4. (a) Let (2, F, P) be a probability space and let A, B e F. Show that P(AB) P(B) + P(A/B) P(BC) = P(A). (b) Let = {1,

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4 A Let 2 F P Be A Probability Space And Let A B E F Show That P Ab P B P A B P Bc P A B Let 1 1 (64.56 KiB) Viewed 31 times

4. (a) Let (2, F, P) be a probability space and let A, B e F. Show that P(AB) P(B) + P(A/B) P(BC) = P(A). (b) Let = {1, 2, 3, 4, 5, 6) and F= P(n). Let X = 1(1,2,3,4,5) +2.1(0) and Y = 2.1(4,6}. Suppose P is the probability measure defined on 2 by P({1})= P({2})=P({{3})=; ) = 1/² P({4})=P({5})=a. P({6}) = 8, with a, ß € [0, 1]. Knowing that X and Y are independent, what are the values of a and 3? 5. Let us consider a stock (called "stock A") whose price today is €12. Suppose that in six months' time the price can only go up to €16 or down to €10. Suppose the interest rate is 10% per annum continuously compounded. Let us consider a call option (called "call option B") written on this stock, with maturity date of 6 months and strike price €13. (a) Determine the fair price c for this call option (show all the steps of your work). Assume that the seller of call options B starts with zero capital, sells 100 call options (each at the fair price c obtained in the previous question), and borrows money to buy A shares of the stock A. (b) How many shares should he buy in order to run no risk at the end of the six months period?