kindly address all the questions. Provide efficient answers

(i) (@) State three features which are desirable when a graduation is performed. (b) Explain why they are desirable. [3] The actuary to a large pension scheme has attempted to graduate the scheme's recent mortality experience with reference to a table used for similar sized schemes in a different industry. He has calculated the standardised deviations between the crude and the graduated rates, zy, at each age and has sent you a printout of the figures over a small range of ages. Unfortunately the dot matrix printer on which he printed the results was very old and the dots which would form the mimis sign in front of numbers no longer function, so you cannot tell which of the standardised deviations is positive and which negative. Relow are the data which you have. Age Standardised deviation 60 2.40 61 0.08 62 0.80 63 0.76 64 1.04 65 0.77 66 1.30 67 1.76 68 0.28 69 0.68 70 0.93 (ii) ) Carry out an overall goodness-of-fit test on the data. (b) Comment on your result. [5] (Lib (@) List four defects of a graduation which the test you have carried out would fail to detect. (b) Suggest, for each of the defects, a test which could be used to detect it. [4] (iv) Carry out one of the tests suggested in part (iii)(b). [3] [Total 15](i) Describe the role that the inflation model plays within the Wilkie model. [3] The Wilkie model proposes an AR(1) process for the continuously compounded rate of inflation I() that can be written as: I(t) = a+ bl(t-1) + =(1) Where a(1) ~ N(O, r-) and a and b are constants with -1 90 = 0 otherwise You may assume that: There exists a risk free asset that earns 5% per month, continuously compounded. The expected effective rate of return on the share is 2% per month. The monthly standard deviation of the log share price is 10%. (i) By using a two period recombining model of future share prices, derive the state price deflators at time 2. The parameters determining the share price after an up-jump and down-jump should be determined by considering the standard deviation of the log share price. [9] (ii) Using the state price deflators from (i) derive the value at time zero of the option. [3] The delta of this derivative at time zero is 7% and the gamma is 10%. The bank which issued the derivative wishes to delta hedge its position in the most efficient manner. Assume that the share price can also be modelled in continuous time with a geometric Brownian motion with volatility (diffusion parameter) of 0.1 consistent with a Black-Scholes framework. (iii) Determine the delta hedging portfolio, as a combination of the risk free asset, the underlying share, and a European Call option on the share with term of 3 months and exercise price of 100. [13] [Total 251