing of a European Call using a self-financing replicating portfolio and Ito’s lemma? Is there any other approach to deriving it?

3.4 The fundamental theorem

What is the fundamental theorem of asset pricing and how is it applied to pricing derivatives?

3.5 Black-Scholes II

How can you solve the Black-Scholes equation derived above? What is the solution and what is the meaning of each term?

3.6 European Call and Put I

Price, Delta, Gamma, Vega, rho for a European Call or : Can you derive them? Can you plot them?

3.7 European Call and Put II

How does the “” of a European affect its sensitivities?

3.8 European Call and Put III

What is the Call and Put Parity and how can it be used in option pricing?

3.9 Skew I

What is the skew or smile and what is the option volatility surface? What do they imply for the price distributions? Basic Finance and Black-Scholes 8

3.10 Volatility Skew II

How does the implied pdf of the price of an asset relate to the corresponding skew? Is there a mathematical relationship between the two?

3.11 Volatility Skew III

Are there any restrictions on the possible shapes of the volatility skew (and surface)?

3.12 Volatility Skew IV

Are the sensitivities of a European Call Option affected by the presence of a Volatility Skew?

3.13 I

How would you price a Binary Option in the context of the Black-Scholes model? How would you price it by replication given the Black-Scholes assumptions hold?

3.14 Binary Option II

In the presence of a Volatility Skew, how can you price a Binary Option? What is the relationship between the prices of a Binary Option with and without a Volatility Skew?

3.15 Empirical and Risk Neutral probabilities

What is the risk-neutral probability and how is it used in derivatives pricing? What is its relationship with the real-world (empirical) probability?

3.16 European VS American

When is it rational to early a European Call Option? A European Put Option? An American Call Option? An American Put Option?

3.17 American Option

What is the exercise boundary for an American option and how can we determine it?

3.18 Numerical

How do you price numerically a European and how an American Put option?

3.19 Trees I

What is a binomial and what is a ? What can they be used for? What are their differences? Basic Finance and Black-Scholes 9

3.20 Trees II

How can you use trees to price European and American Options? What other kinds of payoffs can you possibly price using trees?

3.21 Barriers I

Can you apply the reflection principle to derive the price of a under the Black- Scholes assumptions?

3.22 Barriers II

What are the problems in pricing a Barrier Option as above when we know that, in reality, there is a skew? How can we overcome these problems?

3.23 Combo Options

What is a Combo Option and how can we price analytically in the Black-Scholes framework?

3.24 Quanto Options

What is a Quanto Option and how can we price it analytically in the Black-Scholes framework. What does a replicating portfolio look like for a Quanto Option?

*** 4 Equity and FX Derivatives

4.1 Dividends I

Differences between Black-Scholes assumptions and the real world: How do dividends affect option prices? How do they affect exercise decisions?

4.2 Dividends II

Discuss pricing of vanilla options, European and American, in the presence of dividends. Does it make a difference whether the dividends are discrete or continuous?

4.3 Pricing

How can you price vanilla options numerically with and without dividends?

4.4 I

What are the ideas behind pricing a derivative using a Local Volatility surface? What is the relationship between vanilla option prices and the Local Volatility function σ(K, t)?

4.5 Local volatility II

What are the advantages and what are the shortcomings of the Local Volatility pricing approach? What types of derivatives can you safely price using this approach?

4.6 I

What are the motivations behind using a Stochastic Volatility model for pricing derivatives? What is the specification of the Heston Model and what are the resulting dynamics?

4.7 Stochastic Volatility II

How would you implement the Heston model of Stochastic Volatility to price derivatives and what derivatives would you price using this model?

4.8 Local versus Stochastic

Comparing a Local Volatility with a Stochastic Volatility framework what are the advantages and disadvantages of each approach? What types of derivatives would you price with each model?

4.9 Index Options I

What are Equity Index Options and how can we price Vanilla calls and puts on such indices? Equity and FX Derivatives 11

4.10 Index Options II

What does the skew look like, in general terms, for Equity Options? Is there a skew for Equity Index Options too? If there is one, how does it compare to the skew of options on the individual components? Why?

4.11 Asian Options I

How would you price an in the Black-Scholes framework?

4.12 Asian Options II

How would you price an Asian Option numerically in the presence of skew? What do you expect its sensitivities to be compared to the corresponding vanilla option?

4.13 FX options I

How would you price an FX option in the Black-Scholes framework? What is the use of numeraire in this case?

4.14 Equity Exotics I

What are the Best-of and Worst-of Options? How would you price them? What can you say about the sensitivities of these options? Are you long or short correlation?

4.15 Equity Exotics II

What is the Cliquet Option? How would you price it and what are the main risks? Does it make a difference whether you price it using a local volatility or a stochastic volatility model?

4.16 Equity Exotics III

What is a Himalaya option? How would you price it and what are the main risks?

4.17 Equity Exotics IV

What are Variance Swaps and how are they priced? How about volatility swaps?

4.18 Binary tree

A stock trades at $10. In one year’s time the price of this stock will be either $12 or $8 and the 1-year risk free rate is 10%. What is the value of the European Call option with expiry in 1 year and strike $10? Equity and FX Derivatives 12

4.19 FX options II

Suppose that the 1-year forward exchange rate between currencies A and B is 1.2 A=1 B. Also suppose that the annual interest rates for 1- and 3-year deposits are 1% and 1.2% for currency A and 2% and 2.5% for currency B. How can we calculate the 3-year forward exchange rate between these two currencies?

4.20

What is the difference between Implied Volatility of an option and the future volatility of the underlying stock? Can the Implied Volatility of the option be used to predict the future volatility of the stock?

*** 5 Interest Rate Derivatives

5.1 Markets

What are the most important Interest Rate markets? What are the most liquid instruments? What are the Libor rates? What are the rates?

5.2 Term Structure

What do we call term structure of interest rates and what are some theories that explain it? What is Backwardation and what is in the Interest Rate markets?

5.3 Futures and Forwards

What are the Interest Rate Futures and Forwards? What are the Eurodollar futures? What is the PV01 of a Eurodollar future?

5.4 Yield Curves

What is the Yield Curve and how is it constructed? What are the financial instruments used to build a Yield Curve and what is the process followed?

5.5 Swaps

What is an ? What is a Swap Rate? How do we value a Swap?

5.6 Caps and Floors I

What are Interest Rates Caps and Floors? How are they priced? Is there a skew in the Cap/Floor market and, if so, what does it look like?

5.7 Caps and Floors II

What are the Caplets and Floorlets? What is their relationship to Caps and Floors? Can we obtain the prices of these instruments from the prices of Caps and Floors and, if so, how?

5.8 I

What are the Interest Rate Swaptions? What is a payer and what is a receiver ? How do we price these? Is there an analytic or a numerical method? Interest Rate Derivatives 14

5.9 Swaptions II

How do Swaption prices relate to the prices of Interest Rate Caps and Floors? Can we price a European Interest Rate Swaption if we know the prices of all Caps and Floors? Why or why not?

5.10 Swaptions III

How would you price an American Interest Rate Swaption? What are the important market data and what are the possible implementations? Is there a skew in the Interest Rate Swaption market?

5.11 Bond Pricing I

How do we price a Zero Coupon and how do we price a coupon-bearing bond? Is the price of such bond different when the interest rates are considered stochastic as opposed to deterministic? Why or why not? What is the duration and what is the convexity of a bond?

5.12 Bond Pricing II

How do we price a Call option on an Zero Coupon bond? How about a coupon-bearing bond? What is “Pull-to-par” and what can we say about the price volatility of a bond as we approach expiry?

5.13 Convertible Bonds

What are Convertible Bonds? Can you outline the main ideas behind pricing these instruments?

5.14 Modelling I

What is the short rate and what is the Vasicek model for the short rate? Can you outline a possible implementation of this model? What are the advantages and disadvantages of using such a model?

5.15 Modelling II

What is the Hull and White model for the short rate? Can you outline a possible implementation of this model? What are the advantages and disadvantages of using such a model?

5.16 Modelling III

What is the BGM model and what are its advantages over the short rate models? Can you outline the numerical implementation of such a model?

5.17 Modelling IV

Can we price American Options using the BGM model? What are the problems in such an implementation and what are some possible solutions? Interest Rate Derivatives 15

5.18 Modelling V

How would you calibrate an interest rate model? In particular, how would you calibrate a BGM model? How do you choose the calibrating instruments?

5.19 Modelling VI

What are some possible ways to impart skew to an model such as the BGM?

5.20 Convexity Correction

What is Convexity Correction for interest rate products and when do we need to apply it?

*** 6 Commodities

6.1 Products and Markets

What are the most important commodities economically? What are the producing and what are the consuming countries? What are the main commodity exchanges?

6.2 Forward Curves I

What is the forward curve for a commodity? How are such curves constructed?

6.3 Forward Curves II

What is backwardation and what is contango in a particular commodity market? What are the possible explanations for this phenomenon?

6.4 Skew I

Is there a skew in the commodity markets? Are there any typical skews associated with different commodity markets?

6.5 Skew II

What is the correlation skew? How does it impact derivative pricing?

6.6 Modelling in Commodities I

How would you model a commodity and how do you expect this model to differ from modelling a stock?

6.7 Modelling in Commodities II

What are Jump Diffusions and how would you use them in pricing commodity derivatives? Can you outline a possible implementation? What do you expect the main difficulties to be with your approach?

6.8 Commodity Indices

What are the main Commodity Indices? What are their differences?

6.9 Commodity Indices II

If you had to build a commodities index with 3 components, what would they be and why? Which kind of weighting would you use (production vs. volume weighted)? Also, let’s suppose the index is based on the prompt month roll. What are the issues? Commodities 17

6.10 Cracks and spreads

What is a Crack Option? How do you price such an option in a Black-Scholes-like framework? What are the shortcomings of such an approach and how can you address them?

6.11 Volatility

Commodity markets can become very volatile. How does this affect derivatives pricing and how can you incorporate this knowledge into your derivative pricing?

6.12 Physical VS forward

Describe the considerations/risks associated with running a position of base metals in storage vs short forward contracts (assuming forwards are traded on the relevant liquid exchange).

6.13 WTI vs Brent

How the price dynamics have changed over the past year and why? What is your expectation for next year?

*** 7 Credit Derivatives

7.1 Risky Bonds

How does the price of a risky, “defaultable”, bond compares to price of a risk-free bond? What is the relationship between the prices of these two bonds and the implied Probability of default?

7.2 Credit Default Swaps

What is a ? How is it priced and what does it tell us about the Implied Probability of default? How does the price of a CDS depend on the probability of default and how does it depend on the Recovery rate?

7.3 Term structure

What is the term structure of default probabilities and how can we construct it?

7.4

What is the credit spread and how can we model it? Can you outline a model for the dynamics of the credit spread? How can it be used to price Credit derivatives?

7.5 Option on CDS

What is an option on a CDS? Can you outline a way of pricing such an option?

7.6 Indices I

What are the main Credit Default Swap Indices and what do they consist of? How do they work?

7.7 Indices II

Can you outline a way of pricing options on Credit Default Swap Indices?

7.8 CDO I

What are CDOs and what are CDO tranches? How are CDOs priced? How are the tranches priced?

7.9 CDO II

Can you explain how an option on a CDO tranche work? How does the price of an option on different tranches depend on the correlation of credit spreads of the underlying credits? Credit Derivatives 19

7.10 CDO III

What are bespoke CDOs and how does their option pricing differs from that of standard CDOs?

7.11 Hybrids I

How would you price, in principle, an (a equity call option for simplicity) with a payoff conditional on a risky bond?

7.12 Hybrids II

Suppose that the company that issued the equity in the above example is the same that issued the bond. Would that affect the pricing of the hybrid? Can you suggest any way of taking this effect into account when pricing the hybrid derivative?

*** 8 Credit Exposure and CVA 9 Numerical Methods

9.1 Random Numbers I

How can you produce Uniform random numbers in [0, 1]?

9.2 Random Numbers II

Given a sequence of Uniform random numbers in [0, 1], how can you produce Independent, Normally Distributed random numbers?

9.3 Random Numbers III

Given a sequence of Independent, Normally Distributed random numbers, how can you produce random numbers correlated with a given correlation matrix?

9.4 Monte Carlo I

What is the principle behind pricing derivatives using the Monte Carlo method? Does this method always converge and why? How fast is the convergence?

9.5 Monte Carlo II

When pricing derivatives using a Monte Carlo scheme, how would you calculate the sensitivities Delta and Gamma? What can you say about the convergence of these?

9.6 Monte Carlo III

Can you describe some methods of speeding up pricing in a Monte Carlo scheme? How does each of these methods work?

9.7 Monte Carlo IV

What are quasi-random number sequences and how can they be used in Monte Carlo pricing?

9.8 Monte Carlo V

Do you see any problem in pricing American style options using the Monte Carlo approach? What are some possible solutions?

9.9 Finite Differences I

Can you describe a Finite Difference scheme to price a European Call Option? An American Call option? Numerical Methods 22

9.10 Finite Differences II

How can you adapt the Finite Difference scheme above to use a Local Volatility surface?

9.11 Finite Differences III

Can you price an Asian-type option using a similar scheme?

9.12 Finite Differences IV

What is the most general payoff that you can price using a method such as the one described above?

9.13 Finite Differences V

You think you can adapt a Finite Difference scheme such as the above to accommodate a Stochastic Volatility model? How could this be done?

9.14 Simulating SDEs

What problems do you see in attempting to use a Monte Carlo method when the asset price is described by a Stochastic Volatility model? Can you suggest some ways around these problems?

9.15 Barriers

What do you think are some possible numerical methods for pricing Barrier options in the presence of skew? What are the advantages and disadvantages of each approach?

9.16 PCA

What is Principle Component Analysis? Can you think of some potential applications of this method in finance?

9.17 Copulas

What is a Copula? Can you give a concrete example? How do you generate random numbers using a Copula?

*** 10 Quantitative Risk Management

VAR I

What is VAR and what are its potential uses? What are its limitations?

VAR II

What are some commonly used methods to calculate VAR for a portfolio?

Stress Testing

What is Stress Testing and why is it different than the VAR approach? Can you describe a possible Stress Testing framework for a given portfolio?

Gaps

What is Gap risk? What can you do about it? How can you hedge a Barrier option?

Copulas

Can you outline the potential uses of Copulas for Risk Management?

CAPM I

What is the CAPM? What is α and what is β of a stock in this model?

Sharpe ratio

What is Sharpe Ratio and how is it used in finance?

CAPM II

What are the shortcomings of CAPM and Sharpe ratio approaches in estimating the risk of a portfolio?

Post-Modern Portfolio Theory

What are Downside Risk, Sortino Ratio and Volatility Skewness in the context of Post-Modern Portfolio Theory?

***