Problem Set 1
Note: The time step t used in all the problems is one year. r is a continuously
compounded interest rate.
1.(Trinomial Tree) Consider two assets, one is risky S0 the other is risk-free B0 under
the trinomial tree framework. We want to use mi
Problem Set 4
Question 1 (VBA programming)
Assume that the stock price over one year St , S0 = 100, follows a GBM with = 0.1, = 0.2.
(a) Write a VBA program to simulate 1000 price paths of the stock over the year with the
time step equal to one day, assum
Problem Set 2
Question 1 (VBA programming)
Consider an European put option with the following inputs:
S0 = 100; K = 105; = 7%; = 0.3; r = ln(1.05)
1
The time step t = 3 months or 4 years. The maturity of the option is 3 years. Your task is
to write a VBA
Solution for Assignment 3
Question 1
a) Determine the distribution of Z1 (t).
Solution: It is easy to see Z1 (t) can be written as the Riemann sum of normal random
variables. Using the property that the linear combination of normal random variables is sti
Solution for Problem Set 1
Question 1.
Solution:
a). To show the market does not admit arbitrage, it is equivalent to the condition
V0 = 0, P (V1 0) = 1, P (V1 > 0) > 0
does not hold. It is easy to see
V0 = S0 + B0 = 0 =
S0
B0
We get V1 = (S1 100), and b
Solution for Problem Set 4
Question 2
a). Obviously it can be written as the linear combination of one put option and two call
options both with strike K = 60, i.e
(ST ) = max(60 ST , 0) + 2 max(ST 60, 0)
b). To determine the price, we just need to use Bl
Solution for Problem Set 2
Question 2
(a) Determine the value of d.
To get the value of d, we just apply the formula in the lecture notes.
2 2
2
d = e t + t = 0.7985
(b) Determine the price of this option at time 0;
To determine the option price, you can
Problem Set 3
Question 1
Let Wt be a standard Brownian motion. Dene three stochastic processes Zi (t), i = 1, 2, 3
as
t
t
1
Ws ds, Z2 (t) =
Ws dW s, Z3 (t) = [Wt2 t].
Z1 (t) =
2
0
0
For each xed t > 0,
(a) determine the distribution of Z1 (t);
(b) determi
1. (5 points) Seventy-ve percent of the claims have a normal distribution with mean of 3, 000 and
a variance of 1, 000, 000. The remaining 25% have a normal distribution with a mean of 4, 000 and a
variance of 1, 000, 000. Determine the probability that a
ACT451 Loss Models I
Quiz #1
Name
ID #
10 marks
Loss X has a single parameter Pareto distribution with = 4 and = 20. Let Y = ln(X/). Identify the
density of Y .
ACT451/STA2500 Loss Models
Quiz #2
Name
ID #
10 marks
Loss random variable Y has a two-point mixture with survival function
S(x) = 0.4ex + 0.6ex/2 .
Suppose that Y has V aR = 4 for some 0 < < 1.
Determine and then T V aR .