1a. We observe the following spot rate curve. Assume semiannual compounding and $1000 par value. Calculate the prices of 0.5y, 1y, 1.5y, and 2y zero-coupon Treasury bonds.

period	Years	Annualized Spot Rate
1	0.5	6.0000%
2	1	6.1496%
3	1.5	6.2654%
4	2	6.4227%

Price of 0.5y zero coupon bond = $1000/(1+0.06/2) = $970.87

Price of 1y zero coupon bond = $1000/(1+0.061496/2)^2 = $941.23

Price of 1.5y zero coupon bond = $1000/(1+0.062654/2)^3 = $911.61

Price of 2y zero coupon bond = $1000/(1+0.064227/2)^4 = $881.23

1b. The binomial tree of the annualized 6-month short-term interest rate is in the table below. Each period represents a six-month time interval. Find the risk-neutral probability p1 under which the binomial tree model prices the 1-year zero-coupon bond correctly. Once we know p1, find the risk-neutral probability p2 such that the risk-neutral binomial tree model prices the 1.5-year zero-coupon bond correctly. Last, when we know p1 and p2, , find the risk-neutral probability p3 such that the risk-neutral binomial tree model prices the 2-year zero-coupon bond correctly.

0	1	2	3
		p3
p1	p2		7.5%
		1-p3 7%
	1-p2 6.5%		6.5%
6%		6%
1-p1	5.5%		5.5%
		5%
			4.5%

1c. There is a call option with 6-month expiration. The call option allows you to purchase a 6-month zero-coupon bond with $1000 par value at the price of $970 in the next period. Price the call option using the risk-neutral binomial tree that you developed in 1b. Construct a bond portfolio of 0.5Y and 1Y zero-coupon bonds such that the bond portfolio has cash flows identical to those of the call option. Verify that the price of the bond portfolio is identical to the price of the call option.

1d. There is a 2-year 8% coupon bond with the par value of $1000. Price the bond using (1) the spot rate curve provided in 1a and the risk-neutral binomial tree developed in 1b. Verify that prices obtained from the two methods are identical.

2a. The current spot rate curve is in the table below, which is identical to that in the previous question. Assume is 10%. Find the values for r₁, r₂, and r₃ so that the binomial tree model price the 12-month, 18-month, and 24-month zero-coupon bonds correctly.

period	Years	Annualized Spot Rate
1	0.5	6.0000%
2	1	6.1496%
3	1.5	6.2654%
4	2	6.4227%

2b. Use the binomial tree model that you developed in 2a to price a 2-year 8% coupon bond with the par value of $1000.

2c. Suppose the bond in 2b is callable at the price of $1000 in 6 months, 12 months, and 18 months. Price the callable bond. Does the callable bond have a higher or lower price than the bond in 2b and why?

2d. Suppose the bond in 2b is puttable at the price of $1000 in 6 months, 12 months, and 18 months. Price the callable bond. Does the puttable bond have a higher or lower price than the bond in 2b and why?