Question

A stock is expected to pay a dividend of $1 per share in two months and in five months. The stock price is $56, and the risk-free rate (with continuous compounding) is 8% for all maturities. An investor has just taken a short position in a seven-month forward contract on the stock.`

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

(2) Three months later, the price of the stock is $50 and the risk-free rate is still 8%. What are the forward price and the value of the forward contract?

Answer 1

a). I = $1 x e^{-(0.08 x 2/12)} + $1 x e^{-(0.08 x 5/12)} = $0.9868 + $0.9672 = $1.9540

F0 = (S - PVD)e^{r * t}

= ($56 - $1.9540)e^{(0.08 * 7/12)} = $54.0460 x 1.04778 = $56.63

The initial value of the forward contract is (by design) zero. The fact that the forward price is very close to the spot price should come as no surprise. When the compounding frequency is ignored the dividend yield on the stock equals the risk-free rate of interest.

b). In three months:

I = $1 x e^{-(0.08 x 2/12)} = $0.9868

Forward Price = (S - PVD)e^{r * t}

= ($50 - $0.9868)e^{(0.08 * 4/12)} = $49.0132 x 1.02703 = $50.34

Value of Contract = -[S - PVD - F0e^{-(r * t)}]

= -[$50 - $0.9868 - $56.63e^{-(0.08 * 4/12)}]

= -[$49.0132 - $55.1378] = $6.12