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.
Figure 2.1.1
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.
Figure 2.2.1
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:
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:
Figure 2.2.2
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.
Table 2.2.1
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
= – 17.484
The theta with dt = 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 dt→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.
Table 2.3.1
Figure 2.3.1
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