1. (30 points) Consider a Term Life insurance contract with:
- Policyholder is 45 years old at the start of the contract.
- Single premium P, so the policyholder only pays at the start of the contract (t = 0).
- Benefit of \$100,000, which is paid at the end of the year of death.
- Maturity of the contract is 5 years.
- Condition to receive the benefit is that the policyholder has to die before the maturity
date.
- r = 0.03 is the interest rate the company is expecting to earn each year. We use continuous
compounding, which means that v = er⇥t
.
The actuarial value is used to price the product, meaning to find a premium at t = 0 for the
contract. The premium is equal to:
P = 100, 000 A 1
45:5
= 100, 000 X
4
k=0
vk+1
k|q45
= 100, 000 (v q45 + v2
1|q45 + v3
2|q45 + v4
3|q45 + v5
4|q45)
= 100, 000 (v q45 + v2 p45 q46 + v3
2p45 q47 + v4
3p45 q48 + v5
4p45 q49)
where npx is the probability that somebody aged x will survive for n years and qx is the probability
that somebody aged x will die in the next year.
For the calculation of the premium, you are given the following vector:
q = ⇥
q45 ; 1|q45 ; 2|q45 ; 3|q45 ; 4|q45⇤
q = [0.080780387 ; 0.087867314 ; 0.095821596 ; 0.104749008 ; 0.11477648]
This means, there is a 0.095821596 probability that the policyholder dies in the third year.
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Questions:
a. (5 points) Calculate the single premium the policyholder needs to pay for this contract.
b. (10 points) Set up a Risk Management strategy such that the company is able to pay the
policyholders until time T = 5. Do this first for 1 policyholder.
- At time t = 0 the policyholder is alive and signs the contract. You want to test in
100 000 scenarios if the policyholder survives or dies in the upcoming 5 years, until
the contract ends.
- Liabilities: Consider the random variable Vt, which denotes the value of the liabilities.
For example between t = 0 and t = 1 there are two possibilities: or he dies
and there is a payout at the end of the year, or he doesn’t die and the contract stays
in force without payment.
- Assets: Consider a random variable At, where
At = P eYt where Yt ⇠ N(µY t, 2
Y t)
We assume that µY = r
2
Y
2 . If for example Y
= 0 we have that
At = ert
- Look at all the different time-steps of the contract until you reach year 5 (maturity).
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5
You will have for each time-step a variable V , A and L. You can do this by using a
loop.
- At t = 5, you make a histogram of the loss random variable you have at that time.
- Explain what you see.
c. (5 points) Do the same for when you have 100 policyholders and 100 000 policyholders.
Explain what happens.
d. (10 points) Calculate the premium for which the insurer only has a 30% loss. Don’t do
this by trial and error but create a function that helps you in calculating this.
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2. (50 points) We need the following information for the this question:
Brownian motion (Wiener process):
Definition:
A stochastic process X = {Xt, t
0} is called a standard Brownian motion if
1. X0 = 0
2. X has independent increments: for 0  s1 < t1  s2 < t2, Xt1
Xs1 and Xt2
Xs2 are
independent random variables.
3. X has stationary increments: The statistical distribution of Xt+s
Xs is independent of
s (and so identical in distribution to Xt).
4. Xt+s
Xt ⇠ N(0, s)
5. X is a continuous function of t.
The standard brownian motion is denoted by W = {Wt, t
0}, where W stands for Wiener.
The geometric Brownian motion for the stock price:
It can be shown that the stock price St has a lognormal distribution:
Consider a trader who has a position of 100 000 written call options on a non-dividend-paying
stock. The portfolio is hedged weekly and the time to maturity of the option is one year. We
have the following information:
- Stock price is S0 = \$100
- Strike price is K = \$100
- N = 52 weeks and T = 1 year
- r = 3% (bank)
- µ = 7%
-
= 20%
Do the following steps in R:
a. (5 points) Find the price of 1 call option.
b. (5 points) Plot the option price (y-axis) in function of the underlying stock prices (xaxis).
How can we relate delta with this graph.
c. (5 points) Find the delta of this option at time t = 0.
Plot delta (y-axis) in function of different stock prices (x-axis). Explain the graph.
d. (10 points) Create a dynamic hedging strategy by simulating the stocks with the above
geometric Brownian motion process. You should have something as follows
Stock Shares Cost of Cumulative
Week Price Delta Purchased Shares Purchased Cash Outflow Interest Cost
0 100 0.5987 59,870.6326 5,987,063.26 5,045,723 2,911.834
1 101.63551 0.6291 3,040.0916 308,981.26 5,357,616 3,091.824
– Explain what happens in week 0, 1 and 2.
– Explain what happens in week 52.
– Calculate the total cost of hedging.
e. (10 points) By running this program multiple times (e.g. 100 000), we will get each
time a new stock price process and thus also the corresponding hedging cost. Plot these
different hedging costs into a histogram and explain what you observe.
f. (5 points) Instead of rebalancing once every week, we can also rebalance two times a
week or daily. Create a histogram of the hedging costs for both cases (twice per week
and daily) and explain what you see.
g. (5 points) Do the same as in steps (d. and e.), but now with volatility
= 30%,
= 40%,

= 50%,
= 60%. Explain what you see in the different histograms of the hedging
costs.
h. (5 points) What happens if we change µ?
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3. (5 points) Suppose that the expected return on a stock in the real world is 7%. The stock price
is currently \$25 per share and its volatility is 20%. Assume you own 10,000 shares. How
much could you expect to lose during the next 6 months with a 99% level of confidence.
4. (10 points) A financial institution has just sold 1,000 7-month European call options on the
Japanese yen. Suppose that the spot exchange rate is 0.80 cent per yen, the exercise price is
0.81 cent per yen, the risk-free interest rate in the United States is 8% per annum, the risk-free
interest rate in Japan is 5% per annum, the volatility of the yen is 15% per annum.
Assume that
• Delta = 0.5250
• Gamma = 4.206
• Vega = 0.2355
• Theta = 0.0399
• Rho = 0.2231
Question: Interpret each number (Delta, Gamma, Vega, Theta and Rho).
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5. (15 points) A financial institution has the following portfolio of over-the-counter options on
euros:
Type Position Delta of Option Gamma of Option Vega of Option
Call -1,500 0.7 2.5 1.5
Call -400 0.9 0.55 0.45
Put -1,850 -0.40 1.6 0.65
Put -1,200 -0.70 1.75 1.2
A traded option is available with a delta of 0.3, a gamma of 1.2, and a vega of 0.5.
Questions:
a. (5 points) What position in the traded option and in euros would make the portfolio both
gamma neutral and delta neutral?
b. (5 points) What position in the traded option and in euros would make the portfolio both
vega neutral and delta neutral?
c. (5 points) Suppose that a second traded option, with a delta of 0.1, a gamma of 0.5, and
a vega of 0.6, is available. How could the portfolio be made delta, gamma and vega
neutral?
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6. (15 points) Figure 1 below shows the distribution of daily revenues of the company K in
2018. In total there are 250 observations and the average daily revenue is about \$5.56 million
with a standard deviation of \$9.25 million.
Figure 1: The distribution of the company K’s daily revenues in millions of \$
a. (5 points) What is the 95% 1-day normal VaR?
b. (5 points) What is the 95% 1-day historically simulated VaR? Assume an equal weighting
scheme.
c. (5 points) Suppose that the 95% VaR is breached on 20 trading days. Is the VaR measure
in line with the confidence level set forth?
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7. (15 points) Let us consider the market risk measurement of JPMorgan Chase & Co. during
2007-2008. According to its 10-K financial reports, the average 1-day market risk VaR constituted
\$107 million in 2007, \$196 million in 2008, while reaching \$266 million on December
31, 2008 and \$100 million on December 31, 2007. For simplicity, assume that the average
VaR over 60 days ending on December 31, 2007 and December 31, 2008 coincide with the
average VaR observed over 2007 and 2008, respectively. The number of VaR violations that
JPMorgan reports during 2007-2008 is summarized in Table 1 below:
Year-Quarter Number of VaR violations (per quarter) Number of VaR violations (per year)
2007Q1 0 0*
2007Q2 0 0*
2007Q3 5 5*
2007Q4 3 8
2008Q1 2 10
2008Q2 0 10
2008Q3 1 6
2008Q4 0 3
Table 1: The number of VaR violations reported by JPMorgan Chase & Co. during 2007-2008.
Source: Quarterly (10-Q) and annual (10-K) filings of JPMorgan Chase & Co., available online
*There were no VaR violations during 2006.
a. (5 points) Based on Table 1 above, determine times during which the JPMorgan’s risk
model passed the regulatory backtest and during which it was requested to be revised (or
possibly rejected).
b. (5 points) What would be the 1-day market risk capital requirements of JPMorgan as at
January 2, 2008 and January 2, 2009 under Basel I and II? (assume there is no specific
risk charge)
c. (5 points) What would the 1-day market risk capital requirements of JPMorgan as at January
2, 2008 and January 2, 2009 become under Basel II.5? Since these are all backward
computations, let us consider 2008 as a year of the stressed market conditions in both
cases.

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