All interest rates are annual interest rates with semi-annual compounding. All coupon rates are annual rates paid semi-annually.

Assume today's 6-month spot rate is 9% and in six months the new 6-month spot rate can be either (9+x)% or 8.5%. Assume also that the 1-year spot rate is 9.2% and the price of a 6-month call option on a 1-year zero-coupon bond (with a face value of $1000) with a strike price of $957 is $0.90. Keep at least 7 decimal digits for all your calculations and answers.

a) Find x

b) Without doing explicit calculations (but still referring to appropriate formulas) do you think x will increase, decrease, or remain unchanged if the option price goes down? Assume all other parameters given in these questions remain the same. Explicit calculations without verbal arguments will give you only 1 point.