1.10 Exercises
1. A bettor sells the out-of-the-money Comex Gold upbet at 28.2, $100 per point. What is the potential profit and loss?
2. The S&P Minis on the CME are trading at 1250. A punter fancies the market down. Should the aggressive gambler sell the 1350 upbet or buy the 1150 downbet?
3. The following prices are observed in the Forex $/€ binary options market for September expiry.
If the underlying exchange rate is trading at $119.35, what trade(s) are available to lock in a profit? What will the profit be?
1.11 Answers
1. Potential profit = 28.2 ¥ $100 = $2,820
Potential loss = (28.2 – 100) ¥ $100 = – $7,180
2. Buy the 1150 downbet since the emphasis is on the ‘aggressive’ gambler. Both bets are out-of-the-money and therefore worth less than 50. They both have strikes 100 from the underlying, so assuming a normal distribution, will be worth the same. Just say they were worth 25 each. Then selling the upbet can only ever realise a profit of 25 while buying the downbet at 25, will realise a profit of 75 should it win.
3. Firstly, the underlying in this case is irrelevant. Buying the upbet and the downbet will cost a total of 96.7 to yield a risk-free profit of 3.3, since the upbet and the downbet must aggregate to 100. Since this trade is risk-free ‘fill yer boots’ and do as many as possible, in this case $3,600 of each. Therefore: Profit = 3.3 ¥ $3600 = $11,880
2. Theta & Time Decay
2.0 Introduction
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, market- makers 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.
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
