150 Quant Questions Flashcards
Put options with strikes 30 and 20 on the same underlying asset with the same maturity are trading for $6 and $4 respectively. Can you find an arbitrage?
The value of a put option with strike zero is 0. So we have P(30) = 6, P(20) = 4, and P(0) = 0. These three points are on the line P = (1/5)K. But this contradicts the fact that put options are strictly convex functions of strike price, so there is an arbitrage opportunity. The out with strike 20 is overpriced, and you can buy low, sell high. Buy ⅔ put options with strike 30 and sell 1 put option with strike 20. At maturity T, the value of the portfolio is V(T) = 2max(30 - S(T),0) - 3max(20 - S(T),0)
The number 2^29 has 9 digits, all different. Without computing 2^29, find the missing digit.
For any positive integer n, the sum of the digits is D(n). n - D(n) is divisible by 9. D(2^29) = (sum from 0 to 9 of j) - x = 45 - x. Break 2^29 into factors, with one being as close to 45 as possible: 2^5 * (2^6)^4 = 2^29. Then you have 9 | (45 - x) - 2^5 = 13 - x. x must make it such that 9 is divisible, so x must be 4.
Let W_t be a Wiener process, and let X_t = the integral from 0 to t of W_τ dτ . What is the distribution of X_t? Is X_t a martingale?
dX = Wdt = Wdt + 0dW. Therefore, X is a diffusion process with only drift part W, and thus not a martingale. Using integration by parts:
X_t = tW_t - int(0,t)τ dW_τ
= tint(0,t)dW_τ - int(0.t)τ dW_τ
= int(0,t)(t - τ)dW_τ
~ N(0, int(0,t)(t-τ)^2dτ )
= N(0,t^3/3)
X_t is a normal random variable of mean 0 and variance t^3/3
Alice and Bob stand at opposite ends of a straight line segment. Bob sends 50 ants towards Alice, one after another. Alice sends 20 ants towards Bob. All the ants travel along a straight line. Whenever two ants collide, they simply bounce back and start traveling in the opposite direction. How many ants reach Bob and how many ants reach Alice?
Bob gets 20 ants, Alice gets 50. The number of collisions is 20 * 50 = 1000 collisions.
Find all values of ρ such that
(1 0.6 -0.3
0.6 1 ρ
-0.3 ρ 1)
is a correlation matrix.
The det(M) >= 0. Find det(M) and solve.
How many independent random variables uniformly distributed on [0,1] should you generate to ensure that there is at least one between 0.7 and 0.72 with probability 95%?
N is the smallest number of random variables such that P(at least one rv in [0.70,0.72]) >= 0.95. The probability that a rv uniformly distributed on [0,1] is not in [0.70,0.72] is 0.98. So the probability that none of the N independent variables are in [0.70, 0.72] is 0.98^N. So P(at least one rv in [0.70,0.72]) >= 0.95 = 1 - 0.98^N >= 0.95. Solve for N.
Show that the probability density function of the standard normal integrates to 1
The probability density function of the standard normal is (1/sqrt(2π))e^(-t^2/2)dt. Show that the integral of this from -∞ to ∞ = 1
You travel exactly 1 mile south, 1 mile east, and 1 mile north. You find yourself at the starting point. If you are not at the North Pole, where can you be?
Anywhere that has a latitude circumference of 1near the South Pole. But any mile north of such a latitude with circumference 1/k when you loop the globe k times will put you back where you started.
Solve the Ornstein-Uhlenbeck SDE
The OU SDE is
dr_t = Λ(θ - r_t)dt + σ dW_t. It is used in the Vasicek model for interest rates.
Write a function that computes the n-th Fibonacci number
def fibonacci_iterative(n):
if n <= 0:
return 0
elif n == 1:
return 1
else:
# Initialize the first two Fibonacci numbers
a, b = 0, 1
# Iterate from 2 to n
for _ in range(2, n + 1):
# Compute the next Fibonacci number
a, b = b, a + b
return b
What is the value of i^i (where i is sqrt of -1)?
Recall i = cos(π/2) + isin(π/2) = e^(iπ/2)
i^i = (e^(iπ/2))^i = e^(-π/2)
Which number is larger, π ^e or e^π ?
e^π
Show that
(e^x + e^y)/2 >= e^((x+y)/2)
Let a = e^x and b = e^y. solve for the case is >= 0
Solve x^6 = 64
The six unit roots of z^6 = 1 are:
z_k = exp(2kπi/6) = exp(kπi/3) = cos(kπ/x) + isin(kπ/3)
therefore x_k = 2cos(kπ/3) + 2isin(kπ/3)
What is the derivative of x^x
By chain rule, (x^x)’ - (x^x) * (lnx + 1)
Calculate sqrt(2 + sqrt(2 + sqrt 2 + ….)
2
Find x such that x^x^x^x^x… = 2
sqrt(2)
Which of the following series converge (first two sums from 1 to ∞, second sum from 2 to ∞):
Σ 1/k ; Σ 1/k^2 ; Σ 1/(klnk)
Since all the terms in the middle one are positive, partial sums are uniformly bounded and thus it is a convergent series. It equals π^2/6
Compute integrals:
xln(x) dx and xe^xdx
Use integration by parts: int(udv) = uv - int(vdu) where u is ln(x) and dv is x
Compute integral (x^n)ln(x)dx
Use integration by parts: int(udv) = uv - int(vdu) where u is ln(x) and dv is x^n
Solve the ODE:
y’’ - 4y’ + 4y = 1
y(x) = c1e^(2x) + c2xe^(2x) + ¼
Find f(x) such that
f’(x) = f(x)(1 - f(x))
Use y and integrate:
f(x) = Ce^x/(1 + Ce^x)
Define the Black-Scholes PDE
It estimates the theoretical value of derivatives based on other investment instruments, taking the impact of time and other risk factors into account. It is still regarded as one of the best ways to price an options contract.
The prices of three put options with strikes 40, 50, 70 but otherwise identical, are $10, $20, and $30 respectively. Is there an arbitrage opportunity present? If yes, how can you make a riskless profit?
When plotting, draw a line between the first and last points to see if there is a violation of the convexity criterion. The point at P(50) = 20 is above that line, violating the convexity criterion, so an arbitrage opportunity is available. Buy 3 of the overpriced option, and sell 2 of the $10, and 1 of the $30 options. This will generate a positive cash flow of $10. The value at maturity will be:
V(T) = 2max(40 - S(T),0) + max(70 - S(T),0) - 3max(50 - S(T),0)
The price of stock is $50. In three months, it will either be $47 or $52, with 50% probability. How much would you pay for an at-the-money put?
Real world probabilities do not play any role in valuing an option in a (one period) binomial tree model. Thus, the 50% is just there to throw you off. The value of the option is the discounted expected value of the payoff of the option in the risk-neutral probability measure. Since the interest rates are zero, this can be written as
P(0) = p(RN,up)P(up) + p(RN,down)P(down)
The up and down factors are u = 52/50 and d = 47/50. Thus the risk-neutral probabilities of up and down are
p(RN,up) = (1-d)/(u-d) and
p(RN,down) = (u-1)/(u-d)
The ATM put pays $3 if the stock goes down to $47, so P(down) = 3 and expires worthless if the stock goes up to $52, so P(up) = 0. Putting hte numbers in gives 1.20
The price of stock is $50. In three months, it will either be $60 or $40, with 50% probability. The value of a three months at-the-money put on this stock is $4. Does the value of the three months ATM put increase or decrease, and by how much, if the probability of the stock going up to $60 were 75% and the probability of the stock going down to $40 were 25%?
The probabilities don’t play a role, so the value would still be $4.
What is risk-neutral pricing?
Risk-neutral pricing, or valuation, refers to valuing derivative securities as discounted expected values of their payoffs at maturity, under the assumption that the underlying asset has lognormal distribution with a drift equal to the risk-free rate. It cannot be used for path dependent options such as American options, barrier options, and Asian options (from European perspective).
Describe briefly how you arrive at the Black-Scholes formula
There are several methods. 1) risk-neutral pricing 2) the Black-Scholes PDE solution 3) Binomial tree model pricing. there are 12 different ways to derive it that can be found in Wilmott.
How much should a three months at-the-money put on an asset with a spot price $40 and volatility 30% be worth?
P(ATM) ~ 0.5σS_0sqrt(T)
where S_0 is the spot price and σ is the volatility. T is time in years. Plug and chug.
If the price of a stock doubles in one day, by how much will the value of a call option on this stock change?
You can approximate the value of a deep in the money call on a non-dividend paying asset by using the put-call parity, as C ~ S - Ke^(-rT) where K and T are the strike and maturity of the option, and r is the constant risk free rate. Thus, if the spot prince S doubles, the call option will be even deeper in the money and the value will be approximately 2S - Ke^(-rT). i.e. the value of deep in the month call options will double if the price of the stock doubles.
If the option is around at the money, the percentage generated by doubling the stock price is 1 oom larger since the option will become deep in the money.
If the option is deep out of the money, it trades for fractions of cents. The doubling of the spot price would result in changing of the price of the option by several oom
What are the smallest and larges values that Delta can take?
-1 and 1
What is the Delta of an at-the-money call? What is the Delta of an at-the-money put?
0.5. -0.5
What is the Put-Call parity? How do you prove it?
Show that the time value of a European call option is highest at-the-money
What is implied volatility? What is a volatility smile? How about a volatility skew?
What is the Gamma of an option? Why is it preferable to have a small Gamma? Why is the Gamma of plain vanilla options positive?
When are a European call and a European put worth the same? What is the intuition behind this result?
What is the two year volatility of an asset with 30% six months volatility?
How do you value an interest rate swap?
By how much will the price of a ten year zero coupon bond change if the yield increases by ten basis points?
A five year bond with 3.5 years duration is worth 102. What is the value of the bond if the yield decreases by fifty basis points?
What is a forward contract? What is a forward price?
What is the forward price for treasury futures contracts What is the forward price for commodities future contracts?
What is a Eurodollar futures contract?
What are the most important differences between forward contracts and future contracts?
What is the ten-day 99% VaR of a portfolio with a five-day 98% VaR of $10 million?
How would you compute π using Monte Carlo simulations? What is the standard deviation of this method?
What methods do you know for generating independent samples of the standard normal distribution?
How do you generate Brownian motion stock path using random numbers from a normal distribution?
How do you generate a sample of the standard normal distribution from 12 independent samples of uniform distribution on [0,1]
What is the rate of convergence for Monte Carlo methods?
What variance reduction techniques do you know?
How do you generate samples of normal random variables with correlation ρ?
What is the order of convergence of the Newton’s method?
Which finite difference method corresponds to trinomial trees?
What is the exponential distribution? What are the mean and the variance of the exponential distribution?
If X and Y are independent exponential random variables with mean 6 and 8 respectively, what is the probability that Y is greater than X?
What are the expected value and variance of the Poisson distribution?
A point is chosen uniformly from the unit disk. What is the expected value of the distance between the point and the center of the disk?
Consider two random variables X and Y with mean 0 and variance 1, and with joint normal distribution. If cov(X,Y) = 1/sqrt(2), what is the conditional probability P(X>0 | Y<0)?
If X and Y are lognormal random variables, is there product XY lognormally distributed?
Let X be a normal random variable with mean μ and variance σ^2, and let Φ be the cumulative distribution function of the standard normal distribution. Find the expected value of Y = Φ(X)
What is the law of large numbers?
What is the central limit theorem?
What is a martingale? How is it related to option pricing?
Explain the assumption (dW_t)^2 = dt used in the informal derivation of Itô’s Lemma
If W_t is a Wieder process, find E[W_tW_s]
If W_t is a Wiender process, what is var(W_t + W_s)?
Let W_t be a Wiener process. Find:
integral(0 to t) W_sdW_s and E[integral(0 to t) W_sdW_s and ]
Find the distribution of the random variable
X = integral(0 to 1)W_t dW_t
Let W_t be a Wiener process. Find:
integral(0 to t) W_s^2dW_s
If W_t is a Wiener process, find the variance of X = integral(0 to 1) sqrt(t)e^(W_t^2/8)dW_t
if W_t is a Wiener process, what is E[e^W_t]?
Let W_t be a Wiener process. Find:
integral(0 to t) sdW_s
What is an Itô process?
What is Itô’s Lemma?
If W_t is a Wiener process, is the process X_t = W_t^2 a martingale?
If W_t is a Wiener process, is the process N_t = W_t^3 - 3tW_t a martingale?
What is Girsanov’s theorem?
What is the martingale representation theorem, and how is it related to option pricing and hedging?
Solve dY_t = Y_t dW_t, where W_t is a Wiener process
Solve the following SDEs:
(i) dY_t = μY_tdt + σY_tdW_t
(ii) dX_t = μdt + (aX_t + b)dW_t
What is the Heston model?
A flea is going between two points which are 100 in. apart by jumping either 1 or 2 inches at a time. How many different paths can the flea travel by?
(1/sqrt(5))(φ1^(n+1) - φ2^(n+1))
where n = 100 and φ1/ φ2 = (1+/- sqrt(5))/2
I have a bag containing three pancakes: one golden on both sides, one burnt on both sides, one half golden half burnt. You draw a pancake at random and look at one side. It’s golden. What is the probability that the other side is golden?
Bayes’ formula or out of the six possible sides, 3 are golden. out of these, 2 could be golden on both sides. therefore, the probability is ⅔.
Alice and Bob are playing heads and tails, Alice tosses n+1 coins, Bob tosses n coins. What is the probability that Alice will have strictly more heads than Bob?
½
Alice is in a restaurant trying to decide between three desserts. How can she choose one of the three desserts with equal probability with the help of a fair coin? What if the coin is biased and the bias is unknown?
Toss the coin twice and let three possible outcomes (from TH, HT, TT, and HH) correspond to a dessert. If the outcome you don’t choose shows up, just repeat the procedure.
If the coin is biased with an unknown bias, toss it four times and choose 3 outcomes to correspond to each dessert.
What is the expected number of times you must flip a fair coin until it lands on head? What if the coin is biased and lands on head with probability p?
Fair coin: E[X] = 2
Biased coin: E[X] = 1/p where p is the probability of it landing on heads
What is the expected number of coin tosses of a fair coin in order to get two heads in a row? What if the coin is biased with 25% probability getting heads?
E[X] = (1+p)/p^2 where p is the probability of getting heads. For a fair coin this would be ½
A fair coin is tossed n times. What is the probability that no two consecutive heads appear?
(1/2^nsqrt(5))(φ1^(n+2) - φ2^(n+2))
where φ1/ φ2 = (1+/- sqrt(5))/2
You have two identical Fabergé eggs, either of which will drop if you drop them from the top of a 100-floor building. Your task is to determine the highest floor from which an egg could be dropped without breaking. What is the minimum number of drops required to achieve this? You are allowed to break both eggs in the process.
n(n+1)/2
An ant is in the corner of a 10 x 10 x 10 room and wants to go to the opposite corner. What is the length of the shortest path the ant can take?
10sqrt(5) ~ 22
If the cube has side length a, the shortest path length will be sqrt(5a^2) = a*sqrt(5)
A 10 x 10 x 10 cube is made of 1000 unit cubes. How many unit cubes can you see on the outside?
If you remove the outside unit cubes, you get an 8 x 8 x 8 cube. 8^3 = 512 therefore you see 1000 - 512 = 488 outside cubes.
Mulder is imprisoned by aliens in a large circular field surrounded by a fence. Outside the fence is a vicious alien that can run 4 times as fast as Mulder, but is constrained to stay near the fence. If Mulder can contrive to get to an unguarded point on the fence, he can quickly scale and escape the fence. Can he get to a point on the fence ahead of the alien?
Mulder needs to start at a point x = 1 - π/4 + 0.01 on a circle of radius xR, centered at the center of the field C. Then he needs to run such that he, the alien, and an escape point opposite the alien are co-linear, with C between him and the alien. Once he does that, he can run to the escape point
At your subway station, you notice that two trains running in opposite directions which are supposed to arrive with the same frequency, the train going in one direction comes first 80% of the time while the train going the opposite direction comes first only 20% of the time. What do you think could be happening?
Their set arrival times could be slightly different.
You start with one amoeba. Every minute, it can either die, do nothing, split into two amoebas, or split into three amoebas. Each scenario is equally likely. All further amoebas behave the same way. What is the probability all the amoebas eventually die off?
The probability that the descendants will die out at the beginning is p.
p = ¼(1 + p + p^2 + p^3)
Rearranged: p^3 + p^2 - 3p + 1 = 0. Solve for the roots, and the only one in the interval (0,1) is sqrt(2) - 1.
Given a set X with n elements, choose two subsets A and B at random. What is the probability of A being a subset of B?
(¾)^n
Alice writes two distinct real numbers between 0 and 1 on two sheets of paper. Bob selects one sheet randomly to inspect it. He then has to declare whether the number he sees is bigger or smaller than the other. Is there any way Bob can expect to be correct more than half the times Alice plays this game with him?
How many digits does the number 125^100 have? You are not allowed to use values of log10(2) or log(10)5
For every subset of {1,2,3,…,2013} arrange the number sin the increasing order and take the sum with alternating signs. the resulting integer is called the weight of the subset. Find the sum of the weights of all teh subsets of {1,2,3,…,2013}
Alice and Bob alternately choose one number from the following nine numbers: 1/16, ⅛, ¼, ½, 1, 2, 4, 8, 16 without replacement. Whoever gets three numbers that multiply to one wins the game. Alice starts first. What should her strategy be? Can she always win?
Mr. and Mrs. Jones invite four other couples over for a party. At the end of the party, Mr. Jones asks everyone else how many people they shook hands with and finds that everyone gives a different answer. Of course, no one shook hands with his or her spouse and no one shook the same person’s hand twice. How many people did Mrs. Jones shake hands with?
The New York Yankees and the SF Giants are playing in teh World Series. You would like to bet $100 on the Yankees winning the World Series, but you can only place bets on individual games, and every time at even odds. How much should you bet on the first game?
We have 2 red, 2 green, and 2 yellow balls. In each pair, one ball is heavy and one ball is light. All the heavy balls weigh the same and all the light balls weigh the same. How many weighings on a scale are necessary to identify the three heavy balls?
There is a row of 10 rooms and a treasure in one of them. Each night, a ghost moves the treasure to an adjacent room. You are trying to find the treasure, but can only check one room per day. How do you find it?
How many comparisons do you need to find the maximum in a set of n distinct numbers? How many comparisons do you need to find both the maximum and minimum in a set of n distinct numbers?
Given a cube, you can jump from one vertex to a neighboring vertex with equal probability. Assume you start from a certain vertex. What is the expected number of jumps to reach the opposite vertex?
Select numbers uniformly distributed between 0 and 1, one after the other, as long as they keep decreasing (i.e. stop selecting when you obtain a number that is greater than the previous one you selected)
To organize a chartiy event that costs $100K, an organization raises funds. Independent of each other, one donor after another donates some amount of money that is exponentially distributed with a mean of $20K. the process is stopped as soon as $100K or more has been collected. Find the distribution, mean, and variance of the number of donors needed until at least $100K has been collected.
Consider a random walk starting at 1 and with equal probability of moving to the left or to the right by one unit and stopping either at 0 or at 3.
(i) What is the expected number of steps to do so?
(ii) What is the probability of the random walk ending at 3 rather than 0?
A stick of length 1 drops and breaks at a random place uniformly distributed across the length. What is the expected length of the smaller part?
You are given a stick of unit length.
(i) the stick drops and breaks in two places. what is the probability that the three pieces could form a triangle?
(ii) the stick drops and breaks in one place. then the larger piece is broken again. what is the likelihood that the three pieces could form a triangle?
Why is a manhole cover round?
When is the first time after 12 o’clock that the hour and minute hands of a clock meet again?
Three light switches are in one room, and they turn three light bulbs in another. How do you figure out which switch turns on which bulb in one shot?
Put Option Definition
A put option is an investment tool that gives the buyer the right, but not the obligation, to sell a stock at a specific price (the strike price) before the option’s expiration date. In exchange for this right, the buyer pays the seller a premium.
Strike Price Definition
The price at which the holder can sell the underlying asset. The strike price determines whether an option is in-the-money (ITM) or out-of-the-money (OTM).
Arbitrage Definition
The simultaneous buying and selling of securities, currency, or commodities in different markets or in derivative forms in order to take advantage of differing prices for the same asset.
Wiener Process Definition
If f(t) is a deterministic square integrable function, what is the stochastic integral int(0,t)f(τ)dW_τ?
int(0,t)f(τ)dW_τ ~ N(0,int(0,t)|f(τ)|^2dτ)
Determinant of matrix M, where M is:
(a b c
d e f
g h i)
aei + bfg + cdh - ceg - bdi - afh
Black-Scholes PDE
∂V/∂t + (σ^2S^2/2)(∂^2V/∂S^2) + (r-q)(S∂V/∂S) - rV = 0 where V is the value of the portfolio, t is time, σ is the volatility, S is the spot price, r is the risk-free rate, and q is the continuous dividend pay,ent rate.
Show that any covariance matrix is symmetric positive semidefinite. Show that the same is true for correlation matrices.
Find the correlation matrix of three random variables with covariance matrix
Σx = (1 0.36 -1.44
0.36 4 0.8
-1.44 0.8 9)
If the correlation matrix is Ωx,
Σx = (σ1 0 0
0 σ2 0 * Ωx * the same as the left matrix.
0 0 σ3)
switch Σx and Ωx, the σi become inverted.
σi = sqrt(Σx(i,i)). Multiply the matrices to get Ωx
Assume that all the entries of an n x n correlation matrix which are not on the main diagonal are equal to ρ. Find the upper and lower bounds on the possible values of ρ.
How many eigenvalues does an n x n matrix with real entries have? How many eigenvectors?
n eigenvalues and n eigenvectors
Let A =
(2 -2
-2 5)
i) find a 2 x 2 matrix M such that M^2 = A
ii) find a 2 x 2 matrix M such that A = MM^t
The 2 x 2 matrix A has eigenvalues 2 and -3 with corresponding eigenvectors (1 2) and (-1 3). if v = (3 1), find Av
Let A and B be square matrices of the same size. Show that the traces of the matrices AB and BA are equal
Can you find n x n matrices A and B such that AB - BA = I_n?
A probability matrix is a matrix with nonnegative entries such that the sum of the entries in each row of the matrix is equal to 1. Show that the product of two probability matrices is a probability matrix.
