(i) An investor sells a futures contract an asset when the futures price is $1,500. Each contract is on 100 units of the asset. The contract is closed out when the futures price is $1,540. How much has the investor made of a gain or a loss. (ii) Consider a variable S, that follows the process [2] ds = dt + dz, = where dz is a Wiener process. For the first three years, the next three years 3 and = 4. If the initial value of the variable is 5, what is the probability distribution of the value of the variable at the end of year six? = 2 and = 3; for [6] (iii) If Z is a normal N(0, 1), then the process X, = Z is continuous and is marginally distributed as a normal N(0, t). Is X a Brownian motion? Explain why. (iv) Consider the general stochastic differential equation dG = A(G,t)dX + B(G, t)dt. Use It's lemma to show that it is possible to find a function f(G) which itself follows a random walk but with zero drift. [4] [8]