Hull: Options, Futures, and Other Derivatives, Ninth EditionChapter 23: Estimating Volatilities and Correlations Multiple Choice Test Bank: Questions with Answers 1. How many parameters are necessary to define an EWMA model A. B. C. D. 1 2 3 4 Answer: A One parameter (lambda) is necessary to define an EWMA model. 2. How many parameters are necessary to define a GARCH (1,1) model A. B. C. D. 1 2 3 4 Answer: C Three parameters (omega, alpha, and beta) are necessary to define a GARCH (1,1) model. 3. At the end of Thursday, the estimated volatility of asset A is 2% per day. During Friday asset A produces a return of 3%. An EWMA model with lambda equal to 0.9 is used. What is an estimate of the volatility of asset A at the end of Friday? A. 2.08% B. 2.10% C. 2.12% D. 2.14% Answer: C The variance rate is 0.9×0.022+0.1×0.032 = 0.00045. The volatility per day is the square root of this or 2.12%. 4. At the end of Thursday, the estimated volatility of asset B is 1% per day. During Friday asset B produces a return of zero. An EWMA model with lambda equal to 0.9 is used. What is an estimate of the volatility of asset A at the end of Friday? A. 0.98% B. 0.95% C. 0.92% D. 0.90% Answer: B The variance rate is 0. 0. 0. D.000090. C. During Friday asset A produces a return of 3% and asset B produces a return of zero. Which of the following is true when the parameter lambda equals 0.95 B.1×0. The weights given to observations add up to 0. 7. 0.9 is used. Which of the following is a definition of the covariance between X and Y? A.00009. the estimated covariance between assets A and B is 0.95 .95%. The volatility per day is the square root of this or 0. B. What is an estimate of the covariance at the end of Friday? A. Correlation between X and Y times variance of X times variance of Y B. What does EWMA stand for? A.012+0. 5.1×0.02 = 0. Correlation between X and Y divided by the product of the standard deviation of X and the standard deviation of Y D. The weight given to the most recent observation is 0.000095 Answer: A The covariance is 0.000081 C. Variance of X times the variance of Y C. The weight given to the observation one day ago is 95% of the weight given to the observation two days ago C.9×0. At the end of Thursday. 0.0001+0.000090 B.03×0 = 0.95? A.0001.9×0. An EWMA model with lambda equal to 0. Equally weighted moving average Equally weighted median approximation Exponentially weighted moving average Exponentially weighted median average Answer: C EWMA stands for exponentially weighted moving average 8. 6. Correlation between X and Y times standard deviation of X times standard deviation of Y Answer: D Covariance is the coefficient of correlation multiplied by the product of the two standard deviations.000100 D. 9.D. The implied volatilities of long-dated options tend to move by more than the implied volatilities of short-dated options C.2 0. They calculate data occurring C. All option implied volatilities tend to move by the same about from one day to the next B. 20% B. They calculate B. Which of the following is true of maximum likelihood methods A. 10% C. The implied volatilities of short-dated options tend to move by more than the implied volatilities of long-dated options .25 0. EWMA models do not. 12. D. The weights given to the observation two days ago is 95% of the weight given to the observation one day ago Answer: D The weights given to observations go down by 5% for each day we go back in time.1 or 10%. They involve a the maximum possible values for parameters parameters that give the highest probability of past values for key variables that are most likely to occur in multivariate regression analysis Answer: B Maximum likelihood methods are concerned with estimating parameter values by searching for the values that maximize the chance of observed data occurring. 2% Answer: B The volatility per quarter is 0. If the volatility for a portfolio is 20% per year. B. 11. Which of the following is true GARCH models incorporate mean reversion. EWMA models do not EWMA models incorporate mean reversion. what is the volatility per quarter? A. Which of the following is true A. A. They calculate the future D. 5% D. 10. C. GARCH models do not Both GARCH and EWMA models incorporate mean reversion Neither GARCH nor EWMA models incorporate mean reversion Answer: A GARCH models incorporate mean reversion. 000113. 1. The new volatility is the square root of this or 1. alpha = 0.1) model are: omega =0.04×0. 1. 14. If we observe a change in the value of the variable equal to 2%. The parameters in a GARCH (1. Sometimes C is true and sometimes D is true Answer: C When there is a shock causing an increase or decrease in volatility its impact tends to disappear over time.012 = 0.000002+0.95.04.16% C. 13. 1. Volatilities and correlations increased in the second half of 2008 C.06% 16. All elements of the matrix are positive B. Which of the following is true of a positive semi-definite variancecovariance matrix A.D.1) model are: omega =0. Volatilities and correlations decreased in the second half of 2008 B. Volatilities decreased and correlations increased in the second half of 2008 D. Volatilities increased and correlations decreased in the second half of 2008 Answer: B As the four-index example in the chapter shows.95. The matrix is symmetric D. The current estimate of the volatility level is 1% per day. alpha = 0.95×0. The determinant of the matrix is positive C.) The result is that the increase or decrease volatility has most effect on the implied volatilities of short-dated options.000002. (This is the effect of mean reversion. Which of the following is the closest to the long run .06% D.000002. The parameters in a GARCH (1. how does the estimate of the volatility change A. and beta = 0. 1. Which of the following is true A.02 2+0. The matrix is internally consistent Answer: D The correlations between many different variables are not internally consistent unless the variance-covariance matrix is positive semi-definite.04. and beta = 0.26% B.03% Answer: C The new variance rate is 0. both volatilities and correlations increased in the second half of 2008 15. 1) model are: omega =0.09%. 1.2% C. 0.04.95) 20×(0.0002. The parameters in a GARCH (1. The current estimate of the volatility level is 1% per day.0120. The maximum likelihood is the same for GARCH (1. and beta = 0.000002/(1−0. 1.5% 1. and beta = 0.0% 1. D. Sometimes A is true and sometimes B is true.04+0.11% D.000002.95)=0. C. Which of the following is true A.0% Answer: B per per per per day day day day .0002.average volatility? A. The volatility per day is the square root of this or 1. alpha = 0.95)=0.04−0.000002.4% 17.04−0.10% C. EWMA will always give a maximum likelihood at least as high as GARCH (1.1) must give at least as good a fit to the data as EWMA.1) will always give a maximum likelihood at least as high as EWMA B. 1. GARCH (1.09% B. B. 1. 1.5% 2.1) when the omega parameter is zero.0002+(0. 1. What is the reversion rate for the variance rate implied by the model A.3% D. alpha = 0.95.4% Answer: D The long run average variance rate is 0. It follows that GARCH (1.95. Answer: A EWMA is the particular case of GARCH (1.1) and EWMA D. 1.000002/(1−0.12% Answer: A The long run average variance rate is 0. 18.1) C. The long run average volatility per day is the square root of this or 1.04. What is the expected volatility in 20 days? A. The parameters in a GARCH (1.000118.1% B. 1. The expected variance rate in 20 days is 0. 19.1) model are: omega =0.0002) = 0. This means that the reversion rate is zero. EWMA is a particular case of the GARCH (1. 20.The reversion rate is 1−In this case it is 1−0.1).1) Sometimes EWMA has a higher reversion rate than GARCH (1.1) model where =0.95 = 0. Which of the following is true EWMA is a particular case of GARCH (1.1). but it is not zero EWMA has a higher reversion rate than GARCH (1. A. .04−0.1) and sometimes it has a lower reversion rate than GARCH (1. and .1). C. B. Answer: A The reversion rate is 1− in GARCH (1.01 or 1% per day.1) where the reversion rate is zero EWMA has a lower reversion rate than GARCH (1. D.