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Foreign exchange option

In finance, a foreign exchange option (commonly shortened to just FX option or currency option) is a derivativefinancial instrument that gives the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified date.^[1] See Foreign exchange derivative.

The foreign exchange options market is the deepest, largest and most liquid market for options of any kind. Most trading is over the counter (OTC) and is lightly regulated, but a fraction is traded on exchanges like the International Securities Exchange, Philadelphia Stock Exchange, or the Chicago Mercantile Exchange for options on futures contracts. The global market for exchange-traded currency options was notionally valued by the Bank for International Settlements at $158.3 trillion in 2005.^{[citation needed]}

For example, a GBPUSD contract could give the owner the right to sell £1,000,000 and buy $2,000,000 on December 31. In this case the pre-agreed exchange rate, or strike price, is 2.0000 USD per GBP (or GBP/USD 2.00 as it is typically quoted) and the notional amounts (notionals) are £1,000,000 and $2,000,000.

This type of contract is both a call on dollars and a put on sterling, and is typically called a GBPUSD put, as it is a put on the exchange rate; although it could equally be called a USDGBP call.

If the rate is lower than 2.0000 on December 31 (say 1.9000), meaning that the dollar is stronger and the pound is weaker, then the option is exercised, allowing the owner to sell GBP at 2.0000 and immediately buy it back in the spot market at 1.9000, making a profit of (2.0000 GBPUSD − 1.9000 GBPUSD) × 1,000,000 GBP = 100,000 USD in the process. If instead they take the profit in GBP (by selling the USD on the spot market) this amounts to 100,000 / 1.9000 = 52,632 GBP.

Hedging[edit]

Corporations primarily use FX options to hedge uncertain future cash flows in a foreign currency. The general rule is to hedge certain foreign currency cash flows with forwards, and uncertain foreign cash flows with options.

Suppose a United Kingdom manufacturing firm expects to be paid US$100,000 for a piece of engineering equipment to be delivered in 90 days. If the GBP strengthens against the US$ over the next 90 days the UK firm loses money, as it will receive less GBP after converting the US$100,000 into GBP. However, if the GBP weakens against the US$, then the UK firm receives more GBP. This uncertainty exposes the firm to FX risk. Assuming that the cash flow is certain, the firm can enter into a forward contract to deliver the US$100,000 in 90 days time, in exchange for GBP at the current forward rate. This forward contract is free, and, presuming the expected cash arrives, exactly matches the firm's exposure, perfectly hedging their FX risk.

If the cash flow is uncertain, a forward FX contract exposes the firm to FX risk in the opposite direction, in the case that the expected USD cash is not received, typically making an option a better choice.^{[citation needed]}

Using options, the UK firm can purchase a GBP call/USD put option (the right to sell part or all of their expected income for pounds sterling at a predetermined rate), which:

• protects the GBP value that the firm expects in 90 days' time (presuming the cash is received)
• costs at most the option premium (unlike a forward, which can have unlimited losses)
• yields a profit if the expected cash is not received but FX rates move in its favor

Valuation: the Garman–Kohlhagen model [edit]

As in the Black–Scholes model for stock options and the Black model for certain interest rate options, the value of a European option on an FX rate is typically calculated by assuming that the rate follows a log-normal process.^[2]

In 1983 Garman and Kohlhagen extended the Black–Scholes model to cope with the presence of two interest rates (one for each currency). Suppose that ${\displaystyle r_{d}}$ is the risk-free interest rate to expiry of the domestic currency and ${\displaystyle r_{f}}$ is the foreign currency risk-free interest rate (where domestic currency is the currency in which we obtain the value of the option; the formula also requires that FX rates – both strike and current spot be quoted in terms of "units of domestic currency per unit of foreign currency"). The results are also in the same units and to be meaningful need to be converted into one of the currencies.^[3]

Then the domestic currency value of a call option into the foreign currency is

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