$1.$ Given the following:

$$ The S&P $500$ futures call and put options with exercise prices of $3,000$ expiring at the end of $120$ days $($t $=120/365)$

$$ The S&P $500$ futures contract expiring at the end of $120$ days $($t $=120/365)$

$$ The S&P $500$ call and put spot options with exercise prices of $3,000$ expiring at the end of $120$ days

$$ A current spot index of S$0=3,000.$

$$ The estimated annualized volatility and mean of the spot index$$s logarithmic return are $\backslash$sigma A $=0\mathrm{.}25$ and $\backslash$mu A $=0.$

$$ The annual risk$-$free rate is Rf $=3\%.$

$$ The futures price is determined by the carrying cost model:

$\backslash$Psi $=$ continuous annual dividend yield $=5\%.$

Determine the following option prices using a $30-$period binomial tree and the BOPM Excel programs:

a$.$ American futures call

b$.$ European futures call

c$.$ American spot call

d$.$ European spot call

e$.$ American futures put

f$.$ European futures put

g$.$ American spot put

h$.$ European spot put

$2.$ Given the futures and spot information in Question $2,$ determine the price of European futures call with X $=3,000$ using the Black futures option model $($B$-$S Excel; Black futures$).$ Show in a table the Black futures option model$$s option prices and intrinsic values for different futures prices for the European futures call. Generate your table using spot index values of $2,700,2,800,2,900,3,000,3,100,3,200,$ and $3,300.$