Kindly help answer these questions urgently.

1.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:

F(S) = ln(S - 90)if S > 90= 0otherwise

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.

2.

Q Question 5.20 Can you explain (separately) why 6, and (-S / ) appear in Thiele's equation shown above for a whole life assurance?Q Question 5.21 Write down Thiele's equation for the following contract at duration t (0