A stock is expected to pay a dividend of $1 per share in 2 months and in 5 months. The current stock price is $50, and the continuous compounded risk free interest rate is 8% per annum. An investor has just taken a short position in a 6-month forward contract on the stock. Required:
a. What is the arbitrage free price of the forward contract?
b. What are the forward price and the initial value of the forward contract?
c. Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract?

b) the initial value of the forward contract is $0

we must determine the present value of the dividends that the stock is expected to pay in order to determine the forward price:

present value of dividends = ($1 x e⁽⁻⁰°⁰⁸⁾ ⁽⁰°¹⁶⁷⁾) + ($1 x e⁽⁻⁰°⁰⁸⁾ ⁽⁰°⁴¹⁶⁷⁾) = $1.954

forward price = ($50 - $1.954) · e⁽⁰°⁰⁸⁾ ⁽⁰°⁵⁾ = $50.0068 ≈ $50.01

c) again we first determine the present value of the dividends:

present value of dividends = ($1 x e⁽⁻⁰°⁰⁸⁾ ⁽⁰°¹⁶⁷⁾) = $0.9867

forward price = ($48 - $0.9867) · e⁽⁰°⁰⁸⁾ ⁽⁰°²⁵⁾ = $47.963 ≈ $47.96

short forward contract = -[$48 - $0.9867 - ($50.01 · e⁽⁻⁰°⁰⁸⁾ ⁽⁰°²⁵⁾)] = $2.006 ≈ $2.01

a) in order to determine the arbitrage free forward price, the NPV of our forward price must be 0. It is basically the same answer than (b) only that you calculate it in a different order:

$50 = $1.954 + forward price/e⁽⁰°⁰⁸⁾ ⁽⁰°⁵⁾

$48.046 = forward price/(1 + e⁽⁰°⁰⁸⁾ ⁽⁰°⁵⁾

forward price = $48.046 · e⁽⁰°⁰⁸⁾ ⁽⁰°⁵⁾ = $50.006 ≈ $50.01

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