– 17.484
The theta with dt = 2 days, 1 day and .5 day is –31.947, –19.491 and –17.484 respectively. As the time either side of 2 days to expiry decreases, i.e. as dt→0, the theta approaches the value –16.936, the exact slope of the tangent to the curve at 2 days to expiry.
The next sections on upbet thetas describe how the trader can use this measure of time decay in a practical manner.
2.3 Upbet Theta
Table 2.3.1 provides 1 and 5 day thetas for underlying prices ranging from $99.50 to $99.90 and assumes a strike price of $100 and therefore applies to Fig 2.2.1. The theta for the $99.70 profile with 5 days left to expiry is –6.5057. This value of theta defines how much the upbet will decline in value over one year at the current rate of decay. To gauge how much the upbet will lose in time decay over 1-day divide the theta by 365 so the rough estimate of one-day decay at 5 days would be –6.5057 / 365 = –0.017824. But remember, by convention binary prices are multiplied by 100 to establish trading prices within a range of 0 – 100, so likewise we need to multiply the theta by 100 to get comparable decay. In effect the time decay over 1 day of an upbet with 5 days to expiry is –0.017824 ¥ 100 = –1.7824 points. In fact the upbet with 5.5 and 4.5 days to expiry is worth 28.2877 and 26.2938 respectively, a decay of –1.9939, so it can be argued that a 5-day theta of –1.7824 is a reasonably accurate measure.
Table 2.3.1
Figure 2.3.1
Fig 2.3.1 illustrates how thetas change with the underlying. The assumed strike price is $100 and four separate times to expiry are displayed.
1. It is apparent how little effect time has on the price of an upbet with 50 days to expiry as the 50-day profile is almost flat around the zero theta level.
2. Another point of note is that theta is always zero when the binary is at-the-money. In hindsight this should be reasonably obvious since it has already been pointed out that an at-the-money binary is always worth 50.
3. What may not be so apparent is that totally unlike a conventional option the theta of a binary may be positive as well as negative. This is because an in-the-money binary will have a price moving upwards to 100 as time decays and hence a positive theta, compared to the conventional that always has a negative theta.
As time passes and the upbet gets closer to expiry the absolute value of the theta becomes so high that it fails to realistically represent the time decay of the binary. From Table 2.3.1 the 1-day theta with the underlying at $99.70 is –43.1305 when the upbet value is actually 12.52. The theta is forecasting a decay of:
100 ¥ – 43.1305 / 365 = – 11.8166
which is not so far wide of the mark since it will in fact be –12.52 being the price of this out-of-the-money bet with 1 day to expiry. Should the 0.1 days to expiry profile be included, at an underlying price of $99.92 the theta would be –440.7 and the clarity of Fig 2.3.1 would be destroyed as the graph is drastically rescaled. It would also be suggesting that the upbet would lose:
100 ¥ – 440.7 / 365 = – 120.74 points
over the day when the maximum value of an upbet can only be 100 and, with 0.1 days to expiry this bet would be in fact worth just 16.67.
In general the theta will always underestimate the decay from one day to the next since as can be seen from Fig 2.2.2 the slope of the profiles always gets steeper approaching expiry. This means that the theta, which could be construed as the average price decay at that point, will always over-estimate the time decay that has taken place over the preceding day but will under-estimate the decay that will occur over the following day. When there is less than one day to expiry the theta becomes totally unreliable.
Nevertheless, this mathematical weakness does not render the theta a totally discredited measure. Should a more accurate measure of theta be required when using theta to evaluate one-day price decay, a rough and ready solution would be to subtract half a day when inputting the number of days to expiry. If this offends the purist then another alternative would be to evaluate the bet at present plus with a day less to expiry. The difference when divided by 100 and multiplied by 365 will provide an accurate 1-day theta. This might at first sight appear to defeat the object of the exercise since one is calculating theta from absolute price decay when theta would generally be used to evaluate the decay itself, but it is an accurate and practical method for a marketmaker who is hedging bets with other bets.
The lack of accuracy of thetas close to expiry is not a problem exclusive to binary options but affects conventional options also. Even so conventional options traders still keep a ‘weather-eye’ on the theta, warts and all.
2.4 Downbets over Time
Fig 2.4.1 provides the route over time by which the downbet reaches the expiry profile of Fig 1.5.1. The time to expiry has been expanded in order to include a yearly binary. In this example if the underlying now falls from $100 to $98 the one year downbet only increases to 66.61. The best bit is that if you are long the one year downbet and the market rallies from $100 to $102 the downbet only falls to 35.53. Earlier during the book’s introduction binaries were deemed to be highly dextrous instruments; this example proves that even the most dull, conservative, risk-averse pension fund manager who doesn’t have the stomach for the standard +45° P&L profile of AAA-rated multinational stock can find, in the short term, a more boring, safer way to gain exposure to financial instruments.
Figure 2.4.1
2.5 Downbet Theta
Table 2.5.1
With 10 days to expiry the highest theta in the table occurs when the underlying is $99.25 while, with 5 days to expiry, the highest theta occurs at $99.50. Clearly, unlike a conventional option where the highest theta remains static at the strike over time, with a binary the highest theta shifts towards the strike over time.
Fig 2.5.1 illustrates the downbet thetas where clearly the peak and trough of the theta approach the strike as time erodes. The theta of the 50-day binary is zero across the underlying range indicating that, irrespective of the underlying, the passage of time has zero impact on the price of the bet.
Figure 2.5.1
2.6 Theta and Extreme Time
Extreme time has been introduced as a special case since it should not divert attention away from the ‘normal’ characteristic of theta as outlined in Section 2.2. Nevertheless it would be remiss of a study of binary theta if the following quirk of theta was not acknowledged.
