Find the expected value of the stock’s terminal S value assuming it will fall within the range (i) 50-100; (ii) 0 – 50; (iii) 0 to 100.

Question 1 (30 Marks)
A stock’s terminal value S has a uniform distribution: that is, it is equally likely to
assume any value in the range (0-100) and will not assume any value outside of this
range. The random variable x on which this stock’s value is based has a density
function p(x) =1 for 0 ≤ x ≤ 1 and 0 elsewhere. The stock’s random terminal value is
f(x) =100x.
(a)
Find the distribution function P(x) for p(x)
(4 marks)
(b)
Find the expected value of the stock’s terminal S value assuming it will fall within
the range (i) 50-100; (ii) 0 – 50; (iii) 0 to 100.
(7 marks)
(c)
Find the variance of S in the range 0 – 100 (8 marks)
(d)
What would be the expected future cash flow (contingent on its exercise) of a call
option written on this stock if its exercise price were $50? That is, what is the
expected cash flow of the option conditional on its exercise?