3.1 Suppose that the price of an asset at close of trading yesterday was $350 and its volatility was estimated as 1.4% per day. The price at the close of trading today is $347. Update the volatility estimate using

(a) The EWMA model with = 0.95,

(b) The GARCH(1,1) model with = 0.000003, = 0.05, and = 0.95.

3.2 The number of visitors to websites follows the power law in equation (10.1) with = 2.

Suppose that 1.5% of sites get 550 or more visitors per day. What percentage of sites get

(a) 1,500 more visitors per day

(b) 2,500 or more visitors per day

3.3 Suppose that GARCH(1,1) parameters have been estimated as = 0.000004, = 0.05,and = 0.94. The current daily volatility is estimated to be 1%. Estimate the daily volatility in 30 days.

3.4 Suppose that the current daily volatilities of asset A and asset B are 1.65% and 2.32%, respectively. The prices of the assets at close of trading yesterday were $35 and $52 and the estimate of the coefficient of correlation between the returns on the two assets made at that time was 0.26. The parameter 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 $36 and $53,update the correlation estimate.

3.5 Answer the following questions.

(a) What is the implication of a correlation matric not being positive-semidefinite?

(b) Why are the diagonal elements of a correlation matrix always 1?

(c) Making small changes to a positive-semidefinite matrix with 100 variables will have no

effect on the matrix. Explain this statement?