- Cash Settlement
This pays off nothing if the option is out of the money and pays a fixed amount, Q (in Base Currency – usually USD), if it ends up in the money at the maturity of the option. Let be the forward exchange rate between the valuation date and the maturity of the option. is the volatility of the forward exchange rate. Let also be the time to maturity (settlement) date, the strike price of the option, and the continuously compounded risk-free rate of the base currency at time t. Using Black’s model, the prices of call and put options are given by
(1)
where , (1 or –1) is the call/put indicator, and is the discount factor between the valuation date and the settlement date.
- Asset Settlement
This pays off nothing if the option is out of the money and pays an amount equal to (called Principal % Strike in Atlas) if it ends up in the money at the maturity of the option. Here, is the FX rate at the maturity of the option. Thus, the prices of call and put options are given by
(2)
where .
- Atlas System
In Atlas, the payoff currency is the base currency, which is usually USD.
For direct quote, we have and . The dollar value Delta in USD (called USD Delta) is computed as
USD Delta (3)
where is the spot exchange rate and the perturbation on the spot rate is set to be 0.00005. This dollar value delta is used for daily hedging.
For indirect quote, we have and . The price of the contract and the perturbation of the spot rate in delta calculation is in USD/CAD. Thus, the perturbation of the spot is done as and where . Thus, the USD Delta is defined by
USD Delta = (4)
We created 24 test cases for FX Digital option (12 for direct quote and 12 for indirect quote) and the following tables summarize the results. The valuation date is September 1, 2004 for direct quote and August 31, 2004 for indirect quote. The principal amount, Q, is 1,000,000 USD for both direct / indirect quotes. The same principal applies to other FX derivatives
Direct Quote:
Base Currency: USD
Underlying Currency: EUR
Principal Amount: 1,000,000 USD
Spot Rate: 1.20605 USD/EUR
Indirect Quote:
Base Currency: USD
Underlying Currency: CAD
Principal Amount: 1,000,000 USD
Spot Rate: 1.31895 CAD/USD
Appendix 1. Test Cases (Direct Quote)
Case No. Digital Type Option Type Maturity Date Settlement Date Strike Price 1 Cash Call 31-Dec-2004 04-Jan-2005 1.200 2 Cash Put 31-Dec-2004 04-Jan-2005 1.200 3 Cash Call 01-Sep-2005 06-Sep-2005 1.150 4 Cash Put 01-Sep-2005 06-Sep-2005 1.210 5 Cash Call 15-Aug-2006 17-Aug-2006 1.170 6 Cash Put 15-Aug-2006 17-Aug-2006 1.208 7 Asset Call 31-Dec-2004 04-Jan-2005 1.187 8 Asset Put 31-Dec-2004 04-Jan-2005 1.208 9 Asset Call 01-Sep-2005 06-Sep-2005 1.200 10 Asset Put 01-Sep-2005 06-Sep-2005 1.200 11 Asset Call 10-Jul-2006 12-Jul-2006 1.185 12 Asset Put 10-Jul-2006 12-Jul-2006 1.205
Appendix 2. Test Cases (Indirect Quote)
Case No. Digital Type Option Type Maturity Date Settlement Date Strike Price 1 Cash Call 31-Dec-2004 04-Jan-2005 1.300 2 Cash Put 31-Dec-2004 04-Jan-2005 1.300 3 Cash Call 01-Sep-2005 02-Sep-2005 1.307 4 Cash Put 01-Sep-2005 02-Sep-2005 1.315 5 Cash Call 30-Jun-2006 05-Jul-2006 1.289 6 Cash Put 30-Jun-2006 05-Jul-2006 1.325 7 Asset Call 31-Dec-2004 04-Jan-2005 1.300 8 Asset Put 31-Dec-2004 04-Jan-2005 1.300 9 Asset Call 01-Jun-2005 02-Jun-2005 1.307 10 Asset Put 01-Jun-2005 02-Jun-2005 1.315 11 Asset Call 30-Jul-2006 31-Jul-2006 1.289 12 Asset Put 30-Jul-2006 31-Jul-2006 1.325
Appendix 3. Forward Foreign Exchange Rate
Number of Days from Valuation Date USD/EUR Forward Points (bps) USD/EUR Forward Outright Number of Days from Valuation Date CAD/USD Forward Points (bps) CAD/USD Forward Outright 7 -1.1675 1.205933 7 1.1500 1.319065 14 -2.3350 1.205817 14 2.4500 1.319195 30 -4.6700 1.205583 30 5.4500 1.319495 61 -8.0750 1.205243 59 11.2000 1.320070 91 -10.1000 1.205040 91 17.5000 1.320700 124 -10.6000 1.204990 122 23.0000 1.321250 153 -11.1000 1.204940 153 29.0000 1.321850 181 -11.6000 1.204890 181 34.5000 1.322400 273 -8.6000 1.205190 273 52.0000 1.324150 365 -2.5000 1.205800 365 67.5000 1.325700 546 18.2500 1.207875 546 94.7500 1.328425 730 45.0000 1.210550 730 122.0000 1.331150
Appendix 4. Forward Foreign Exchange Rate Volatilities
Number of Days from Valuation Date USD EUR FX Volatility Number of Days from Valuation Date USD CAD FX Volatility 0 0.10000 0 0.08750 5 0.10000 6 0.08750 12 0.09900 13 0.08630 28 0.09700 29 0.08500 57 0.10000 58 0.08450 89 0.10100 90 0.08250 177 0.10400 178 0.08100 268 0.10475 269 0.08050 363 0.10550 364 0.08000 728 0.10540 729 0.08000 1,093 0.10520 1,094 0.07950 1,457 0.10490 1,458 0.07950 1,822 0.10490 1,823 0.07950
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