Time is both an advantage and a problem in option trading. Time to expiration acts like gravity — continuously tugging option values toward final price premiums of zero. Those who buy and sell options try to take advantage of temporary differences between market prices and values that are determined by theoretical or market-based time-related pricing models.
For the past three decades, exchange-traded options in the United States have been priced primarily by theoretical calculations, typically based on the Black-Scholes pricing model. One of the key assumptions in these pricing models is that market price variations are related to the square root of time.
The decline of option prices and time premiums through a 360-day period is shown in “Loss in time premium” (above). The chart of declining value as a function of the square root of time and option premiums falling through time to expiration as computed by the Black-Scholes model shows why option premiums face a continuous downward force as time to expiration gets closer. First, prices vary in relation to the square root of time. Second, options in the U.S. markets are valued mainly by Black-Scholes models that are based on declining values corresponding to the square root of time.
The price curves shown on “Loss in time premium” show the acceleration in decline of value with the shrinking of time to expiration. In their book, The Complete Guide to Option Selling (McGraw-Hill, 2004), James Cordier and Michael Gross recommend selling options that have slightly more than 90 days to expiration. They term this 90-day area the “sweet spot” for option sellers. Counting back nine bars from the right side of either time premium chart indicates that this area is near the beginning of a more rapid decline in premium as the option’s expiration date approaches. At 90 days or more, the market may permit a profitable premium to the seller and still yield an advantage due to the effect of time.
THE TIME FACTOR
Actual market prices vary as to how well they relate to the square root of time and to theoretical pricing by Black-Scholes. In “Time Effect” (above) Black-Scholes prices are computed for Eurodollar calls on Sept. 12, 2006.
By use of a constant standard deviation, futures price and strike price, the only variable is time. In this sample, theoretical prices are closer to market prices for longer times to expiration. Eurodollar option prices are listed as interest rates. Unusually low rates for short-term options are related to temporary rate and price differences caused by Federal Reserve policy actions during 2006.
A device that may be used to find overvalued and undervalued options is the LLP pricing model. Calculations for put and call options on a single underlying asset at one point in time are shown in “How to price Eurodollar options,” July 2006 (link below). Excel download files for LLP calculations are available at www.futuresmag.com.
The LLP model generates a price curve that includes a number of different strike prices. Option market prices are primarily established by computer models that produce thousands of theoretical prices in a continuous process, and market prices for the most part lie along parabolic curves that are similar to those calculated by the LLP log-log parabola pricing model.
The euro September 2006 call data from Aug. 28, 2006 provided the data for “Euro FX calls” (below). Using 10 of 37 strike prices for which option prices were listed by the Chicago Mercantile Exchange (CME) on Aug. 28, the LLP model was used to compute the option price curve.
As shown by “Euro FX calls,” predicted prices are close to actual market prices. The option pricing equations generated by the LLP model permit forecasting future prices, given changes in the underlying asset. An LLP price curve may be used for a number of days to provide information for hedging and trading options having a single expiration date.
One method may be used with assets that have option prices listed by various exchanges for multiple expiration dates. The system compares several option curves, from longer to shorter times to expiration. Two underlying assets are used to illustrate this process: S&P 500 futures and live cattle futures, which are both listed by the CME.
On Aug. 31, 2006, formulas for LLP price curves were computed for CME S&P 500 futures calls expiring on five future dates. These regression formulas were used to produce the price curves shown on “S&P futures call options” (below). To compute comparative price curves, each formula assumed an exercise price equal to 1.00, with ratios of the futures price/strike price ranging from 0.995 to 1.008. Intrinsic values are equal to the ratios of the futures price/strike price minus 1.0.
The chart of S&P call price curves shows that the curves increase in height with longer times to expiration. Spaces between the curves reflect the time premiums computed for each expiration date. With the passage of time, each curve will in turn subside approximately in accordance with the decreasing values shown on “Loss of time premium” and eventually be worth either nothing or its intrinsic value. As each expiration date occurs, a new longer-term option curve will enter at the top of the chart.
Option price curves do not always match the organized time sequences shown on the S&P 500 chart. Exceptions to the rule are included in “Live cattle calls” (below). This set of LLP price curves computed Aug. 28, 2006 includes six expiration dates from October 2006 through August 2007. On this chart, the ratios of the futures price/strike price are increased by increments of only 0.001. The small increases in ratios result in a look at option prices in which the differences between price curves are highly magnified, and in which the curves appear almost as straight lines due to the brief price range.
The structure of time differences is still apparent, but two of the curves (April and June, 2007) are almost the same height. The small difference between the two is shown by the time premiums where the futures price equals the strike price. The time premium percentage for April is 4.07 and the premium percentage for June is 4.13. Price curves calculated on Sept. 12, 2006 showed that the two curves had reversed positions according to the time to expiration, with April 2007 having a time premium of 4.13% and June 2007 having a premium of 4.07%.
Any number of variables might cause time premium reversals such as those for live cattle options on Sept. 12. They could be due to temporary supply and demand conditions for the two expirations or more permanent seasonal effects. It is still evident, however, that when April 2007 contracts expire a time premium should remain for June 2007 calls.
On Nov. 21, 2006, another look at the option price curves for April 2007 and June 2007, live cattle calls showed time premiums more in line with expected pricing. April call options had a premium of 3.61% while the June premium was 4.08%. The underlying futures prices for both expiration months had increased: by 0.55% in June and by 0.30% in April. However, for the strike price nearest to the futures price on Sept. 12, the June call option price increased by 0.45 while April declined by 0.455. A spread trade on Sept. 12, selling April and buying June, would have resulted in a gain of approximately 90 points.
Just as there are differences between the expected heights of price curves and actual market premiums, there are usually small variations between prices predicted along LLP curves and corresponding market prices. These differences may be used to increase chances for profitable trades.
Becoming acquainted with option price curves and time patterns shown on price charts may allow option traders to avoid relatively overpriced options and to find those that are underpriced. The best news for option traders and option writers is that the market is well organized, with very small differences between market prices and theoretically computed prices. Unusual price effects tend to stand out. Using the flow of data from the CME, the Chicago Board of Trade and other providers in computation models that isolate the differences can assist in finding the best option trading and writing opportunities.
Paul D. Cretien is a retired professor of finance at Baylor University and a chartered financial analyst. He wrote The Basics of Bank Investments (Graduate School of Banking at LSU, 2004). E-mail him at PaulDCretien@aol.com.