Question

Question 1 - (25 points) (a) Consider a 2-year forward contract to buy a coupon-bearing bond that will mature 2 year from today. The current price of the bond is $102. Sup- pose that on that bond 4 coupon payments of $6 are expected after 6 months, 12 months, 18 months and 24-months. We assume that the 6-month, 12- month, 18-month and 24-month risk-free interest rates (continuously com- pounded) are, respectively, 1%, 1.3%, 1.6% and 1.9% per annum. Determine the strike price, the forward price and the value of the forward contract. (b) 15 months later, the price of the bond is $105 and the risk-free interest rates for maturities 3-month, 9-month, are, respectively, 0.6% and 0.9% per annum. What are the strike price, the forward price and the value of the forward contract?

Answer 1

(a) The price of forward is determined as

FP = (S - PVC).(1+R_f)^T

where S = current price of the bond = $102

PVC = Present value of coupons = 6 * e^{(R₁*0.5 + R₂*1 + R₃*1.5 + R₄*2)}

R_{1 =} 1%, R_{2 =} 1.3%, R_{3 =} 1.6%, R_{4 =} 1.9%

PVC = 6 * e^0.8 = 6.5

Rf = risk-free rate over the maturity tenor of forward i.e. 1.9%

FP = (102 - 6.5).(1 + 1.9%)² = 99.16

At the origination of Forward, there is no gain or loss as either party can't start the contract will a loss.

Thus, Strike Price = 99.16 and value of the future contract = 0

(b) After 15months, the bond price is $105 and remaining time to maturity of future contract is 9months. At this time, forward price is

FP_t = (105 - PVC_t).(1+Rf)^9/12

PVC_t = 6 * e^{(R₁*0.25 + R₂*0.75}); R_{1 =} 0.6%, R_{2 =} 0.9%

PVC_t = 6.05

Forward price after 15 months, FP_t = 99.617

Value of forward contract at time t is given by below formula:

V_t = (S_t - PVC_t) - FP / (1+R_f)^(T-t)

t = 15 months or 1.25; T = 2

V_t = (105 - 6.05) - 99.16 / (1.009)^(0.75) = 0.454

Value of forward contract after 15 months = 0.454

Strike price = Forward price - Value of forward = 99.617 - 0.454 = 99.163