Please answer the following questions, show full calculations, explanations and proofs:

1. Bond pricing In class, we have discussed the pricing of bonds with finite maturity, deriving the bond formula Here, B is the current bond price, T is the number of periods to maturity, r> 0 is the per-period discount rate, C is the compon, payment, and P is the principal payment due at maturity. A variation of the bond is a perpetuity, which perpetually makes per-period coupon nayments and thus never mitte. It follows that the price of the perpetuity is B- (a) Consider a growing perpetuity, that makes the payment after one period, C(1 +9) after two periods, C(1+919 after three periods, etc., where 0 0 is the per-period discount rate, C is the coupon payment, and P is the principal payment due at maturity. A variation of the bond is a perpetuity, which perpetually makes per-period coupon payments and thus never matures. It follows that the price of the perpetuity is (a) Consider a growing perpetuity, that makes the payment C after one period, C(1 +9) after two periods, C(1 + 9) after three periods, etc., where 0 O order Taylor expansion of the portfolio valve around S shows that if the stock price moves to S + AS, then the portfolio value moves to approximately V(S + AS) hs(S + AS) + hp (P(S) + AS), so the change in value is approximately (hs + hp) 45. Technically, v(s + 48) - V(5) - (ns + Mapy (ns + hopelons) as se As scras), (1) for some constant C> We define the number Ap-which is denoted by the delta" of the hedging asset (with respect to stock price risk). A delta hedge is implemented by purchasing an amount of the hedging asset such that the portfolio is teutral to small movements of the stock price, i... such that hs + hpAr = 0: We arrive at hy hp for the delta hedge, where positivity follows from the assumption that Ap