Editing Foreign exchange option

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{{Short description|Derivative financial instrument}}
{{Foreign exchange}}
In finance, a '''foreign exchange option''' (commonly shortened to just '''FX option''' or '''currency option''') is a [[derivative (finance)|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.<ref>"[http://au.ibtimes.com/articles/111913/20110213/foreign-exchange-fx-terminologies-forward-deal-and-options-deal.htm Foreign Exchange (FX) Terminologies: Forward Deal and Options Deal]" Published by the [http://au.ibtimes.com/forex International Business Times AU] on February 14, 2011.</ref> See [[Foreign exchange derivative]].<ref>{{cite web |last1=Hammad |first1=Muhammad |title=Fastest Currency Exchange Company |url=https://www.linkexchange.com.pk/ |access-date=10 June 2023 |website=Link International Exchange Company |publisher=Farhan}}</ref>

==Valuation: the Garman–Kohlhagen model {{Anchor|Garman–Kohlhagen model}} ==
As in the [[Black–Scholes model]] for [[stock options]] and the [[Black model]] 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.<ref>{{cite web|title=British Pound (GBP) to Euro (EUR) exchange rate history|url=http://www.exchangerates.org.uk/GBP-EUR-exchange-rate-history.html|website=www.exchangerates.org.uk|access-date=21 September 2016}}</ref>

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 <math>r_d</math> is the [[risk-free interest rate]] to expiry of the domestic currency and <math>r_f</math> 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.<ref>{{cite web|title=Currency options pricing explained|url=http://www.derivativepricing.com/blog/bid/59633/Currency-options-pricing-explained|website=www.derivativepricing.com|access-date=21 September 2016}}</ref>

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

:<math>c = S_0e^{-r_f T}\mathcal{N}(d_1) - Ke^{-r_d T}\mathcal{N}(d_2)</math>

The value of a put option has value
:<sub><math>p = Ke^{-r_d T}\mathcal{N}(-d_2) - S_0e^{-r_f T}\mathcal{N}(-d_1)</math></sub>

where :
:<math>d_1 = \frac{\ln(S_0/K) + (r_d - r_f + \sigma^2/2)T}{\sigma\sqrt{T}}</math>
:<math>d_2 = d_1 - \sigma\sqrt{T}</math>

:<math>S_0</math> is the current spot rate
:<math>K</math> is the strike price
:<math>\mathcal{N}(x)</math> is the cumulative normal distribution function
:<math>r_d</math> is domestic risk free [[simple interest]] rate
:<math>r_f</math> is foreign risk free simple interest rate
:<math>T</math> is the time to maturity (calculated according to the appropriate [[day count convention]])
:and <math>\sigma</math> is the [[Volatility (finance)|volatility]] of the FX rate.

==References==
{{Reflist}}

{{Derivatives market}}

[[Category:Options (finance)]]
[[Category:Derivatives (finance)]]