Fin 500Q – Quantitative Risk Management Homework #3
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Fin 500Q – Quantitative Risk Management
Homework #3 Solutions
1. Suppose that the price of an asset at close of trading yesterday was $300 and its volatility was estimated as 1.3% per day. The price at the close of trading today is $298. Update the volatility estimate using:
(a) The EWMA model with λ = 0.94.
Answer: Using the EWMA model, the variance is updated to
0.94 × 0.0132 + 0.06 × (−2/300)2 = 0.00016153
so that the new daily volatility is ^0.00016153 = 0.01271 or 1.271%.
(b) The GARCH(1,1) model with ω = 0.000002,α = 0.04, and β = 0.94.
Answer: Using GARCH(1,1), the variance is updated to
0.000002 + 0.94 × 0.0132 + 0.04 × (−2/300)2 = 0.00016264
so that the new daily volatility is ^0.00016264 = 0.01275 or 1.275%.
2. Suppose that the parameters in a GARCH(1,1) model are α = 0.03,β = 0.95, and ω = 0.000002.
(a) What is the long-run average volatility?
Answer: The long-run average variance, VL , is
ω 0.000002
1 − α − β 0.02
The long-run average volatility is ^0.0001 = 0.01 or 1% per day.
(b) If the current volatility is 1.5% per day, what is your estimate of the volatility in 20, 40, and 60 days?
Answer: The expected variance in 20 days is
0.0001 + 0.9820 (0.0152 − 0.0001) = 0.000183.
The expected daily volatility is therefore ^0.000183 = 0.0135 or 1.35%. Similarly, the expected volatilities in 40 and 60 days are 1.25% and 1.17% per day, respectively.
(c) What volatility should be used to price 20-, 40, and 60-day options?
Answer: We have a = ln ( ) = 0.0202. The variance used to price 20-day options is 252 × [0.0001 + 1 (0.0152 − 0.0001)] = 0.051
so that the annual volatility is 22.61%. Similarly, the volatilities that should be used for 40- and 60-day options are 21.63% and 20.85% per annum, respectively.
(d) Suppose that there is an event that increases the current volatility from 1.5% per day to 2% per day. Estimate the effect on the volatility in 20, 40, and 60 days.
Answer: The expected variance in 20 days is now
0.0001 + 0.9820 (0.022 − 0.0001) = 0.0003.
The expected daily volatility in 20 days is therefore ^0.0003 = 0.0173 or 1.73%. Similarly, the expected daily volatilities in 40 and 60 days are 1.53% and 1.38%, respectively.
(e) Estimate by how much the event increases the volatilities used to price 20-, 40, and 60-day options. Answer: When today’s volatility increases from 1.5% per day to 2% per day, the volatility used to price a 20-day option changes by
1 − e −0 .0202×20 0.015
or 6.88% on an annual basis, bringing the volatility up to 29.49%. Similarly, the 40- and 60-day volatilities increase to 27.63% and 26.10%, respectively.
3. The probability density function for an exponential distribution is λe −λx where x is the value of the
variable and λ is a parameter. The cumulative probability distribution is 1 − e −λx . Suppose that two
Gaussian copula to define the correlation structure between V1 and V2 . You can use the file “bivar.xls”
to compute values of the cumulative bivariate normal distribution function.
(a) What is the probability that V1 ≤ 1?
Answer: The probability is 1 − e −1 .0·1 = 0.632.
(b) What is the probability that V2 ≤ 1?
Answer: The probability is 1 − e −2 .0·1 = 0.865.
(c) With a copula correlation of 0, what is the probability that V1 ≤ 1 and V2 ≤ 1?
Answer: The probability that V1 ≤ 1 is transformed to the normal value U1 = Φ −1 (0.632) = 0.337. This probability can be calculated in Excel with the formula =NORM .INV(0.632, 0, 1). Similarly, the probability that V2 ≤ 1 is transformed to the normal value U2 = Φ −1 (0.865) = 1.102. With a copula correlation of 0, we can use the provided Excel file to find that the joint probability is M(0.337, 1.102, 0) = 0.547 (= 0.632 × 0.865).
(d) With a copula correlation of 0.5, what is the probability that V1 ≤ 1 and V2 ≤ 1?
Answer: With a copula correlation of 0.5, the joint probability is M(0.337, 1.102, 0.5) = 0.591.
(e) With a copula correlation of −0.2, what is the probability that V1 ≤ 1 and V2 ≤ 1?
Answer: With a copula correlation of −0.2, the joint probability is M(0.337, 1.102, −0.2) = 0.531.
4. Suppose that a bank has made a large number of loans of a certain type. The one-year probability of default on each loan is 1.2%. The bank uses a Gaussian copula for time to default. It is interested in estimating a 99.97% worst case for the percent of loans that default on the portfolio. Show how this worst case percentage varies with the copula correlation, using copula correlations of 0 .2, 0.4, 0.6, and 0.8.
Answer: The WCDR with a 99.97% confidence level is
Φ ( ) ,
where Φ is the CDF of a standard normal distribution. We compute that Φ −1 (0.012) = −2.257 (Excel:
WCDR = Φ ( −2.25 · 3.432 ) = Φ(−0.807) = 0.210,
which is computed in Excel as =NORM .DIST(−0.807, 0, 1, 1).
The table below gives the WCDR for different values of the copula correlation.
ρ |
WCDR(%) |
0.2 |
21.0 |
0.4 |
45.5 |
0.6 |
73.7 |
0.8 |
96.5 |
2023-04-29