# Futures Contract

A futures contract is a standardized exchange-traded contract on a currency, a commodity, stock index, a bond etc. (called the underlying asset or just underlying) in which the buyer agrees to purchase the underlying in future at a price agreed today. It differs from a forward in that it is traded on an exchange, it requires an upfront margin to be paid to the exchange and that it is periodically marked to market.

The buyer of a futures contract is said to have a long position to the underlying asset (also called underlying) while the seller has a short exposure. Future contracts are important risk management tools that allows companies to hedge their interest rate risk, exchange rate risk and some business risks associated with commodity prices. However, they are also used by investors to obtain exposure to a stock, a bond, a stock market index or any other financial asset. Because futures contracts are standardized, there is an active market in which participants can trade their futures contracts before their expiry date. New York Mercantile Exchange (NYMEX), Chicago Board of Trade (CBOT) and Chicago Board Options Exchange (CBOE) are the main exchanges on which futures can be traded.

Parties looking to purchase or sell futures contracts are required to maintain a margin with the exchange. It gives the exchange an assurance that they have necessary funds to honor their obligation in event of any adverse price movement. Exchanges calculate the net payoff on futures contracts periodically, typically daily, and credits the differential amount in the margin account if a party gains on the futures contract and draws on the margin balance if the party has lost money due to price movement.

## Formula

The spot price is the price of the underlying asset at the inception of futures contract, i.e. time 0. The forward price is the price of the underlying at which the futures contract stipulates the exchange to occur at time T.

The value of a futures contract is different from the future price. It is the value of the long or short position in the futures contract itself and it depends on whether the spot price of the underlying asset at the time of valuation is higher or lower than the agreed futures price and the risk-free interest rate.

The futures price i.e. the price at which the buyer commits to purchase the underlying asset can be calculated using the following formulas:

$$ {\rm FP}_0=S_0\times{(1+i)}^t $$

Where FP_{0} is the futures price, S_{0} is the spot price of the underlying, i is the risk-free rate and t is the time period.

The formula is a little different for futures contract in which the underlying asset has cash inflows or outflows during the term of the futures contract, for example stocks, bonds, commodities, etc.

The value of futures contract for the buyer (i.e. the party with long position) at inception is zero and the value at expiration equals the difference between the associated spot rate at the expiration date minus the futures price i.e. the price at which the futures contract was purchased.

$$ V_T=S_T-F_0 $$

Where V_{T} is the value at expiration, S_{T} is the spot price at expiration and F_{0} is the futures price looked through the futures contract.

Because futures contracts are traded on an exchange, parties might sell them any time between the inception date and the expiration date. In such an event, the value of the futures contract equals the difference between the spot price at that time (denoted as S_{t}) minus the present value of the futures price locked at time t. The value of futures future F_{0} that we expected to get at time T can be worked out by discounting the futures price (F_{0}) at risk-free rate r for the remaining time period (i.e. T minus t). The value of the futures contract can be worked out as follows:

$$ V_t=S_t-\frac{F_0}{{(1+r)}^{(T-t)}} $$

## Example

Your company gives its customers an option to pay in bitcoin. Roughly 0.5% of your monthly revenue is collected in bitcoins so you havenâ€™t felt the need to hedge the exposure. However, your sales team tells you that recently they recently signed a contract that will result in a particularly large payment of 200 bitcoins in 3-months. Due to extreme volatility in the cryptocurrencies, you decide to hedge the exposure.

CBOE offers bitcoin futures with a multiplier of one and because you current have a long position in bitcoin, so you must get a short position in bitcoin futures, i.e. you must sell 200 futures.

You sell 200 XBT/K8 contracts with a future value of $6,820. The contract matures in 90 days.

The value of your contract at inception is zero. At the settlement date, if the bitcoin price has dropped to $6,400, the payoff to the party with long position is as follows:

$$ V_T=S_T-F_0=200\times$6,400-200\times$6,820=-$84,000 $$

Because you have short position, you will payoff will be exactly opposite to the payoff to the long position. Your profit amounts to $84,000. By selling the futures you have guaranteed that you get at least $6,820 per bitcoin no matter what happens to the bitcoin price. You can sell the 200 bitcoins at the spot price at expiration of $6,400 for proceeds of $1,280,000 and receive $84,000 as profit on the futures contract. Hence, your net cash flows are $1,364,000; which equals the price you locked i.e. 200 multiplied by $6,820.