Foreign exchange option
Encyclopedia
In finance, a foreign-exchange option (commonly shortened to just FX option or currency option) is a derivative
financial instrument that gives the owner 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.
The FX 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 contract
s. The global market for exchange-traded currency options was notionally valued by the Bank for International Settlements
at $158.3 trillion in 2005.
, 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 at 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 they immediately convert the profit into GBP this amounts to 100,000/1.9000 = 52,631.58 GBP.
The difference between FX options and traditional options is that in the latter case the trade is to give an amount of money and receive the right to buy or sell a commodity, stock or other non-money asset. In FX options, the asset in question is also money, denominated in another currency.
For example, a call option on oil allows the investor to buy oil at a given price and date. The investor on the other side of the trade is in effect selling a put option on the currency.
To eliminate residual risk, match the foreign currency notionals, not the local currency notionals, else the foreign currencies received and delivered don't offset.
In the case of an FX option on a rate, as in the above example, an option on GBPUSD gives a USD value that is linear in GBPUSD using USD as the numéraire (a move from 2.0000 to 1.9000 yields a .10 * $2,000,000 / 2.0000 = $100,000 profit), but has a non-linear GBP value. Conversely, the GBP value is linear in the USDGBP rate, while the USD value is non-linear. This is because inverting a rate has the effect of
, which is non-linear.
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.
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:
for certain interest rate option
s, the value of a European option on an FX rate is typically calculated by assuming that the rate follows a log-normal process.
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
is the risk-free interest rate
to expiry of the domestic currency and
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.
Then the domestic currency value of a call option into the foreign currency is
The value of a put option has value
where :
is the current spot rate
is the strike price
is the cumulative normal distribution function
is domestic risk free simple interest rate
is foreign risk free simple interest rate
is the time to maturity (calculated according to the appropriate day count convention
)
(as for example the Vanna-Volga method
). Although the option price 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
, 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.
Derivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...
financial instrument that gives the owner the right but not the obligation to exchange money denominated in one currency
Currency
In economics, currency refers to a generally accepted medium of exchange. These are usually the coins and banknotes of a particular government, which comprise the physical aspects of a nation's money supply...
into another currency at a pre-agreed exchange rate
Exchange rate
In finance, an exchange rate between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency...
on a specified date.
The FX options market is the deepest, largest and most liquid market for options of any kind. Most trading is over the counter (OTC)
Over-the-counter (finance)
Within the derivatives markets, many products are traded through exchanges. An exchange has the benefit of facilitating liquidity and also mitigates all credit risk concerning the default of a member of the exchange. Products traded on the exchange must be well standardised to transparent trading....
and is lightly regulated, but a fraction is traded on exchanges like the International Securities Exchange
International Securities Exchange
International Securities Exchange Holdings, Inc. is a wholly owned subsidiary of German derivatives exchange Eurex. It is a member of the Options Clearing Corporation and the Options Industry Council . Historically, responsibility for organizing the Options Industry Conference is rotated amongst...
, Philadelphia Stock Exchange
Philadelphia Stock Exchange
Philadelphia Stock Exchange , now known as NASDAQ OMX PHLX, is the oldest stock exchange in the United States, founded in 1790. It is now owned by NASDAQ OMX and located at 1900 Market Street, in Center City Philadelphia.-History:...
, or the Chicago Mercantile Exchange
Chicago Mercantile Exchange
The Chicago Mercantile Exchange is an American financial and commodity derivative exchange based in Chicago. The CME was founded in 1898 as the Chicago Butter and Egg Board. Originally, the exchange was a non-profit organization...
for options on futures contract
Futures contract
In finance, a futures contract is a standardized contract between two parties to exchange a specified asset of standardized quantity and quality for a price agreed today with delivery occurring at a specified future date, the delivery date. The contracts are traded on a futures exchange...
s. The global market for exchange-traded currency options was notionally valued by the Bank for International Settlements
Bank for International Settlements
The Bank for International Settlements is an intergovernmental organization of central banks which "fosters international monetary and financial cooperation and serves as a bank for central banks." It is not accountable to any national government...
at $158.3 trillion in 2005.
Example
For example a GBPUSD FX option 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 rateExchange rate
In finance, an exchange rate between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency...
, or strike price
Strike price
In options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, the strike...
, 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
Call option
A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...
on dollars and a put
Put option
A put or put option is a contract between two parties to exchange an asset, the underlying, at a specified price, the strike, by a predetermined date, the expiry or maturity...
on sterling
Pound sterling
The pound sterling , commonly called the pound, is the official currency of the United Kingdom, its Crown Dependencies and the British Overseas Territories of South Georgia and the South Sandwich Islands, British Antarctic Territory and Tristan da Cunha. It is subdivided into 100 pence...
, 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 at 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 they immediately convert the profit into GBP this amounts to 100,000/1.9000 = 52,631.58 GBP.
Terms
- CallCall optionA call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...
—The right to buy an asset at a fixed date and price - Put optionPut optionA put or put option is a contract between two parties to exchange an asset, the underlying, at a specified price, the strike, by a predetermined date, the expiry or maturity...
—The right to sell an asset a fixed date and price - FX option—The right to sell money in one currency and buy money in another currency at a fixed time and relative price.
- StrikeStrike priceIn options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, the strike...
—The asset price at which the investor can exercise an option - SpotSpot priceThe spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...
—The price of the asset at the time of the trade - ForwardForward priceThe forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
—The price of the asset for delivery at a future time - NotionalNotionalNotional is an American Thoroughbred racehorse. He was sired by In Excess and out of the mare Truly Blessed. His damsire, French Deputy, is a son of the 1997/98 Leading sire in North America, Deputy Minister....
—The amount of each currency that the option allows the investor to sell or buy - Ratio of notionals—The strike, not the current spot or forward
- Non-linear payoff—The payoff for a straightforward FX option is linear in the underlying currency, denominating the payout in a given numéraireNuméraireNuméraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
. - NuméraireNuméraireNuméraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
—The currency in which an asset is valued - Change of numéraire—The implied volatility of an FX option depends on the numéraire of the purchaser, again because of the non-linearity of
.
The difference between FX options and traditional options is that in the latter case the trade is to give an amount of money and receive the right to buy or sell a commodity, stock or other non-money asset. In FX options, the asset in question is also money, denominated in another currency.
For example, a call option on oil allows the investor to buy oil at a given price and date. The investor on the other side of the trade is in effect selling a put option on the currency.
To eliminate residual risk, match the foreign currency notionals, not the local currency notionals, else the foreign currencies received and delivered don't offset.
In the case of an FX option on a rate, as in the above example, an option on GBPUSD gives a USD value that is linear in GBPUSD using USD as the numéraire (a move from 2.0000 to 1.9000 yields a .10 * $2,000,000 / 2.0000 = $100,000 profit), but has a non-linear GBP value. Conversely, the GBP value is linear in the USDGBP rate, while the USD value is non-linear. This is because inverting a rate has the effect of
, which is non-linear.Hedging with FX options
Corporations primarily use FX options to hedgeHedge (finance)
A hedge is an investment position intended to offset potential losses that may be incurred by a companion investment.A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, many types of...
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
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
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
Forward contract
In finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed today. This is in contrast to a spot contract, which is an agreement to buy or sell an asset today. It costs nothing to enter a...
to deliver the US$100,000 in 90 days time, in exchange for GBP at the current forward rate
Forward rate
The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a forward rate.-Forward rate calculation:...
. 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.
Using options, the UK firm can purchase a GBP call/USD put option (the right to sell part or all of their expected
Expected
Expected may refer to:*Expectation*Expected value*Expected shortfall*Expected utility hypothesis*Expected return*Expected gainSee also*Unexpected...
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 modelBlack model
The Black model is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing bond options, interest rate caps / floors, and swaptions...
for certain interest rate option
Interest rate option
Interest rate option is a derivative financial instrument.Interest Rate Options are a form of Exchange Traded Derivative whose underlying value is the rate on various Financial Interest rates,including treasury bills, and bonds. The exchange of these is monitored and facilitated by the CME Group.An...
s, the value of a European option on an FX rate is typically calculated by assuming that the rate follows a log-normal process.
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
is the risk-free interest rateRisk-free interest rate
Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....
to expiry of the domestic currency and
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.Then the domestic currency value of a call option into the foreign currency is
The value of a put option has value
where :
is the current spot rate is the strike price is the cumulative normal distribution function is domestic risk free simple interest rate is foreign risk free simple interest rate is the time to maturity (calculated according to the appropriate day count conventionDay count convention
In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements . This determines the amount transferred on interest payment dates, and also the calculation of...
)
- and
is the volatilityVolatility (finance)In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
of the FX rate.
Risk Management
A wide range of techniques are in use for calculating the options risk exposure, or GreeksGreeks (finance)
In mathematical finance, the Greeks are the quantities representing the sensitivities of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of...
(as for example the Vanna-Volga method
Vanna Volga
The Vanna-Volga method is a technique for pricing first-generation exotic options in FX derivatives.It consists in adjusting the Black–Scholes theoretical value by the cost of a portfolio which hedges three main risks...
). Although the option price 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
Volatility Smile
In finance, the volatility smile is a long-observed pattern in which at-the-money options tend to have lower implied volatilities than in- or out-of-the-money options. The pattern displays different characteristics for different markets and results from the probability of extreme moves...
and interest rate curves
Yield curve
In finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
.
After Garman-Kohlhagen, the most common models are SABR
SABR Volatility Model
In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "Stochastic Alpha, Beta, Rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the...
and local volatility
Local volatility
A local volatility model, in mathematical finance and financial engineering, is one which treats volatility as a function of the current asset level S_t and of time t .-Formulation:...
, although when agreeing risk numbers with a counterparty
Counterparty
A counterparty is a legal and financial term. It means a party to a contract. A counterparty is usually the entity with whom one negotiates on a given agreement, and the term can refer to either party or both, depending on context....
(e.g. for exchanging delta, or calculating the strike on a 25 delta option) Garman-Kohlhagen is always used.