Help me work the following;

A pension scheme awards deferred annuities to members who leave service before retirement. The deferred annuities fall under the Social Security Act (1985) and must be increased over the period from leaving service to the members' normal retirement dates in line with increases in the Retail Prices Index (RPI) but subject to a maximum of 5%% per annum compound. While the rate of inflation as measured by the RPI may be negative from one period to another the amount of the annuity on the vesting date cannot be less than that awarded on leaving service. (i) Consider a member who has just left the pension scheme and has been awarded a deferred annuity of El per annum, deferred for a years. Let Q(0) denote the value of the RPI at time / with the current value being Q(0) = 1. Let R(n) = 1.05"_ (a) Write down an expression for the annualised annuity benefit re- valued after n years. (b) Explain how the benefit can be viewed as combining two European options and describe these options. [6] (ii) Let & denote the uniform force of interest that can be earned on fixed money investments and let n denote the uniform force of interest earned on investments linked to the RPI Suppose that the RPI follows the following diffusion process: d(log Q(1)) = u(t, Q(1)) dt + adz where H(t, Q()) is the mean rate of change o is the standard deviation (a constant) of the diffusion process Z is a unit Normal (Brownian motion) process(a) Show that the value of the deferred annuity at time t, V, is given by: V, =Qe"(1 - old,) - 0(f)) + Ae "Off,) + Be "(d.) where Q1 = Q(0) D () is the standard cumulative Normal distribution function T=n-1 log( Ae " (Qe "]) GVT GVT 214 = log(Ae " ( Que ") GVT GVT 2 A, lower limit of the annuity ($1 per annum) B, upper limit of the annuity (R(n) = 1.05") d, = log(Que " / Be GVT 2 d. = log(Que / Be & GVT 2 (b) Comment on the suitability of the assumptions used in this model. (c) Compare the results that would be obtained using the more traditional and simpler approach of assuming a constant (such as 5%) rate of growth for the RPI