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Question 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 / years and &, 82,... are a sequence of independent and identically-distributed random variables from a standard normal distribution. The initial volatility Go equals 0.15. (i) Determine the long-term distribution of of . [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, - ")di + BdW, where o, is the volatility at time / years, W, is standard Brownian motion and the parameters o, / and / all take positive values. (ii) (a) Show that for this model: (b) Hence determine the numerical value of / and a relationship between the parameters o and / if it is required that o, has the same long-term mean and variance under each model. (c) State another consistency property between the models that could be used to determine precise numerical values for o and B . [7] The derivative pricing formula used by the trader involves the squared volatility V, =0, , which represents the variance of the returns on the index. (iii) Determine the SDE for , in terms of the parameters o, / and /. [2] [Total 12]Question This question is based on Subject 103 September 2004 Question 10(i) and (ii). Assume that the spot rate of interest at time /, S(), can be modelled by S()=e-20), where W(1) is a Brownian motion with drift coefficient / and volatility coefficient 1 such that W(O) = 0. (i) Write down an expression for W() in terms of a standard Brownian motion, BO) . [1] (ii) Show that {S(1) :/ > 0) is a continuous-time martingale. [4] [Total 5]