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10.13 A company uses the GARCH(1,1) model for updating volatility. The three parameters are ω, α, and β. Describe the impact of making a small increase in each of the parameters while keeping the others fixed. 10.14 The parameters of a GARCH(1,1) model are estimated as ω = 0.000004, α = 0.05, and β = 0.92. What is the long-run average volatility and what is the equation describing the way that the variance rate reverts to its long-run average? If the current volatility is 20% per year, what is the expected volatility in 20 days? 10.15 Suppose that the daily volatility of the FTSE 100 stock index (measured in pounds sterling) is 1.8% and the daily volatility of the dollar–sterling ex- change rate is 0.9%. Suppose further that the correlation between the FTSE 100 and the dollar–sterling exchange rate is 0.4. What is the volatility of the FTSE 100 when it is translated to U.S. dollars? Assume that the dollar–sterling exchange rate is expressed as the number of U.S. dollars per pound sterling. (Hint: When Z = XY, the percentage daily change in Z is approximately equal to the percentage daily change in X plus the percentage daily change in Y.) 10.16 Suppose that GARCH(1,1) parameters have been estimated as ω = 0.000003, α = 0.04, and β = 0.94. The current daily volatility is estimated to be 1%. Estimate the daily volatility in 30 days. 10.17 Suppose that GARCH(1,1) parameters have been estimated as ω = 0.000002, α = 0.04, and β = 0.94. The current daily volatility is estimated to be 1.3%. Estimate the volatility per annum that should be used to price a 20-day option.
3 Chapter 11 Questions 11.1 If you know the correlation between two variables, what extra information do you need to calculate the covariance? 11.2 What is the difference between correlation and dependence? Suppose that y = x2 and x is normally distributed with mean zero and standard deviation one. What is the correlation between x and y? 11.3 What is a factor model? Why are factor models useful when defining a correlation structure between large numbers of variables? 11.4 What is meant by a positive-semidefinite matrix? What are the implications of a correlation matrix not being positive-semidefinite? 11.5 Suppose that the current daily volatilities of asset A and asset B are 1.6% and 2.5%, respectively. The prices of the assets at close of trading yesterday were $20 and $40 and the estimate of the coefficient of correlation between the returns on the two assets made at that time was 0.25. The parameter λ used in the EWMA model is 0.95.(a) Calculate the current estimate of the covariance between the assets. (b) On the assumption that the prices of the assets at close of trading today are $20.50 and $40.50,