As an analyst at Bank of America Merrill Lynch$,$ you are evaluating European call futures option and European put futures options. A futures price is currently $$50.$ It is expected to move either to $$55$ or down to $$45$ over the next three month. The risk$-$free interest rate is $8\%$ per annum with continuous compounding. Please answer the following questions related to the futures call options and put options with a strike price of $51.$

a$.$ What is the probability of an up movement in a risk$-$neutral world? Blank $1($sample answer: $35\mathrm{.}50\%)$

b$.$ What is the futures call option delta? Blank $2($sample answer: $0\mathrm{.}45)$

c$.$ What is the futures put option delta? Blank $3($sample answer: $-0\mathrm{.}45)$

d$.$ What is the value of the call option?Blank $4($sample answer: $$1\mathrm{.}45)$

e$.$ What is the value of the put option? Blank $5($sample answer: $$1\mathrm{.}45)$