Options on currency can be somewhat confusing to price; particularly to someone who isn't used to the terminology of the market, particularly with the units.
In this post; we will break down the steps to pricing a FX option using a couple different methods. One is to use the Garman Kohlhagen model (which is an extension of the Black Scholes models for FX) and the other is to use Black '76 and price the option as an option on a future. We can also price this option either as a call option or as a put option.
We're assuming you have an option pricer to do these calculations. You can download a free trial of ResolutionPro for this purpose.
- Put option on GBP, Call option on USD
- Valuation Date: Dec 24, 2009
- Maturity Date: January 7, 2010
- Spot price as of Dec 24: 1.599
- Exercise price: 1.580
- Volatility: 10%
- GBP risk free rate: 0.42%
- USD risk free rate: 0.25%
- Notional: £1,000,000 GBP
Put option on FX example
First, we'll look at the Put option. The current spot price of the currency is 1.599. This means 1 GBP = 1.599 USD. So the USD/GBP rate must drop to below the strike of 1.580 for this option to be in-the-money.
We now put the inputs above into our option pricer. Note our rates above are annually compounded, Act/365. Though generally these rates would be quoted as simple interest, Act/360 for USD, Act/365 for GBP and we'd need to convert them to whatever compounding/daycount our pricer uses. We're using a Gereralized Black Scholes pricer, which is the same as Garhman Kohlhagen when used with FX inputs.
Our result is 0.005134. The units of the result are the same as our input which is USD/GBP. So if we multiple this by our notional in GBP we get our result in USD as the GBP units cancel out.
0.005134 USD/GBP x £1,000,000 GBP = $5,134 USD
Call option on FX example
Now let's run the same example as a call option. We invert our spot price and exercise to be GBP/USD rather than USD/GBP.
This time the units are in GBP/USD. To get the same result in USD, we multiple 0.002032 GBP/USD x 1,580,000 USD (the notional in USD) x 1.599 USD/GBP (current spot) = $5,134 USD.
Note in the inputs to our pricer, we are now using the USD rate as domestic and GBP as the foreign.
The key point of these examples is to show that it's always important to consider the units of your inputs as that will determine how to convert them into the units you require.
FX Option on Future example
Our next example is to price the same option as an option on a future using the Black '76 model.
Our forward price for the currency on the expiry date is 1.5991
We will use this as our underlying in our Black option pricer.
We get the same result when we priced using the Black-Scholes / Garman Kohlhagen models. $5,134 USD.
For details on the math behind these models please see help.derivativepricing.com.
Learn more about Resolution's support for foreign exchange derivatives.
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