Theta & Time Decay

Theta is a ratio that measures how much the bet price will change due
to the passing of time.
Theta is probably the easiest ‘greek’ to conceptually grasp and is possibly
the most easily forecast since the passage of time itself moves in a
reasonably uniform manner.
Bets on many financial instruments are now always ‘in-running’, i.e. there
is always a market open on which to trade. These days there is a 24-hour
market in foreign exchange trading so any bet on the future level of the
$/£ rate is always ‘in-running’ with the theta constantly impacting on the
price of bets. On other markets which operate in discrete time periods,
where the market is open for a limited period of five days a week, marketmakers
will often use Monday’s theoretical prices on a Friday afternoon
in order not to get too exposed to the weekend’s three-day time decay.
An understanding of time decay and theta is thus critical to the trading
and risk management of binary options. The remainder of this section on
theta will analyse the effect of time decay on upbets and downbets, and
how this impact on the price of a bet is measured.
2.1 Upbets v the Underlying over Time
This section discusses time decay and its effect on the price of upbets as
time to expiry decreases, ultimately resulting in the profile of Fig 1.2.1.
Fig 2.1.1 shows the profile of upbets with a strike price of $100 and a
legend indicating the time to expiry. A unique characteristic of the binary
is that, irrespective of whether upbet or downbet or time to expiry, each
profile travels through the price 50 when the underlying is at-the-money,
i.e. the underlying is exactly the price of the strike. This is because a
symmetrical bell-shaped normal probability distribution is assumed so
that when the underlying is at-the-money there is a 50:50 chance of the
underlying going up or down. This feature of the binary immediately
distinguishes it from the conventional option where the at-the-money can
take any value.

2.2 Price Decay and Theta
Fig 2.2.1 describes the prices of upbets with a strike price of $100 and
time to expiry decreasing from 50 days to zero. In Fig 2.1.1 if one were
to imagine a vertical line from the underlying of $99.70 intersecting 50
price profiles (instead of just the five listed in the legend) then in Fig 2.2.1
the middle graph would reflect those upbet prices against days to expiry.
The $99.90 profile is always just 10 cents out-of-the-money and is always
perceived to have good chance of being a winning bet. Only over the
last day does time erosion really take effect with a near precipitous price
fall from 35 to zero. The $99.50 profile paints a different picture as this
upbet is always 50¢ out-of-the-money and the market gives up on the bet
at an earlier stage. On comparing the gradients of the $99.90 and $99.50
profiles, the former has a shallower gradient than the $99.50 profile for
most of the period but then as expiry approaches, this relationship
reverses as the gradient of the $99.90 profile increases and becomes more
steeply sloping than the $99.50 profile.
This gradient that we are referring to in Fig 2.2.1 is known as the theta.
The theta of an option is defined by:
θ = δP / δt
where: P = price of the option, and
t = time to expiry
so that δP = a change in the value of P
and δt = a change in the value of t.
The theta is therefore the ratio of the change in the price of the option
brought on by a change in the time to expiry of the option.
To provide a more graphic illustration Fig 2.2.2 illustrates how the slopes
of the time decay approach the value of the theta as the incremental
amount of time either side of the 2 days to expiry is reduced to zero. The
gradient can be calculated from the following formula:
θ = (P0 – P1)/(T1 – T0) × 365/100
where: P0 = Price at time 0
P1 = Price at time 1
T0 = Time to expiry at time 0
T1 = Time to expiry at time 1

Table 2.2.1 shows the value of the bet as the days to expiry decreases
from 4 to 0 with the underlying at $99.80. The theta with 2 days to expiry
is actually –16.936 and this is the gradient of the tangent of the curve
’$99.80 Underlying’ in Fig 2.2.2 with exactly 2 days to expiry.

Thus, the 4-Day Decay line runs from 35.01 to zero in a straight line and
has an annualised gradient of:
Gradient of 4-Day Decay = (0 – 35.01) / (4 – 0) × 365 / 100
= –31.947
Likewise for the 2-Day & 1-Day Gradients:
Gradient of 2-Day Decay = (22.18 – 32.86) / (3 – 1) × 365 / 100
= –19.491
Gradient of 1-Day Decay = (26.56 – 31.35) / (2.5 – 1.5) × 365 / 100
= –17.48
The theta with δt = 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 δt→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.
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 atthe-
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. FromTable 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 underestimate
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.