A certain stock has low volatility on some days and high volatility on other days

Chapter 9, Problem 36

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A certain stock has low volatility on some days and high volatility on other days. Suppose that the probability of a low volatility day is p and of a high volatility day is q = 1 p, and that on low volatility days the percent change in the stock price is N (0, 2 1), while on high volatility days the percent change is N (0, 2 2), with 1 < 2. Let X be the percent change of the stock on a certain day. The distribution is said to be a mixture of two Normal distributions, and a convenient way to represent X is as X = I1X1+I2X2 where I1 is the indicator r.v. of having a low volatility day, I2 = 1I1, Xj N (0, 2 j ), and I1, X1, X2 are independent. (a) Find Var(X) in two ways: using Eves law, and by calculating Cov(I1X1 + I2X2, I1X1 + I2X2) directly. (b) Recall from Chapter 6 that the kurtosis of an r.v. Y with mean and standard deviation is defined by Kurt(Y ) = E(Y ) 4 4 3. Find the kurtosis of X (in terms of p, q, 2 1, 2 2, fully simplified). The result will show that even though the kurtosis of any Normal distribution is 0, the kurtosis of X is positive and in fact can be very large depending on the parameter values.