Читать книгу Binary Options - Hamish Raw - Страница 39

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.