Part 1:  Forwards and Interest Rate Swaps Suppose that  you are given the following term structure of zero coupon yields (spot rates).

Maturity  r 

0.5   0.02 

1         0.025 

1.5      0.03 

2   0.04 

Semi-annually compounded interest  rates

1. From the given spot rates, calculate the forward rates, f(0, 0.5, 1), f(0, 1, 1.5), and f(0,1.5, 2).

2. Suppose that  f(0, 1, 1.5) quoted by a FRA dealer is 4.5%. (This should be almost 0.5 percentage points higher than  what you found above.)  Using f(0, 1, 1.5) and the given zero coupon yields, construct  an arbitrage  trading  strategy  to take  advantage  of this mispricing.

3. What  is the  fair fixed rate  in a 2-year interest  rate  swap in which fixed and floating payments  are exchanged every six months? Note that this is a little different from the lecture slides where payments were quarterly and the reference rate was the three-month spot rate.

4. Suppose that half a year passes. The term structure of zero coupon yields is now

Maturity       r

0.5              0.01

1                0.015

1.5              0.02

Semi-annually compounded interest rates

If you entered  into  a fixed-for-floating contract  previously (with  a notional  of $100), what is the value of your position now?

Part 2:  Treasury Bond Futures

It is July  30, 2015.  The  cheapest-to-deliver bond  in a September  2015 Treasury bond futures contract  is a 13% coupon bond with maturity at August 4, 2035. Delivery is expected to be made on September  30, 2015. Coupon payments  on the bond are made on February 4 and August 4 each year.  The term structure is flat, and the rate of interest  with semiannual compounding  is 12% per annum.

1. Calculate  the current (full) price of the cheapest-to-deliver bond.

2. Calculate  the conversion factor for the bond.

3. Calculate  the quoted futures price for the futures contract.

Part 3:  Fixed Income Options

Suppose that  you are given the following interest  rate  tree with semi-annually  compounded interest  rates.

Suppose that  you are also told that  σ = 0.015 and ? = 0.5.

1. Using the tree above, calculate  the price of a 1.5-year cap with a strike of 3% and a notional of $100.

2. Calculate  the price of a 1.5-year bond with a coupon rate  of 3% (coupons paid semi-annually)  and a face value of $100.

3. What  is the price of a 1.5-year floor with a strike of 3% and a notional of $100?

Part 4:  Callable Bonds

Now, let’s consider the  pricing of callable bonds.   We will compare  this  with  non-callable bonds.

1. Using the tree from Part  2, calculate the value of a 1.5-year bond with a coupon rate of 3% (coupons paid semi-annually)  that  is callable at $100 at each coupon date (starting at t = 0.5) just after the coupon is paid.  Recall that  a callable bond price is equal to an equivalent non-callable bond price minus the value of the call.

2. Add 0.001 to all of the  interest  rates  in the  tree from Part  2.  Calculate  the  price of the callable bond in (1) using this tree.

3. Subtract 0.001 from all of the interest  rates in the tree from Part  2. Calculate the price of the callable bond in (1) using this tree.

4. Using the callable bond prices in (1), (2), and (3), calculate  the modified duration  of the callable bond using

B(y + ?y) − B(y − ?y) 1 2 × ?y × B(y)

5. Calculate  the modified duration  of the non-callable version of the 1.5-year bond.  You should already have the prices you need for this in (1), (2), and (3).