A stock index is currently $1,500.$ Its volatility is $18\%.$ The risk$-$free rate is $4\%$ per annum

$($continuously compounded$)$ for all maturities.

a$.$ Calculate values for u$,$ d$,$ and p when a six$-$month time step is used.

b$.$ What is the value of a $12-$month European call option with a strike price of $1,480$

given by a two$-$step binomial tree?

c$.$ Suppose a trader sells $10$ contracts. What position in the stock is necessary to hedge

the trader$$s position at the time of the trade?

d$.$ What is the value a $12-$month European put option with a strike price of $1,480$

given by a two$-$step binomial tree.

e$.$ Verify that the European call and European put prices satisfy put$$call parity.

f$.$ Assume instead that the dividend yield on the index is $2\mathrm{.}5.$ What is the value a $12-$

month American put option with a strike price of $1,480$ given by a two$-$step

binomial tree.