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

Problem 2 is solved below please solve Problem 3:

Problem 2: Consider a 1-year zero-coupon bond with $1,000 face value. Assume todays 6-month forward rate is 6% and in six months the new 6-month forward rate can be either 5% or 7%. Assume also that the current price of the 6-month call option with a strike price of $970 on this bond is $2.17. Find the 1-year spot rate today. Keep at least 4 decimal digits.

The interest rates movement, value of bond and value of call option after 6 months are in the table:

	7%	966.1835749	0
6%	5%	975.6097561	5.6097561
today	after 6 months	Bond price after Six months	Value of Call option after 6 months

So our neutral risk probability of interest (7%) be x

(0 + 5.609756 - 5.609756x) / (1.030) = 2.17

x = 0.6015691235

From this we find that the expected forward rate: 0.07*0.601569 + 0.05*0.3984 = 0.062031 or 6.2031%

And now we get our 1 year Spot rate today: 1.03*1.031 - 1 = 0.061946 or 6.1946%

Problem 3: Use all the information in Problem 2 plus assume that every 6 months the 6-month forward rate can either increase or decrease by 1%. I.e., todays 6-month forward rate is 6%; in 6 months the 6-month forward rate can be either 5% or 7%; and in 1 year the 6-month forward rate can be either 4%, 6%, or 8%. Assume also that the 1.5-year spot rate is 6.25%. Like in lectures, assume that the risk-neutral probability depends on the time period but not on the exact node. Find the price of a 1-year call option with a strike price of $960 on the 1.5-year zero coupon bond. Keep at least 4 decimal digits. You may use the results of Problem 2 if you find any of them useful.

(Please Solve Problem 3 and Show all Steps)

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