A certain stock has low volatility on some days and high volatility on other days Suppose that the probability of a low

Question

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 o while on high volatility days the percent change is N 0 0 with a 02 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 I is the indicator r v of having a low volatility day I2 1 4 X N 0 03 and I1 X1 X2 are independent a Find Var X in two ways using Eve s law and by calculating Cov I X 12X2 11X1 12X2 directly b Recall from Chapter 6 that the kurtosis of an r v Y with mean and standard deviation is defined by E Y 04 Kurt Y 3 Find the kurtosis of X in terms of p q oi o 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

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