Question 1;

An analyst is investigating the extent to which the price of a stock at the beginning and end of a time interval can be used to provide information about its price during the interval. The model used is S, = Satur+ GB, where B, is a standard Brownian motion. The analyst wishes to investigate the difference between S, and the price S, which would be predicted given only So and ST. given by S, = (T - D)So + DT OSIST. T (i) Write down E(S, - So), Var(S, - So). E(Sr - S,), Var(Sy-5,) and Cov(S, - 5p. ST - 5,), for O SIST. [3] (ii) Calculate the expectation and variance of S, - S,. [4] (iii) Find the value of re [0, 7] where Var(S, -S,) is greatest. Comment on your answer. [2] (iv) Give two reasons why the model S, = Spexp(ur + GB,) might be more suitable than the one used by the analyst and discuss whether the results of (iii) might have been substantially different if the analyst had used this model in the first place. [3](i) Describe the three elements of a linear congruential generator, and set out the recursive relationship used to generate the pseudo-random number sequence. [3] (ii) The first three numbers produced by a linear congruential generator are 0.954, 0.462 and 0.628. Use these to generate three pseudo-random numbers from the Pareto distribution with o = 2 and 2 = 1. [The density of the Pareto distribution is f(x) = - (2+ x)a+ 1 (x > 0).] [4] An exponential random variable / with rate parameter ). may be simulated using the formula T= --log U where U is uniformly distributed on [0, 1]. Explain how this can be used to simulate a path of a Markov jump process (X,: /2 0) which has two states, H (healthy) and $ (sick), with transition rates o from H to $ and p from S to H. Assume X = H. [4]Let (x,: 1