Forward Contracts
Options and Futures
Lorenzo Naranjo
Spring 2024
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: \[ F = S e^{(r - q) T} \] where \(S\) is the spot price of the asset, \(T\) is the maturity of the futures contract, \(q\) 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 \[ 10{,}000 \times (1.2806 - 1.3000) = -193.5 \text{ forward-points.} \]