Garman–Kohlhagen model
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In finance, a foreign exchange option (commonly shortened to just FX option or currency option) is a
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
financial instrument that gives the right but not the obligation to exchange money denominated in one
currency A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" is a standardization of money in any form, in use or circulation as a medium of exchange, for example banknotes and coins. A more general ...
into another currency at a pre-agreed exchange rate on a specified date. 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 International Securities Exchange Holdings, Inc. (ISE) is a wholly owned subsidiary of American multinational financial services corporation Nasdaq, Inc. It is a member of the Options Clearing Corporation (OCC) and the Options Industry Council ( ...
,
Philadelphia Stock Exchange Philadelphia Stock Exchange (PHLX), now known as Nasdaq PHLX, is the first stock exchange established in the United States and the oldest stock exchange in the nation. The exchange is owned by Nasdaq, which acquired it in 2007 for $652 million, a ...
, or the Chicago Mercantile Exchange for options on
futures contract In finance, a futures contract (sometimes called a futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between parties not yet known to each other. The asset ...
s. The global market for exchange-traded currency options was notionally valued by the
Bank for International Settlements The Bank for International Settlements (BIS) is an international financial institution owned by central banks that "fosters international monetary and financial cooperation and serves as a bank for central banks". The BIS carries out its work thr ...
at $158.3 trillion in 2005.


Example

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 In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set ...
, 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 or Calls may refer to: Arts, entertainment, and media Games * Call, a type of betting in poker * Call, in the game of contract bridge, a bid, pass, double, or redouble in the bidding stage Music and dance * Call (band), from Lahore, Paki ...
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.


Terms

* Call option – the right to buy an asset at a fixed date and price. * Put option – the right to sell an asset at a fixed date and price. * Foreign exchange option – the right to sell money in one currency and buy money in another currency at a fixed date and rate. *
Strike price In finance, the strike price (or exercise price) of an option is a fixed price at which the owner of the option can buy (in the case of a call), or sell (in the case of a put), the underlying security or commodity. The strike price may be set ...
– the asset price at which the investor can exercise an option. *
Spot price In finance, a spot contract, spot transaction, or simply spot, is a contract of buying or selling a commodity, security or currency for immediate settlement (payment and delivery) on the spot date, which is normally two business days after the ...
– the price of the asset at the time of the trade. *
Forward price The forward price (or sometimes forward rate) 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, the forward price can be expressed in t ...
– the price of the asset for delivery at a future time. * Notional – 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''. *
Numéraire The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, actin ...
– the currency in which an asset is valued. * Non-linear payoff – the payoff for a straightforward FX option is linear in the underlying currency, denominating the payout in a given
numéraire The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, actin ...
. * Change of numéraire – the
implied volatility In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equ ...
of an FX option depends on the numéraire of the purchaser, again because of the non-linearity of x \mapsto 1/x. * In the money – for a put option, this is when the current price is less than the strike price, and would thus generate a profit were it exercised; for a call option the situation is inverted.


Trading

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, traders match the ''foreign'' currency notionals, not the local currency notionals, else the foreign currencies received and delivered do not 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 The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, actin ...
(a move from 2.0000 to 1.9000 yields a 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 x \mapsto 1/x, which is non-linear.


Hedging

Corporations primarily use FX options to
hedge A hedge or hedgerow is a line of closely spaced shrubs and sometimes trees, planted and trained to form a barrier or to mark the boundary of an area, such as between neighbouring properties. Hedges that are used to separate a road from adjoin ...
''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 The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the European mainland, continental mainland. It comprises England, Scotlan ...
manufacturing firm expects to be paid for a piece of engineering equipment to be delivered in 90 days. If the GBP strengthens against the over the next 90 days the UK firm loses money, as it will receive less GBP after converting the into GBP. However, if the GBP weakens against the , 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 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 on at the time of conclusion of the contract, making it a type of derivat ...
to deliver the in 90 days time, in exchange for GBP at the current
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 To extract the forward rate, we ne ...
. 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: * 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 The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black†...
for
stock options In finance, an option is a contract which conveys to its owner, the ''holder'', the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified da ...
and the
Black model The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions. It ...
for certain interest rate options, the value of a
European option In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These optionsâ ...
on an FX rate is typically calculated by assuming that the rate follows a log-normal process. The earliest currency options pricing model was published by Biger and Hull, (Financial Management, spring 1983). The model preceded the Garman and Kolhagen's Model. 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 r_d is the
risk-free interest rate The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ra ...
to expiry of the domestic currency and 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. Then the domestic currency value of a call option into the foreign currency is :c = S_0e^\mathcal(d_1) - Ke^\mathcal(d_2) The value of a put option has value :p = Ke^\mathcal(-d_2) - S_0e^\mathcal(-d_1) where : :d_1 = \frac :d_2 = d_1 - \sigma\sqrt :S_0 is the current spot rate :K is the strike price :\mathcal(x) is the cumulative normal distribution function :r_d is domestic risk free simple interest rate :r_f is foreign risk free simple interest rate :T is the time to maturity (calculated according to the appropriate
day 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 (FRAs). This determines the number of days ...
) :and \sigma is the volatility of the FX rate.


Risk management

An earlier pricing model was published by Biger and Hull, Financial Management, spring 1983. The model preceded Garman and Kolhagen Model. A wide range of techniques are in use for calculating the options risk exposure, or
Greeks The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group and nation indigenous to the Eastern Mediterranean and the Black Sea regions, namely Greece, Cyprus, Albania, Italy, Turkey, Egypt, and, to a lesser extent, oth ...
(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 Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given expi ...
and interest rate curves. After Garman–Kohlhagen, the most common models are
SABR The Society for American Baseball Research (SABR) is a membership organization dedicated to fostering the research and dissemination of the history and record of baseball primarily through the use of statistics. Established in Cooperstown, New ...
and
local volatility A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level S_t and of time t . As such, it is a generalisation of the Black–Sch ...
, although when agreeing risk numbers with a
counterparty A counterparty (sometimes contraparty) is a legal entity, unincorporated entity, or collection of entities to which an exposure of financial risk may exist. The word became widely used in the 1980s, particularly at the time of the Basel I deliberat ...
(e.g. for exchanging delta, or calculating the strike on a 25 delta option) Garman–Kohlhagen is always used.


References

{{Derivatives market Options (finance) Derivatives (finance)