Economic problems, particularly those in Europe, have increased the public’s awareness of how fragile the relationships are between different currencies, and how difficult it may be to keep an international currency such as the euro afloat. With its institutional limitations, the European Central Bank is not able to act with the full force of a central bank characterized by the U.S. Federal Reserve — buying and selling government securities and providing liquidity to member banks as a lender of last resort.
Although the problems the euro faces seem severe, it is not at the top of the list in terms of implied volatility. “Forex option price curves” (below) shows that the options market on Nov. 4, 2011 put the March 2012 euro calls in second place following the Swiss franc, and slightly higher than the Japanese yen. The Swiss franc may have been temporarily in the top position because of its relatively steep 11.3% decline from Sept. 2 to Nov. 4. Higher price curves indicate predicted larger price spread variations between Nov. 4 and expiration in March.
Percentage price changes for March 2012 futures over 90 days ending on Nov. 18, 2011 show a similar pattern with minus signs as the U.S. dollar index at plus 4.67% strengthened against other currencies: Swiss franc – 14.29%, euro – 5.37, British pound – 4.27%, Canadian dollar – 3.57%, Australian dollar – 2.64%, and Japanese yen – 0.58%.
“A futures pair” (below) shows the possibility of spreads between the euro and Australian dollar options. For the purpose of this chart, the space between prices for the two currencies is reduced by multiplying Australian dollar prices by 1.37 (because the average price of the March euro over the Sept. 1 to Nov. 18 period is approximately 37% larger than the Australian dollar futures). Variations in the March 2012 call prices indicate that the Australian dollar options were more variable than the euro during this period.
As long as the options price relationship holds, spreads between the pair could be profitable.
For example, buying relatively undervalued Australian dollar futures at 0.9241 on Oct. 4 along with a sale of euro futures at 1.3239, and reversing the spread trade on Oct. 14, would have produced a gain of 9.9% for the Australian dollar vs. 4.7% for the euro’s March contract.
Testing hypotheticals
Prices for currency options trades may be compared to the hypothetical prices shown by Barchart.com. These theoretical prices are based on the Black-Scholes option pricing model for a wide range of put and call strike prices. Barchart.com also shows actual trades and the timing of them at specific strikes. This permits a look at individual trades relative to the hypothetical price curve when the trades were made. Prices are shown with a 10-minute delay.
For example, on Nov. 16, 2011, a trade for March 2012 euro calls at 1.40 strike is priced at 0.0266 at 10:07 a.m. when hypothetical prices for puts and calls are shown for latest options at 10:20 a.m. As explained later, when a smoothed option price curve is computed, the trade at 0.0266 is estimated to be underpriced by approximately $64 compared with the theoretical pricing model.
“March 2012 euro calls” (below) charts variations from published prices. As shown on the lower graph (“Euro option price curve”), the regression model is a close fit to Barchart.com prices, with market prices falling almost on the predicted prices at each strike price. The upper chart contains the variations from the smoothed price curve in dollars instead of option points. Because each option point for euro calls is worth $125,000, the variations are greatly magnified from those along the lower curve.
A similar pair of charts is shown on “March 2012 Australian dollar calls” (below). The smoothed curve of prices in option points related to strike prices exhibits a few points that deviate slightly from the curve, but overall the fit is very close. It is only when the variations from the curve are multiplied by $100,000 (the value of one option point for Australian dollar options) that the differences are measured in terms of U.S. dollars.
The small dollar variations from zero in both of the upper charts for euro and Australian dollar calls are rounding differences. While the price quotes for options are limited to four decimal places, the smoothed curve data is calculated on a computer spreadsheet. Computer calculations are carried to as many as 18 decimal places; thus, with the magnification of 100,000 or 125,000, variations around the curve are relatively small.
The larger peak variations — at strikes of 1.05 and 1.10 for Australian dollar calls or 1.40, 1.45 and 1.50 for euro calls — represent actual trades. The dollar variations for these trades suggest that many actual trades take place at prices that are overpriced (with a spike down) or underpriced (with an upward spike) relative to the Black-Scholes hypothetical prices. That there are differences from the theoretical prices is not surprising. An analogy might be a football player on the field not knowing exactly where the first-down line is, while the television audience can clearly see the yellow line on the TV image.
Real-time anomalies
Options traders are trying to find the correct price in real time with shifting parameters, so missing the hypothetically correct price by $50 or $60 may not be a bad estimate. It would be interesting to see how many traders may look back at a source such as Barchart.com to find out how well their actual buying or selling prices compared with the hypothetical price curve. Black-Scholes option prices depend on an estimate of the volatility of the underlying futures contract; differences in estimates can lead to over- or underpricing.
The charts on page 28 show option prices and strike prices for the euro on Nov. 16 and the Australian dollar on Nov. 8. Option prices are those listed by Barchart.com, while the predicted prices are computed by a regression equation that creates a parabolic option price formula. The five actual trades all are undervalued, depicted as positive variances of predicted prices less market prices. To take advantage of the lower-than-theoretical prices, credit spreads are suggested in which an adjacent call at the hypothetical price is sold while the undervalued option is bought.
Two tables showing option and strike prices on Nov. 16 for the euro and Nov. 8 for the Aussie are included online. Option prices are those listed by Barchart.com, while the predicted prices are computed by a regression equation that creates a parabolic option price formula. The five actual trades are all undervalued, as positive variances of predicted prices less market prices. To take advantage of the lower-than-theoretical prices, credit spreads are suggested in which an adjacent call at the hypothetical price is sold while the undervalued option is bought.
The credit spreads in this sample are expected to be completed in the same day as the original trade, from mid-morning until the afternoon close. Results of the trades also are shown online. Because the underlying euro futures price declined on Nov. 16, the actual trades that were bought as undervalued calls resulted in losses, while the corresponding sales of adjacent calls gained. For Australian dollar calls on Nov. 8, buying the undervalued options produced an overall gain for the day as the futures price increased.
The analysis of options pricing and the results of actual trades make it apparent that success — particularly in the short-term — may depend on knowledge of hypothetically correct prices as well as the significant magnification of small variances from those prices. Forex futures are influenced by world markets and political climates, while their puts and calls are priced by surgically precise mathematical models.
