Question

The current spot price of gold is $1,780 per ounce (oz.). Storing gold costs $12 per oz. per year, and the storage costs are paid when the gold is taken out of storage. The risk-free rate is 5% per annum (continuously compounded).

i) What is the futures price for gold delivery in three-month?

ii) If the current futures price for delivery of gold in three-month is $1810 per oz., identify a riskless arbitrage strategy

Answer 1

Spot Price = $1,780 per ounce | Storage costs = $12 per ounce per year

Risk-free rate = 5%

i) Storage costs of $12 per ounce is per year, hence, we need to convert it for 3 months.

As there are 12 months in a year, 3 months becomes 3 / 12 years or 1/4 years.

Storage costs for 3 months = 12 * 1 / 4 = $3 per ounce

Futures Price formula = Spot Price * e^RT + Storage costs to be paid when gold is taken out

Futures Price for Gold in 3-month = 1,780 * e^{5% * 1/4} + 3

Futures Price for Gold in 3-month = 1,780 * 1.012578 + 3

Futures Price for Gold in 3-month = 1,802.39 + 3

Futures Price for Gold in 3-month = $1,805.39

ii) Futures Price for Gold in 3-months = $1,810

As a riskless arbitrage strategy, we can follow below steps:

a) Go short on the gold Futures contract and promise to sell gold at price $1,810 in 3 months which leads to inflow of $1,810

b) Use $1,780 out of $1,810 to buy gold at spot price of $1,780 per ounce.

c) Invest remaining $30 in risk-free security

d) Pay storage costs of 3$ at the end of 3 months.

Profit at the end of 3 months = (1,810 - 1,780) * e^{5% * 1/4} - 3 = 30 * 1.012578 - 3 = 30.38 - 3 = $ 27.38

This strategy is also known as Cash-and-carry arbitrage strategy.