no implicit calls or puts further in this market there are no bid ask spreads and no 599210

You are given the following default-free long bond:

Face value: 100

Issuing price: 100

Currency: USD

Maturity: 30 years

Coupon: 6%

No implicit calls or puts. Further, in this market there are no bid-ask spreads and no trading commissions. Finally, the yield curve is flat and moves only parallel to itself.There is, however, a futures contract on the 1-year Libor rate. The price of the contract is determined as

V_t= 100(1-f_t)

where f_tis the “forward rate” on 1-year Libor.

(a) Show that if the yield of the 30-year bond is y_t, then at all times we have

y_t= f_t

(b) Plot the pricing functions for V_tand the bond.

(c) Suppose the current yield y₀is at 7%. Put together a zero-cost portfolio that is delta-neutral toward movements of the yield curve.

(d) Consider the following yield movements over 1-year periods:

9%7%9%7%9%7%

What are the convexity gains during this period? (e) What other costs are there?