# Foreign exchange option

In finance, a foreign exchange option (commonly shortened to just FX option or currency option) is a derivative financial 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.

## Hedging

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

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

${\displaystyle c=S_{0}e^{-r_{f}T}{\mathcal {N}}(d_{1})-Ke^{-r_{d}T}{\mathcal {N}}(d_{2})}$

(proof?) The value of a put option has value

${\displaystyle p=Ke^{-r_{d}T}{\mathcal {N}}(-d_{2})-S_{0}e^{-r_{f}T}{\mathcal {N}}(-d_{1})}$

where :

${\displaystyle d_{1}={\frac {\ln(S_{0}/K)+(r_{d}-r_{f}+\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}}}$
${\displaystyle d_{2}=d_{1}-\sigma {\sqrt {T}}}$
${\displaystyle S_{0}}$ is the current spot rate
${\displaystyle K}$ is the strike price
${\displaystyle {\mathcal {N}}(x)}$ is the cumulative normal distribution function
${\displaystyle r_{d}}$ is domestic risk free simple interest rate
${\displaystyle r_{f}}$ is foreign risk free simple interest rate
${\displaystyle T}$ is the time to maturity (calculated according to the appropriate day count convention)
and ${\displaystyle \sigma }$ is the volatility of the FX rate.

## Risk management

A wide range of techniques are in use for calculating the options risk exposure, or Greeks (as for example the Vanna-Volga method). Although the option prices produced by every model agree (with Garman–Kohlhagen), risk numbers can vary significantly depending on the assumptions used for the properties of spot price movements, volatility surface and interest rate curves.

After Garman–Kohlhagen, the most common models are SABR and local volatility[citation needed], although when agreeing risk numbers with a counterparty (e.g. for exchanging delta, or calculating the strike on a 25 delta option) Garman–Kohlhagen is always used.

## References

1. ^ "Foreign Exchange (FX) Terminologies: Forward Deal and Options Deal" Published by the International Business Times AU on February 14, 2011.
2. ^ "British Pound (GBP) to Euro (EUR) exchange rate history". www.exchangerates.org.uk. Retrieved 21 September 2016.
3. ^ "Currency options pricing explained". www.derivativepricing.com. Retrieved 21 September 2016.