1) (35 pts) The Black-Scholes model of stochastic asset price interest rate St is given by stochastic differential equation = dStSdt + S +dW+ Let = 0.15, and = 0.1.At_t=2_calculate a) Pr(St 0.2) b) Pr(0.15 St 0.25) c) Pr(St > 0.26) St=Soe-t+oW 6- (15%) Assume that X= The final grade of a student in Stochastic Modeling class and Y=The final grade of a student in Structural Analysis class The joint probability of the grades earned by a student is P(X,Y) which is as follows. Structural Analysis Grades Stochastic Modeling Grades 95 80 90 0.2 0.4 75 0.3 0.1 How the grades of Stochastic and Structural Analysis classes are related to each other? (Positive. Negative or no relationship?) *Hint: No need for the final value of p. Do necessary calculations to show how X and Y are related to each other (Positive, Negative or no relationship) Consider the following bivariate regression model for 1,..., n. The regressor is stochastic and exhibits variability in the sample, i.e., (-2)0. The error term u; is i.i.d. with mean zero and constant variance of o for any value of x. Given a random sample of observations {(,, )}from the population of interest, consider the following estimator . (a) (3 marks) Under which assumption is B an unbiased estimator of 8? (b) (5 marks) Let us assume that the assumption in (a) is satisfied so that is unbiased. Will this assumption also ensure that is consistent? In your answer, discuss the difference between unbiasedness and consistency. Consider the following bivariate regression model yi = a + xi + ui, for i = 1,..., n. The regressor is stochastic and exhibits variability in the sample, i.e., Pn i=1(xi -x") 26= 0. The error term ui is i.i.d. with mean zero and constant variance of 2 for any value of xi. Given a random sample of observations {(yi,, xi)} n i=1 from the population of interest, consider the following estimator