Question

A stock is expected to pay a dividend of $1.50 per share in three months and in six months. The stock price is currently $45and the risk-free rate is 6% per annum with continuous compounding for all maturities. An investor has just taken a short position in seven-month forward contract on the stock.

#1) What is the forward price for no arbitrage opportunity?

#2) What is the initial value of the forward contract?

4 months later.  Now, the price of the stock is $50 and the risk-free rate is still 6% per annum with continuous compounding.

#3) What is the new forward price for no arbitrage opportunity?

#4) and what is the value of the short position in the forward contract?

		#1) forward price initially = $44.55
		#1) forward price initially = $43.57
		#1) forward price initially = $46.52
		#2) initial value of the contract = $0
		#2) initial value of the contract =cannot determine without further information
		#2) initial value of the contract = -$1.52
		#3) forward price 4 months later = $52.21
		#3) forward price 4 months later = $48.85
		#3) forward price 4 months later =  $49.25
		#3) forward price 4 months later =  $50.75
		#4) value of the short forward contract 4 months later = $0
		#4) value of the short forward contract 4 months later = -$5.60
		#4) value of the short forward contract 4 months later = $5.60
		#4) value of the short forward contract 4 months later = cannot determine without further information

Answer 1

Formula to calculate forward price is:

F0 = (S0 - I) * e^rt

where F0 = current forward price,

S0 = current spot price

r = continuously compounded rate

t = time of the contract

I = PV of income (interest or dividend arising from the asset)

Part 1:

Given information

S0 = $45

r = 6%

dividend = $1.5

We have two dividends one after 3 months and other after 6 months. We will have to calculate the present value of the dividends using the interest rate provided.

Present value of dividend = dividend * e-^rt

hence I = 1.5 * e^{-0.06/12 * 3} + 1.5 * e^{-0.06/12 * 6} ( we have to convert the interest rate to monthly and then multiply time in months)

Substituting I in the forward formula provided above we get

F0 = (45 - 1.5 * e^{-0.06/12 * 3} - 1.5 * e^{-0.06/12 * 6} ) * e^{0.06/12 * 7} = 43.57

(Note we have used 7 months in the last term because the contract is for 7 months)

Part 2: Initial value of the forward contract is zero.

At the time on entering into a forward contract, long or short, the value of the forward is zero. This is because the delivery price of the asset and the forward price today are equal. The value of the forward is basically the present value of the difference in the delivery price and the forward price

Part 3:

4 months later, S0 = 50. One dividend has been paid and one dividend is pending 2 months from now. The life of contract remaining is 3 months

Hence we substitute the new values in the forward formula above:

F0 = (S0 - I) * e^rt

F0 = (50 - 1.5 * e^{-0.06/12 * 2}   ) * e^{0.06/12 * 3} = 49.25

Part 4:

Value of the short position of the forward contract = - (F0 - K) * e^-rT

where K = value of forward contract at the time of initiation = 43.57 (calculated earlier)

= - (49.25 - 43.57) * e ^{-0.06/12 *3}  

= - 5.60