A trader owns gold as part of a longterm investment portfoli

A trader owns gold as part of a long-term investment portfolio. The trader can buy gold for $250 per ounce and sell gold for $249 per ounce. The trader can borrow funds at 6% per year and invest funds at 5.5% per year. (Both interest rates are expressed with annual compounding.) For what range of one-year forward prices of gold does the trader have no arbitrage opportunities? Assume there is no bid-offer spread for forward prices.

Solution

Arbitrage will be possible if the forward price is to low or to high. If it is to high, you short a forward contract and then replicate the long forward position by buying 1 ounce of gold at the spot_ask price = 250 and borrowing 250*exp(-.06*T) at 6%. If it is to low, you long a forward contract and then replicate the short forward position by selling 1 ounce of gold at the spot_bid price = 549 and lending 249*exp(-.05*T) at 5%. So arbitrage is not possible so long as

249*exp(.05*T) <= Forward Price <= 250*exp(.06*T)