Finance 567 Homework 4: Expected shortfall and EVT

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1. Expected shortfall using the Normal distribution. (total 3 points) You have written 60 calls
and 60 puts on ABC stock, and 40 calls and 40 puts on DEF stock. Each option is on 100 shares.
The ABC call and puts have strike prices of $100, and the DEF calls and puts have strike prices
of $150. All of the options expire in one month (21/252 year). The prices of ABC and DEF are
$101.17 and $148.97 per share, respectively. The continuously compounded interest rate is 0.01
or 1% per year, and neither ABC nor DEF will pay dividends during the next month. The
implied volatility of ABC is 0.45 or 45% and the implied volatility of DEF is 0.37 or 37%.

(a) (1 point) The expected log returns on ABC and DEF are 0.0005 (0.05%) and 0.0004 (0.04%)
per day, respectively, and the standard deviations of the log returns are 0.028 (2.8%) and 0.023
(2.3%) per day. The correlation between their returns is 0.40 (40%). Assume that the log returns
of the two stocks are normally distributed. Consider a simple model in which the only market
factors are the returns of the two stocks, and use the Monte Carlo method with a probability of
5% to compute the VaR of the portfolio. What is your estimate of the 5% VaR?
Remark: Be sure to work with log or continuously compounded returns, not simple returns.
(b) (2 points) Use VaR(5%) denote the VaR you computed in part (a). Using the probability of
5%, what is the expected shortfall? That is, what is E[Loss | Loss ≥VaR(5%)]?
2. Expected shortfall using the bivariate t distribution. (total 3 points) Consider again the
portfolio in Question 1, and once again assume that the VaR horizon is one trading day. The
mean of the log returns is
�
𝜇𝜇1
𝜇𝜇2
� = �
0.0005
0.0004�,
as in Question 1. The covariance matrix C is
𝐶𝐶 = � 𝜎𝜎1
2 𝜎𝜎1𝜎𝜎2𝜌𝜌
𝜎𝜎1𝜎𝜎2𝜌𝜌 𝜎𝜎2
2 � = � (0.028)2 (0.028)(0.023)(0.4)
(0.028)(0.023)(0.4) (0.023)2 �,
and the degrees of freedom is ν = 4.
(a) (1 point) Use these parameters, the Monte Carlo method, and a one-day horizon to compute
the 5% VaR of the portfolio.
2
(b) (2 points) Use VaRt(5%) denote the VaR you computed in part (a). Using the probability of
5%, what is the expected shortfall? That is, what is E[Loss | Loss ≥VaRt(5%)]?
3. EVT (total 4 points) You are interested in using Extreme Value Theory (EVT) to model the
distribution of daily stock index returns above some threshold u.
(a) (2 points) You first estimated that the threshold u = 0.02 and you also discovered that 250 of
the 8,000 observations you use exceed 0.02. You then use the method of maximum likelihood to
estimate that ξ = 0.36, β = 0.008. Based on the estimates u = 0.02, ξ = 0.36, and β = 0.008, what
is the probability that a daily return R is greater than 0.03?
(b) (2 points) Let R denote a return, and consider the mean excess function E[R – u | R ≥ u].
Suppose that for a threshold of u = 0.02 the mean excess is E[R – 0.02 | R ≥ 0.02] = 0.012, and
for a threshold of u = 0.03 the mean excess is E[R – 0.03 | R ≥ 0.03] = 0.016. If u = 0.02 is a
reasonable choice of a threshold, then what should be the mean excess at a threshold of u = 0.04,
that is what is E[R – 0.04 | R ≥ 0.04]?
Remark: Part (b) does not use the parameter estimates from part (a).