MTH5510: QRM - Exercises Set 2
Due date: August 12, 2024;12:00pm;
Analysis the output of the Matlab code is mandatory. I am not interested just to the Matlab code.
Hand in stapled hardcopy at the beginning of the tutorial session
Note: You might want to use Matlab for this exercise; adequately report and comment on your
results (For a quick introduction to Matlab visit http://www.mathworks.com/access/helpdesk/
help/pdf_doc/matlab/getstart.pdf)
Exercise I: This question deals with a portfolio of five stocks. At time t, the values of the stocks
are S1,t = 100, S2,t = 50, S3,t = 25, S4,t = 75, and S5,t = 150. The portfolio consists of 1 share
of S1, 3 shares of S2, 5 shares of S3, 2 shares of S4, and 4 shares of S5. These risk factors are
logarithmic prices and the factor changes have mean zero and standard deviations 10−3, 2 · 10−3,
3 · 10−3, 1.5 · 10−3, and 2.5 · 10−3, respectively. The risk factors are independent.
1. Compute VaRα, VaR
mean
α , and ESα using Monte Carlo with 10,000 simulations. Do this for
α = {0.90, 0.91, . . . , 0.99}. Also use the following distributions for the risk factor changes:
• For each i ∈ {1, 2, 3, 4, 5}, Xi,t+δ ∼ t(3, µ, σ) for appropriate values of µ and σ
• For each i ∈ {1, 2, 3, 4, 5}, Xi,t+δ ∼ t(10, µ, σ) for appropriate values of µ and σ
• For each i ∈ {1, 2, 3, 4, 5}, Xi,t+δ ∼ t(50, µ, σ) for appropriate values of µ and σ
• For each i ∈ {1, 2, 3, 4, 5}, Xi,t+δ ∼ N (µ, σ2)
Plot the results.
2. Comment on the following:
• The value of VaRα compared to VaRmeanα
• The value of VaRα and ESα as compared between the four distributions. Are the results
what you expected?
Exercise II: This question deals with delta hedged call option. The following are the Black-
Scholes parameters for a European call option at time t = 0:
T = 0.5
rt = 0.05
σt = 0.2
St = 100
K = 100.
1
The portfolio consists of long position on the call option, and the corresponding position in the
stock which makes the portfolio delta neutral. Let ∆ = 1day, Z1 = log(S), and Z2 = σ (r
will be considered in this problem). The risk factor changes are normally distributed with mean
zero. Their standard deviations over one day are 10−3 and 10−4 and their correlation is −0.5.
(a) Compute V aRα, V aR
mean
α , and ESα for α = 0.95 and α = 0.99 using the following
methods:
• Monte Carlo full revaluation with 10,000 simulations
• Monte Carlo on the linearized loss with 10,000 simulations
• Variance-covariance method
Do not neglect the time derivative in any linearizion in this question.
Exercise III: Let L have the Student t distribution with ν degree of freedom. Derive the
formula
ESα(L) =
(
gν(t
−1
ν (α))
1− α
)(
ν + (t−1ν (α))2
ν − 1
)
.
You will need to look up the probability density function of the distribution at hand.