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.

2.5 Downbet Theta
The downbet theta is the negative of the same strike upbet theta. Table
2.5.1 provides thetas for $100 strike downbets. They are all positive. This
is because with the given range of underlying every downbet is in-themoney
since the strike is $100 and the underlying is less than the strike.
All in-the-money bets settle at 100 therefore each of the downbets in the
table will increase in value as time passes.
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.

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.
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 outof-
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.
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.