Forward Contracts

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

Definitions

Forward Positions

• The party that has agreed to buy has a long position whereas the party that has agreed to sell has a short position.
• A long forward requires the buyer to purchase the asset at expiration for the futures price prevailing when the contract was first bought, which we denote by K.
• If the asset price at maturity is S, then the payoff of the long position is S - K, whereas the payoff of a short position is K - S.

Payoffs

Example: Currency Forward

• On May 24, 2010, the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of $1.4422 per British pound.
• This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2010.
• The payoff of this contract is 1{,}000{,}000 \times (S_{T} - 1.4422).
• The table below shows the payoff for different values of the exchange rate in six months.

S_{T}	1.2000	1.3000	1.4000	1.5000	1.6000
Payoff	-242,200	-142,200	-42,200	57,800	157,800

Forward Contract Payoff

• The figure shows the payoff of a long forward written on £1,000,000 with delivery price $1.4422 per £.

Forward Price
Non-Dividend Paying Assets

The Forward Price

• The forward price of of a non-dividend paying asset with maturity T years is given by: F = S e^{r T} where S denotes the spot price of the asset and r is the risk-free rate expressed per year with continuous compounding.

Example: Forward Price of a Non-Dividend Paying Stock

• Consider a non-dividend paying stock trading at $40.
• The risk-free rate is 5% per year with continuous compounding.
• What is the 3-month forward price? F = 40 e^{0.05 \times 3/12} = \$40.50
• What if the forward price was higher or lower than $40.50?

Example: Forward Price Arbitrage (1)

• Suppose that the spot price of a non-dividend-paying stock is $40, the 3-month forward price is $43 and the 3-month interest rate is 5% per year with continuous compounding.
• Is there an arbitrage opportunity?

	T = 0	T = 3/12
Short forward	0.00	43 - S_{T}
Borrow	42.47	-43
Long stock	-40.00	S_{T}
Total	2.47	0

• Yes, the forward price is too high!

Example: Forward Price Arbitrage (2)

• Suppose that the spot price of non-dividend paying stock is $40, the 3-month forward price is $39 and the 3-month interest rate is 5% per year with continuous compounding.
• Is there an arbitrage opportunity?

	T = 0	T = 3/12
Long forward	0.00	S_{T} - 39
Invest	-38.52	39
Short stock	40.00	-S_{T}
Total	1.48	0

• Yes, the forward price is too low!

Example: Forward Price on Gold

• Suppose that gold spot is currently $1,870.60, and consider a forward contract on gold expiring in one year.
• Assume that the cost of storing gold is negligible and there are no additional benefits accruing from owning gold.
• The risk-free rate is 5% per year with continuous compounding.
• Then, the no-arbitrage forward price of gold is: F = 1870.60 e^{0.05} = \$1{,}966.51.

Valuing an Existing Forward Contract

• A forward contract is worth zero when it is first negotiated.
• Afterwards it may have a positive or negative value.
• Suppose that K is the delivery price and F is the forward price for a contract that would be negotiated today.
• By considering the difference between a contract with delivery price K and a contract with delivery price F we can deduce that:
  • The value of a long forward contract is (F - K) e^{-r T}.
  • The value of a short forward contract is (K - F) e^{-r T}.

Example: Valuing an Existing Forward Position

• You entered into a short forward contract some time ago on a non-dividend paying asset when the forward price was $200.
• Today the contract has 6 months until maturity and the current forward price is $190.
• The current risk-free rate is 5% per year with continuous compounding.
• If we buy a forward today, that would lock-in a certain cash flow in six months of 200 - 190 = \$10.
• The present value today of this cash flow is 10 e^{-0.05 \times 6/12} = \$9.75 which is the value of the short forward contract.

Forward Price
Stocks Paying Dividends

Assets Paying Cash Dividends

• The forward price of an asset paying cash dividends is given by: F = (S - D) e^{r T} where D is the present value of the dividends or income earned during life of forward contract.
• Note that D could be negative if the asset requires to pay for storage and does not provide any other source of income.

Example: Forward Price of a Dividend Paying Stock

• Consider a stock that currently trades at $50.
• The stock is expected to pay dividends of $1.15 and $1.20 in two and five months, respectively.
• The risk-free rate is 5% per year with continuous compounding.
• The present value of the dividends paid during the life of the forward contract is: D = 1.15 e^{-0.05 \times 2/12} + 1.20 e^{-0.05 \times 5/12} = 2.32
• The 6-month forward price of the stock is: F = (50 - 2.32) e^{0.05 \times 6/12} = 48.89

Example: Forward Price Arbitrage (3)

• Suppose that in the previous example the observed forward price is $50.20.
• Is there an arbitrage opportunity?

	T = 0	T = 2/12	T = 5/12	T = 6/12
Short forward	0.00			50.20 - S_{T}
Loan 1	1.14	-1.15
Loan 2	1.18		-1.20
Loan 3	48.96			-50.20
Long stock	-50.00	1.15	1.20	S_{T}
Total	1.28	0	0	0

• Yes, the forward price is too high!

Assets Paying a Dividend Yield

• The futures price of a dividend-yield paying asset is given by: F = S e^{(r - \delta) T} where S is the spot price of the asset, T is the maturity of the futures contract, \delta is the continuous dividend or convenience yield, and r denotes the continuously compounded interest rate.

Forward Price
Foreign Currencies

Currencies and Exchange Rates

• The exchange rate between two currencies is usually defined as the number of domestic currency units per unit of foreign currency.
• Note that you could always define it the other way around (indirect-quotes), but that could lead to mistakes.
• Consider the EUR/USD exchange rate:
  • The quote currency is the US dollar (USD)
  • The base currency is the Euro (EUR)
• If the EUR/USD exchange rate is $1.47/€
  • For a US investor, 1 Euro is worth $1.47
  • For a European investor \$1 = 1/1.47 = \text{€} 0.68.

Direct Quotes for Exchange Rates

• Remember the street market convention:
  • A direct quote is the price of 1 unit of base currency expressed in the quote currency
  • For example, the direct quote of the EUR/USD could be S = \$ 1.4380 / \text{€} and represents the price in USD of 1 EUR.
    
• The market convention of calling this exchange rate EUR/USD might be misleading since it represents the number of USD per EUR, i.e. \$1.4380 \Leftrightarrow \text{€}1.
• Some currency pairs such as EUR/USD or GBP/USD use the USD as the quote currency.
• However, most currency pairs are expressed using the dollar as the base currency, i.e., USD/JPY, USD/CNY, USD/CLP, etc.

Currency Forward

• A foreign currency is analogous to a security providing a yield.
• The yield is the foreign risk-free interest rate.
• It follows that if r^{*} is the foreign risk-free interest rate F = S e^{(r - r^{*}) T}

Example: Currency Forward

• The current GBP/USD exchange rate is 1.30.
• The interest rates in USD and GBP are 1% and 3% per year with continuous compounding, respectively.
• The 9-month GBP/USD forward price is then F = 1.30 e^{(0.01 - 0.03) \times 9/12} = 1.2806, or 10{,}000 \times (1.2806 - 1.3000) = -193.5 \text{ forward-points.}