Question

Given no arbitrage opportunities in the futures market, please estimate the 12-month futures price given the...

Given no arbitrage opportunities in the futures market, please estimate the 12-month futures price given the following information.

3-month futures price: \$100

6-month futures price: \$105

futures price (F)= spot price * e^(r*t)

rate (r) = annual rate of interest

time (t) = time in years

for the 3-month futures, time is 3/12 , or 0.25 years

for the 6-month futures, time is 6/12 , or 0.5 years

let spot price be S

F = S*e^(r*t)

S * e^(0.25 * r) = 100 ==>  e^(0.25 * r) = 100 / S (3-month futures price)

S * e^(0.5 * r) = 105 ==>  e^(0.5 * r) = 105 / S (6-month futures price)

the 6-month futures price = 3-month futures price * e^(0.25 * r)

the 6-month futures price is 100 / S

(100 / S) * e^(0.25 * r) = (105 / S)

e^(0.25 * r) = 1.05

0.25r = Ln(1.05)

r = 0.1951

1 year futures price = 6-month futures price * e^(r * 0.5) - because 6 months in a year, time period is 0.5 years

1 year futures price = 105 * e^(0.1951 * 0.5)

1 year futures price = \$115.76

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