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?
2. If the underlying is $50 and the theta of the $49 strike downbet is –5.1, roughly what will the theta of the $51 strike downbet be?
3. If there are eight days to expiry and one attempts to roughly approximate the time decay of a bet over the forthcoming single day, which theta would provide the most accurate estimate?
a) the current 8-day theta
b) the 7-day theta, or
c) a theta somewhere in between the seventh and eighth day?
4. For each bet find the time decay over the requested number of days using the associated theta.
5. The following table provides the prices of three bets at five separate points in time (t) where t–1 has one day less to expiry than t. Find an approximation to theta for each bet at time t.
2.11 Answers
1. Thetas are negative when out-of-the-money, positive when in the money and zero when at-the-money. Therefore,
a) Positive
b) Negative
c) Zero
d) Positive
2. Since the $49 strike downbet and the $51 strike downbet are equidistant around the underlying they will have roughly the same absolute theta, i.e. 5.1. Therefore the $51 strike downbet will have a theta of roughly +5.1 since it is in the money and will therefore increase in value as time passes..
3. Both a) and b) provide average thetas at the single points of 8 days and 7 days to expiry exactly. The former will underestimate the one day time decay while the latter will overestimate it; therefore c) will provide the most accurate forecast.
4. Using the formula:
Time Decay = 100 x (No. of Days x Theta ) / 365
Bet 1: 100 ¥ ((1 ¥ 5) / 365) = 1.37 points
Bet 2: 100 ¥ ((2 ¥ 5) / 365) = 2.74 points
Bet 3: 100 ¥ ((5 ¥ 10) / 365) = 13.70 points
5. Ignore t–2 and t+2, as the most accurate theta is obtained from making the increment in time as small as possible, i.e. as dt→0. Therefore subtracting the price at t+1 from the price at t–1 gives the difference in price over two days, and then multiplying by 365 and dividing by 100 provides the correct theta:
Theta = [( Pt–1 – Pt+1 ) / ((t + 1) – (t – 1))]*365/100
Bet 1 = [( 50.280 – 50.331 ) / (( t + 1 ) – ( t – 1))] ¥ 365 / 100 = –0.0931
Bet
