PPlease address the following questions fully. Thank you

An insurer's portfolio consists of three independent policies. Each policy can give rise to at most one claim per month, which occurs with probability 0 independently from month to month. The prior distribution of 0 is beta with parameters o = 2 and B = 4. A total of 9 claims are observed on this portfolio over a 12 month period. (i) Derive the posterior distribution of 0. [2] (ii) Derive the Bayesian estimate of 0 under all or nothing loss. [4] [Total 6] Individual claim amounts on a particular insurance policy can take the values 100, 150 or 200. There is at most one claim in a year. Annual premiums are 60. The insurer must choose between three reinsurance arrangements: no reinsurance individual excess of loss with retention 150 for a premium of 10 proportional reinsurance of 25% for a premium of 20 Complete the loss table for the insurer. (4] Reinsurance Claims A B C 0 100 150 200 (li) Determine whether any of the reinsurance arrangements is dominated from the vicwpoint of the insurer. [2] (iii) Determine the minimax solution for the insurer. [1] [Total 7]An investment bank has issued a derivative on a share (with share price, S, of 100) that provides for the following payoff after two months: MS) = In(5 -90) ifS > 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. 191 (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