Question

Today is January 3. Your friend David has just bought a futures contract on a stock index, and the contract specifies one year to expiration. The current share price is $80, and the annually compounded interest rate is 10%. The stock will pay quarterly dividends of $2 during the next year, with dividends payments on the following dates:

• January 25
• April 25
• July 25
• October 25

Assume that this is a non-leap year.

a.       What is the futures price on this contract on January 3?

b.       What is the cost-of-carry on this futures contract on January 3?

c.       What is the value of this futures contract on February 17 if the spot price turns out to be $90 on that day?

d.       Suppose that, instead of paying a quarterly dividend of $2, the stock index has an annual dividend yield of 10%. The continuously compounded interest rate is 10.52%. What is the price of the index futures on January 3? What is its value on February 17 when the index turns out to be $90?

Answer 1

Answer a) The future price

F₀ = (S₀ – I)e^rT

I = PV(d1)+PV(d2)...+PV(d4) = d1e^-rt1 +d2e^-rt2 + d3e^-rt3+d4e^-rt4

Future price at 3 Jan,

Dividend	Date	Time(days)	Time (year)	e^rt	PV(d)
2	25-Jan	22 days	0.060	1.01	2.01
2	25-Apr	90 days	0.247	1.02	2.05
2	25-Jul	91 days	0.249	1.03	2.05
2	25-Oct	91 days	0.249	1.03	2.05
				Total (I)	8.16

F₀ = (80-8.16)* e^0.10*1 = $ 79.397 = $ 79.40.

Answer b) Futures price = Spot price + cost of carry

Cost of carry = F_o - S_o = 79.40-80 = (- 0.60)

Answer c) Value Future contract , f = (F₀-K)*e^-rt = (91.55-79.40)*e^-0.1*0.877 = $ 11.13.

as K = $ 79.40 ,

at 17 Feb , F₀ = (90 - 6.14)*e^0.10*0.877 = 91.55

present value of Income at 17 feb , and time for contract = 365-45= 320 days (0.877 years)

Dividend	Date	Time(days)	Time (year)	e^rt	PV(d)
0	25-Jan	22 days	0.060	1.01	0.00
2	25-Apr	67 days	0.184	1.02	2.04
2	25-Jul	91 days	0.249	1.03	2.05
2	25-Oct	91 days	0.249	1.03	2.05
				Total (I)	6.14

Answer d) The price in future at 3 jan,  F₀=S₀e ^(r–y)T = 80*e^{(0.1052-0.10)*1}= $ 80.42

At 17 Feb , f = (F₀-K)*e^-rt = (90.412-80.42)*e^-0.1*0.877 = $ 9.16

K = $ 80.42

at 17 Feb , F₀ = 90*e^{(0.1052-0.10)*0.877} =90.412