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An Example of Put-Call Parity for Currency Options

Assume the user has purchased a European vanilla USD Call / NZD Put with a face value of USD $1 million. The USD riskless rate is 5%, the NZD riskless rate is 7%, exchange rate volatility is 20%, and the option will expire in 1 year. The spot rate is 0.52 (indirect quote) and the strike rate is 0.51.


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 7% and 5%, respectively. Because the NZD is the domestic CCY, 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 (1/0.52 = 1.9231). 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 0.1099 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 0.1099 * 1,000,000 = NZD 109,865 (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 5% (7%), the spot rate is 0.52 and the strike rate is 0.51. With these inputs, the model returns a value per unit FX of USD 0.2914 and the total value of the premium on the deal is USD 57,130. This amount is equivalent to the NZD premium computed above when it is converted to NZD at the spot rate of 0.52. That is, USD 57, 130 / 0.52 = NZD 109,865.


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