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Any currency option deal may be equivalently valued as either a call or a put using a parity condition that is specific to currency options. This condition states that:
Let’s suppose a foreign trader has purchased USD (1%) OTM Call / NZD (1%) ITM Put of 1Y expiry in European vanilla style with a face value of USD $1 million.
The risk-free rate in US is 0.24% (3M US T-bill),
The risk-free rate in NZ is 2.33% (3M NZ T-bill),
FX rate volatility is at around 20%, and let’s say option expiry would be 1 year.
The spot FX is 0.6851 (indirect quote) and the strike rate is 0.6786.
Consider NZD as the domestic CCY, so the spot must be quoted as the number of units of NZD required to purchase 1USD – this is the reciprocal of how the rate is quoted above (1/0.6851= 1.4596).
To value the option as a call, we treat the NZD as the domestic CCY and the USD as the foreign CCY. The domestic and foreign interest rates are therefore 2.33% and 0.24%, respectively.
Note that once both the spot and strike quotes are converted, the call option is out of the money.
When these parameters are passed to the Garman-Kohlhagen routine, a value of NZD 0.0396 is returned.
This is a per unit value, and since a total of 1 million US dollars underlie the deal, the total premium value is 0.0396 * 1,000,000 = NZD 39,600 (with allowance for rounding).
In order to demonstrate the parity condition, we could also value this deal as a put. In this case, the domestic (foreign) CCY is the USD (NZD), the domestic (foreign) interest rate is 0.24% (2.33%), the spot rate is 0.6851 and the strike rate is 0.6891.
With these inputs, the model returns a value per unit FX of USD 1.9370 and the total value of the premium on the deal is USD 57,801.
This amount is equivalent to the NZD premium computed above when it is converted to NZD at the spot rate of 0.6851. That is, USD 57, 801*0.6851= NZD 39,600.
If the spot FX is expressed in a consistent approach to the strike price, then the computed total premium for the call leg will be the same as the computed total premium for the put leg, assuming both values are expressed in the same currency.
Source for rates:
