Consider a two$-$period binomial share price model. Let the current share

price be $$100$ and the appropriate risk$-$free rate be $4\%$ per period. Across a

period, the share price can either increase by $8\%$ or decrease by $8\%.$ The

share does not pay dividends.

a$)$ Find the current price of an at$-$the$-$money European style put option on

the shares. Illustrate your answer using a diagram of the two$-$period

binomial tree.

b$)$ Using your answer to part $($a$),$ find the current price of an at$-$the$-$money

European style call option on the shares assuming there are no arbitrage

opportunities.

c$)$ Explain how you would set up a hedged portfolio for the first period of

the model, using some combination of the shares, puts and risk$-$free

bonds, and demonstrate that it works.

$2)$ What determines any difference between the prices of otherwise identical

American and European style call options? Briefly explain your answer.