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)
