One of the least-understood aspects of option pricing is how to account for holidays and weekends when looking at short-dated option prices. Intuitively options have low premiums if there is a smaller window of opportunity to extract value from the gamma. The question is: how do traders calculate this exactly?
Look At The Business Days
The price of an at-the-money option is given by the following formula:
The longer the maturity, the more an option is worth. Usually a one-week option contains five business days and two weekend days. What if the week is over the Easter holidays and contains only three business days, the other four being holidays and weekends? This option should be worth less, but it is not immediately obvious how much less.
One solution to this problem is to translate calendar days (regular 7 day weeks composed of 5 business days and 2 weekend days) into business days. In this scenario, one normal week is considered to be around 5.2 business days long. This includes the 5 regular business days and an additional .1 day for each weekend day to account for the occasional shock event that would cause Monday's Sydney opening price to be very different from Friday's New York closing price.
The link between calendar time and business days comes through the option premium: the amount an option is worth must be constant across both regimes. Therefore we can transform calendar volatility, what we trade, into business day volatility. The current situation in one-week options provides a timely example:
Example
April 22's overnight euro/dollar implied volatility was 10.5%. There were three normal business days, two weekend days and two holidays in Europe but not Tokyo or New York, worth .35 of a day.
There are, therefore, 3.9 business days in the seven calendar days. A little rearranging of the formula in the first paragraph gives the following:
Therefore the fair value for one-week volatility can be calculated as 7.85%.
This approach has many more applications. It is desirable to consider the number of business days when comparing implied volatility to historical volatility. It is also possible to reverse-engineer the above calculation to calculate the importance of holidays currently being priced into the market.
The FX Options Market Microstructure
The above procedure makes theoretical and intuitive sense. But, does the above analysis match with the empirical evidence from the foreign exchange options market? These types of questions fall into the broad topic of 'market microstructure.' That is, how the options market actually trades.
Using our foreign exchange options database, we address this question looking at two different possible effects:
1 the weekend effect
2 the holiday effect, and more specifically the Easter holiday.
Weekly Seasonality
The presence of two non-trading days in a calendar week, Saturday and Sunday, should affect any option with a maturity of less than a week. The so-called overnight volatility--the implied volatility associated to an option expiring one day ahead--is one of the quoted volatilities. Figure 1 shows the time series of euro/dollar one-day implied volatility. One characteristic is striking. On Friday, implied volatility tends to trade at a sharp discount with respect to both Thursday and Monday levels.
Indeed, Friday overnight volatility takes into account the three calendar days of which only one is an active business day. As the chart shows, implied volatility drops on average four vol figures on Friday and recovers by Monday. The pattern observed in euro/dollar is common to all major currencies. Using a simple regression technique we estimated the average discount for three currency pairs, euro/dollar, dollar/yen and euro/yen. Using daily data back to 1996 we estimated the following regression:
Equation 3 is a simple mean reverting model augmented with a set of dummies to take into account the day of the week effect.
Figure 2 shows the results for the three currency pairs under study. For all currencies the Friday's and Monday's coefficients were of the correct sign and highly statistically significant. An overall test for the existence of seasonality is statistically accepted. Figure 3 shows the estimated gamma coefficients in the three currencies. On Fridays implied volatilities decline between 4.26% and 4.66% from the Thursday levels and recover on Monday.
This weekly seasonality is specific to the way in which the FX option market trades and is not related to the actual volatility of spot: options traders maintain constant premium, assuming market conditions remain the same by varying the level of implied volatility.
The Easter Holiday Effect On Implied Volatility
As argued in the previous paragraph the weekend effect can be detected only on foreign exchange options contracts with a maturity of less then seven days. Nevertheless the effect of major holidays like Christmas, New Year, and Easter could potentially affect even implied volatilities with longer maturities. In this section we analyze the effect of the Easter holiday on the one-week implied volatility. We selected this contract for two main reasons: (1) the maturity is short enough to be eventually sensitive to such a short holiday period (2) this maturity is not affected by the weekend, hence any pattern we are able to highlight would be specifically to the Easter holiday and not related to any other calendar effect.
Detecting an Easter holiday effect on implied volatility is slightly more tricky than the weekend effect described above. Unfortunately, as the Easter weekend changes from one year to the next, simple regression techniques cannot be used. However, an event study analysis can help us in our investigation.
We looked back over the past 10 years at one-week implied volatility for the major currencies. For each currency we calculated the change from the Thursday, one week before Easter to the Thursday immediately before Easter, the last trading day. Subsequently we calculated the change from Easter Thursday to the following Thursday. Table 1 shows the main results of this analysis.
One striking characteristic is that in all currency pairs the one-week implied volatility initially decreased ahead of Easter and subsequently increased during the days immediately after the holiday.
Immediately ahead of Easter one-week implied volatility dropped on average between -0.5% and -1.34%.
Subsequently volatilities reversed, with the increase ranging from a massive 1.37% to a more modest 0.07%. What is more, the hit ratio--the number of times vol moved higher--ranges between 62-100%. Euro/dollar, Aussie dollar/U.S. dollar, dollar/Swissie and cable seem to be more sensitive to the Easter holiday period.
Figure 4 shows the Easter dynamics for all currencies. All in all this analysis supports the view that options traders adjust implied volatility to take into account for the number of business days in the contract.
Conclusions
Short-dated option pricing contains an element of art as well as science. There is no clear way to ascertain the exact worth of any given holiday. For example, an American bank may attribute more weight to a U.S. holiday than a European or Japanese bank would. These differences of opinion present potential pockets of value in the market.
This week's Learning Curve was written by Giovanni Battista Pillitteri, foreign exchange strategist at Deutsche Bank in London.
