Adress all the questions effectively

Let R; denote the return on security i given by the following multifactor model K; =di + bill + bal2 + ... + bizz + c; a; and c; are the constant and random parts respectively of the component of the return unique to security i. 1, ... It are the changes in a set of L indices. bax is the sensitivity of security i to factor k. (i) State the category of the above model where: (a) index I is a price index index 2 is the yield on government bonds index 3 is the annual rate of economic growth (b) index I is the level of Research and Development expenditure index 2 is the price earnings ratio [2] (ii) Determine the number of parameters to be estimated in a single index model and in a multifactor model. [4]Suppose that under the unique Equivalent Martingale Measure, Q, for a term structure model, the SDE satisfied by the instantaneous interest rate / is air, - a(u - r,) di+ 3dz, where c > 0. u and c are fixed parameters and. under O. Z is a standard Brownian Motion. The process X is defined by X, =r,b(T-n) + [r ds. where the function b is given by b(s) = (1 -e-35) / a. The function f is given by fux. () = expia(7 - n) - x). where a is a differentiable function. (i) Apply Ito's formula to /(X,, (). [G] [Hint: First show that dX, = Adr + BdZ, where A, = oub(T -t) and B, = ob(T -()] (ii) (a) Find a differential equation which the function a must satisfy for XX,. f) to be a martingale. (b) Determine an additional condition on a which is necessary for a bond with unit payoff at time 7 to have a price given by the formula B(L D)= AX,, D) exp [ ras ). [4] [Total 10] The following model has been suggested for the short term interest rate at time t, 7: ar = ur,di + or, dZ where o and u are fixed parameters and Z, is a standard Brownian motion. (i) Outline three properties of this model and comment on their desirability. [3] (ii) Outline the properties of the following two models for interest rates: the une-factor Vasicek model (b ) the Cox-Ingersoll-Ross model [3] [Total 6]