Exam Questions

Question 1

How to compure an embedded jump chain (based on uniformization)?

Answer 1

# 1. Choose η s.t. η ≥ sup | Q(i,i)|
2. P = I + 1/η Q

Note: you do not have to say what η is, just how we define it.

Question 2

How to compute a pure jump chain?

Answer 2

Set all diagonal values to zero, divide all others by the absolute value of the diagonal element (per row).

Question 3

How to describe a simulation algorithm of a continuous Markov chain?

Answer 3

“We sunulate the embedded jump chain. We let X₀=x₀ as the initial state, and repeat the following:

1. Simulate the holding time T_n which is exponentially distributed with mean 1/η
2. From state i X_n, we reach new state X_n+1 = j from probability distribution P(i, ·).

For all t ≥ 0, X_t=X_n, Σ_k=0ⁿT_k ≤ t ≤ Σ_k=0ⁿ⁺¹T_k​, with T₀=0

Question 4

How to calculate the variance?

Answer 4

Var(x) = E[(x-E(x))(x-E(x))’] (‘ only if matrix)

Question 5

What are the requirements of ANOVA?

Answer 5

1. Normally distributed data
2. Equal variances
3. Y₁, …, Y_n are independent

Question 6

How to test for equal variances?

Answer 6

A statistical test e.g., Levene-test

Question 7

What to do after ANOVA suggests different means?

Answer 7

Post-hoc analysis, e.g. Tukey test (where you just perform a two-sample test on all pairs of Y_i and Y_j)

Question 8

How to provide a SPSA descent algorithm?

Answer 8

A generalized optimization algorithm goes as follows:

θ_n+1 = θ_n + ε_nG(θ_n)

When using the SPSA descent algorithm we choose ε_n = 1 / (n + 1), and (G(θ_n))_i = - (J(θ_n + η_nΔ_n) - J(θ_n - η_nΔ_n))/(2η_nΔ_n(i)), i = 0, 1, 2, .., k

We need to choose an η_n s.t.:

∑ε_k²/η_k² = Σ 1/ ((k+ 1)η_k²)≤∞

Question 9

What are the requirements of Δ_k(i) of the SPSA?

Answer 9

Δ_k(i) and 1/Δ_k(i) need to be bounded, symmetric around zero and mutually independent.

Question 10

How to describe the clock structure of a DES model?

Answer 10

Question 11

Answer 11

Question 12

How to check the mean of a RNG?

Answer 12

1. Get SE = sqrt(Var(x)/n)
2. Get 95% (or other) conf. interval (x^bar - 1.96SE, x^bar + 1.96SE)
3. Check if 0.5 in the interval.

Question 13

Answer 13

Question 14

Try to generate a random standard normal distribution

Answer 14

Question 15

λ from an exponential distribution

Answer 15

Question 16

How to calculate the sample variance of a bernoulli distribution, with just the sample mean?

Answer 16

S² = n /(n-1) Y^bar(1 - Y^bar)

Question 17

Answer 17

Question 18

Answer 18

Note: important to use t-dist.

Question 19

Answer 19

Question 20

Answer 20

Question 21

How to compute a Euler scheme? How to get an algorithm?

Answer 21

Set dt := h, dX(t) := X(t + h) - X(t), (and somtimes useful dW(t) = Zsqrt(h), where Z ~ N(0, 1)).

Solve for X(t + h). If you need an algorithm, then simulate while T≥t.

Question 22

How to construct a replicating portefolio?

Answer 22

Set

Δbest value + (1+r)b = besvt value - strike price (K)

Δworst value + (1 +r )b = 0 (usually or worst value - K if larger then 0)

Question 23

How to verify the no arbitrage policy?

Answer 23

Compute the profit in best case, and worst case. Note that always it is exactly possible to pay all parties involved.

Question 24

How to compute option prices using the risk-neutral probability?

Answer 24

best price = u S₀ => u = best price / current price (S₀)

worst price = d S₀ => d = worst price / current price (S₀​)

p* = (1 + r - d)/(u-d)

Thus C = 1/(1+r) E*[(S₁-K)⁺] = p* · (best price - strike price(K))