Question

A short forward contract with exactly 360 days to maturity on a stock is entered into when the stock price is $9.00 and the risk-free interest rate is 15.00% per annum with continuous compounding for all maturities.  The stock is certain to pay dividends per share of 20 cents in 60 days-time and 30 cents in 270 days-time. Assume one year is 365 days.

Required:

1. What are the forward price and the initial value of the forward contract?

QUESTION 2 continued:

2. Exactly 180 days later the stock price is $8.00 and the risk-free interest rate is 12.50% per annum (continuously compounded) for all maturities.  What are the forward price and the value of the short position in the forward contract?

Answer 1

(a) The spot price of a stock is $9, a dividend of $0.2 is due in t1 = 60/365 = 0.164 years and another dividend of $0.3 is due in t2 = 270/365 = 0.740 years. The interest rates are 15%. Since there are no storage costs, the forward price of a contract that settles in T = 360/365 = 0.986 year is

FO(0) = S(0)e^rT − d1e^{r(T −t1)} − d2e^{r(T −t2)}

= 9e^0.15*0.986 − 0.2e^{0.15*(0.986-0.164)} − 0.3e^{0.15*(0.986-0.740)}

= $9.897

Also, the initial value of the forward contract is set at zero.

(b) The forward price of a contract that settles in T' = 180/365 = 0.493 year is

FO(180) = S(180)e^rT' − d2e^{r(T' −t2)} = 8e^0.125*0.493 - 0.3e^0.125(0.247) = $8.199

Given that, a while back, you entered into a short forward contract to sell an asset for K ($9.897). Today, the same forward contract has a price F0($8.199) that happens to be lower than K($9.897). The older futures contract that you entered allows you to receive K($9.897) instead of the lower F0($8.199) at expiration, a value or benefit at expiration equal to K-F0 ($1.698). Thus the present value today of that benefit is:

Value of the short position in the forward contract, f = (K − F0)e^−rT = $1.696*e^-0.125*0.493 = $1.597