Question $1$

Consider a two$-$period binomial model for a non$-$dividend paying stock whose current

price is $S_0=100.$ Assume that:

over each six$-$month period, the stock price can either move up by a factor $u=1\mathrm{.}2$

or down by a factor $d=0\mathrm{.}8$

the continuously compounded risk$-$free rate is $r=5\%$ per six$-$month period

$($i$)($a$)$ Prove that there is no arbitrage in the market. $[2]$

$($b$)$ Construct the binomial tree. $[3]$

$($ii$)$ Calculate the price of a standard European call option written on the stock $S$

with strike price $K=100$ and maturity one year. $[5]$

Consider a special type of call option with strike price $K=100$ and maturity one

year. The underlying asset for this special option is the average price of the stock

over one year, calculated as the average of the prices at times $0,0\mathrm{.}5$ and $1$ measured in

years.

$($iii$)$ Calculate the initial price of this call option assuming it can be exercised only

at time $1.[5]$