Please refer to the attachment.

Assume that the spot rate of interest at time r. S(), can be modelled by 5() -e"V, where W() is a Brownian motion with drift coefficient of and volatility coefficient I such that HIO) = 0. Write down an expression for W(f) in terms of a standard Brownian motion. B(). [1] (in) Show that (S(r):/ > 0) is a continuous-time martingale. [4] [Total 5]A derivatives trader is modelling the volatility of an equity index using the following discrete-time model : Model 1: Of = 0.12+0.40,-1 +0.056, , 1= 1,2.3.... where o, is the volatility at time f years and 6 ,62.-.. are a sequence of independent and identically-distributed random variables from a standard normal distribution. The initial volatility To equals 0.15. Determine the long-term distribution of a, . (3] The trader is developing a related continuous-time model for use in derivative pricing. The model is defined by the following stochastic differential equation (SDE): Model 2: do, -a(0, - p )it + Bdw, where o, is the volatility at time f years, W, is standard Brownian motion and the parameters cr, / and & all take positive values. (ii) (a) Show that for this model: (b) Hence determine the numerical value of a and a relationship between the parameters of and # if it is required that o, has the same long-term mean and variance under each model. (e) State another consistency property between the models that could be used to determine precise numerical values for o and B. The derivative pricing formula used by the trader involves the squared volatility V, =of , which represents the variance of the returns on the index. (iii) Determine the SDE for , in terms of the parameters or, S and #. [2] [Total 12]