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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