Binary Options - Hamish Raw

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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.

      Figure 2.6.1

      When there is a large amount of time to the expiry of the bet then theta behaves in an unusual manner. Fig 2.6.1 is Fig 2.2.1 but with a different time scale along the horizontal axis. The horizontal axis is now expressed in years and what the graph illustrates is that as time to expiry increases for an out-of-the-money upbet, the value of the upbet decreases. This implies that the curious situation would exist whereby an investor could buy the upbet with years to expiry, hope that the underlying does not rise, and still see his investment increase in value over time. In effect, the out-of-the-money upbet with sufficient time remaining to expiry has a positive theta.

      The more ambitious reader may wish to shut their eyes and try and figure this one out, but for those of whom want to push on to the next subject here’s the intuitive answer. This out-of-the-money upbet is constrained by the prices zero and 50. However close the underlying gets to the strike and irrespective of how much time is specified in the contract, the upbet cannot breach 50. And on the downside the probability of an event can never be negative so the upbet is restricted to zero. Increasing the time to expiry therefore has a decreasing effect on the price of the upbet close to the strike, as the probability of the upbet travelling through the strike cannot exceed 50%. But at the same time the increased time increases the probability of the underlying travelling to zero thereby ensuring a losing bet. Obviously this extreme case applies to downbets as well.

      Is this quirk of any relevance? Probably not a lot. But consider an insurance contract (binary option) written at Lloyd’s of London…a contract with a lengthy ‘tail’. Food for thought?

      2.7 Bets v Conventionals

      Fig 2.7.1 provides a comparison of thetas for upbets, downbets and conventional calls and puts.

      Points of note are:

      1. Downbets and upbets mirror each other across the horizontal axis.

      Figure 2.7.1

      2. Whereas the theta of the conventional call and put are the same and are always negative, the theta of upbets and downbets each take on both positive and negative values.

      3. The theta of the conventional is at its greatest absolute value where the theta of upbets and downbets are both zero, i.e. when the options are at-the-money.

      2.8 Formulae

      2.9 Summary

      The theta is an immensely important ‘greek’ since it is always impacting on the value of a bet even should the underlying market be closed for trading. As a tool the theta provides:

      1. for the premium writer, a measure to indicate the bet with the greatest time decay;

      2. for the punter looking for gearing, the bet with the highest theta may well be the bet to avoid; while

      3. for marketmakers, thetas provide the ability to hedge one bet with another in order to be theta neutral in front of, say, a long weekend.

      Apart from these common features of thetas, binary thetas have little in common with conventional thetas. In particular they can take positive as well as negative values which can prove a major headache for the premium writer should the underlying travel through the strike so that writers who are expecting to take in premium now find they are paying out as the theta swings from negative to positive time decay.

      Both binary and conventional theta are prone to the same increasing inaccuracy as time to expiry approaches zero. Nevertheless, they provide an essential parameter provided the deficiencies are clearly understood.

      2.10 Exercises

      1. Is the theta for the following prices of upbets and downbets positive or negative?

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