Introduction
Vega is a ratio that measures how much the bet price will change due to
a change in implied volatility.
All option pricing needs to make assumptions about the behaviour of
certain variables. Some are clearly inappropriate, e.g. two of the main
assumptions behind the Black-Scholes option-pricing model are that
volatility remains a constant and that continuous hedging is feasible, both
of which are plainly not true. But financial theorem in general often needs
to make these assumptions in order to draw comparisons between one
instrument and another for evaluation purposes. The gross redemption
yield of a bond is a case in point. In calculating the g.r.y. there is an
implicit assumption that all dividends will be reinvested in the same bond
at the same gross redemption yield through to maturity. Totally unfeasible
of course, but nevertheless the g.r.y. has provided one of the more
enduring measures in the relative evaluation of bonds. By the same token,
some of the underlying assumptions with respect to the pricing of stock,
currency and bond options can also be considered a bit flaky and this
chapter covers, possibly, the most tenuous of them all, volatility.
Volatility is a measure of the movement in the underlying. There are two
measures of volatility, historic and implied. Historic volatility is a measure
of past price movements, while implied is a number thrown out of the
pricing algorithm if the option price is an input instead of volatility.
There will be no inclusion of Wiener processes or Brownian motions in
this chapter so non-mathematicians need not bolt for the door just yet. An
understanding of the normal and lognormal distribution would be an
asset though.
3.1 Normal Distribution
Fig 3.1.1 illustrates two bell-shaped normal distributions. They are
symmetrical about the mean, zero in this case, while the standard
deviation, or volatility in options parlance, is ‘low’ or ‘high’.
If the share price at any time is assumed to be the mean of the
distribution, then the volatility measures the number of times the price has
traded, or maybe settled, away from the mean. If readings were taken
over a year measuring the difference in the price of a stock from one day
to the next, then graphically the readings may be represented as a normal
bell-shaped distribution. If the stock has low volatility then it would be
expected that for a high number of incidences the stock will close at
around the same price the following day. The frequency of the stock
having a big move in one day will be rare and this distribution is reflected
in the ‘Low Volatility’ graph. On the other hand if the stock is volatile
then the incidence of a big move will be greater and the distribution will
look more like the ‘High Volatility’ graph.
In Fig 3.1.2 the area under the graph and above the horizontal axis at
zero frequency is 1, reflecting the total probability of an event taking
place. In this illustration the arithmetic mean has been replaced by the
current underlying price of $100 which is directly below the highest point
of the distribution. If a vertical line was drawn from the peak to the
horizontal axis then 50% of the total area would be to the left of this line
and 50% to the right. This reflects the probability that there is a 50:50
chance of the underlying going up or down. To the right of $100 there is
a vertical line at the strike price of $101. If the area B to the right of the
$101 strike contains 40% of the total area A+B, then the probability of the
underlying ending above the strike is 40% and it would be reasonable to
expect the market price of the $101 upbet to be trading at 40.
If this normal distribution is an accurate fit of the movement of the
underlying, then since the graph is symmetrical at about $100, the
probability of the $99 downbet ending in-the-money, i.e. the underlying
below $99, is also 40%.
3.2 Lognormal Distribution
There is a flaw with using the normal distribution for modelling the price
movements of financial and commodity instruments and that is that
absolute price movements are assumed. For example, if a healthy
company has a share price trading at £1, should the probability of the
share going down £1 be the same as the share going up £1? In effect, are
the odds of the healthy company going bust with absolutely no assets
producing a zero share price the same as the odds of the healthy
company doubling the share price to £2? The answer is no, and in order
to circumvent this problem, the lognormal distribution is used in financial
engineering.
If the underlying moves from $100 to $80 then it has fallen 20%. If the
underlying falls $20, from $120 to $100, then it has only fallen 16.667%.
If the underlying falls from $125 to $100, then that is a 20% fall. In effect
a distribution is required that gives a move from $100 down to $80 the
same probability as a move from $100 up to $125. The lognormal
distribution fits the bill since –log($80/$100)=log($125/$100).
The lognormal distribution is shown in Fig 3.2.1 where there is a fatter
skew to the right of the mean, so that when a lognormal distribution is
used the $80 strike downbet is now worth less than the $120 strike upbet,
reflecting a higher probability of an absolute move of $20 upwards rather
than downwards.
3.3 Historic Volatility & Implied Volatility
Historic volatility is a measurement derived from studying the past price
behaviour of the underlying, while inputting the option price into the
pricing model (instead of volatility) will generate the implied volatility.
Historic Volatility
The numbers required to calculate the ‘fair value’ of an option are the
underlying price, the strike price, the time to expiry, volatility plus interest
rates and yield (which are both assumed to be zero for the present in
order to keep things simple). Since time to expiry, the current underlying
price and the strike price are given, plus zero interest rates and yield are
assumed, the volatility number in evaluating ‘fair’ value is the all important
input. There are many statistical methods of calculating the historic
volatility ranging from simple weighted averages through to more complicated
stochastic methodologies. Many books have been written on this
subject alone and this book is not going to compete in that space, but
one observation can be made.
When assessing an accurate historic volatility for use in an options
pricing, model the algorithm employed will generally be of less
importance than the length of time used over each reading. For example,
assume a trader wishes to trade a twenty-day option, should a 1-day, 20-
day or 1-year moving average with respectively a 5%, 6% and 7% ‘vol’
be used? The twenty-day ‘vol’ would be the most appropriate because if
the one-day 5% volatility is used then there is the ever-present danger of
getting caught short when the market moves. Alternatively, the 7% oneyear
‘vol’ could be too expensive since it is perfectly feasible that the
market may be stagnant for 20 days and then become volatile for the
remaining 232 trading days in the year.
‘Scalpers’ will take an ultra short-term view on ‘vol’ while funds look at
the bigger picture adopting a range of various measures of stochastic
volatilities. There is no such thing as the historic volatility but a myriad of
techniques and timescales all deriving their own interpretation of historic
volatility.
Implied Volatility
Implied volatility is the measure of standard deviation that is an output
from Black-Scholes and other mathematical models available for evaluating
options.
By entering the underlying, strike, time to expiry and option price one
can generate the implied volatility. This volatility can then be used as the
input to create a whole range of option prices for different strikes and
underlying. With conventional options the preferable strike to use for
calculating the implied volatility is the at-the-money option. This is
because at-the-monies are generally more liquid so a keener price is
available, but more importantly they have the most time value which is
the factor on which volatility works. But with binary options the at-themoney
is always 50 so it is necessary to generate two implied volatilities
either side of the at-the-money and, bearing this in mind, the greatest
liquidity will be found in an out-of-the-money upbet and out-of-themoney
downbet.
Historic v Implied Volatility
An options marketmaker will always be bidding and offering around a
volatility level where he feels the weight of options buying equates to the
weight of options selling. To that end historical volatility is of little concern
to him, just a constant assessment as to where the ‘implied’ is at any one
time. Alternatively an options hedge fund may specialise in studying
historic volatility in order to go ‘long’ or ‘short’ premium. The relative
importance of historical and implied volatility is very much dependent on
the trader’s view on random walk theory and Chartism. If the reader is by
nature a chartist then it is more likely that the historic volatility concept
will weigh more heavily than fundamentals. Alternatively the
fundamentalist is more likely to study the schedule of upcoming
announcements on economic data and events and attempt to gauge what
kind of reaction they are likely to precipitate in the marketplace.
A brief anecdote might better describe the gulf that can occur between the
two separate views of volatility. In 1992 the author was trading
conventional options on 90-day Sterling LIBOR futures. The future was
trading at 89.25 with the 89.00 Puts on offer at 1 tick a thousand contracts.
The options had two and a half days to go to expiry. The implied volatility
of the 89.00 Puts was about 14.8% with the at-the-money 89.25 straddle
trading at around 12%. On the (arguable) assumption that the straddle was
a more accurate reflection of historic volatility than the 89.00 Put, even at
1 tick the 89.00 Puts looked expensive with an implied volatility of 14.8%.
An associate bought the thousand on offer. Two days later sterling came
under such pressure that the UK government was forced to raise interest
rates by 2% to 12% leading to the future expiring that day at 88.00 with the
89.00 Puts expiring worth 100. The trader who bought the 89.00 Puts
grossed £1.25m on a £12,500 outlay. Basically the implied ‘vol’ was too
cheap and was not accurately reflecting the tumult that was going to occur
in the next two days. Implied volatility is important, so too is historic
volatility, but what is most important is forecasting where historic volatility
is going to be because that is where the real money is.
Dienstag, 23. Juli 2013
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.
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.
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.
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.
Downbet
The random walk model in Fig 1.4.1 describes when downbets win and
lose. The starting point is yet again $100 with twenty-five days to expiry,
except here the strike is $1 below at $99. In this instance the downbet is
‘out-of-the-money’ when the underlying is above the strike of $99 and ‘inthe-
money’ below the strike.
1. After day three RW1 falls to $99.01 and bounces up. This is the closest
RW1 gets to the strike and is trading at around $99.75 at the downbet
expiry. Consequently RW1 closes out-of-the-money and is a losing bet.
2. RW2 initially falls to the $99 level in tandem with RW1 but breaches the
strike. After ten days the underlying travels back up through the strike to
trade alongside RW1 at expiry. Therefore this too is a losing bet.
3. RW3 trades down to the $99 level with seven days left. With three
days to go RW3 trades back up to $99 from below the strike before
making a final downward move on the last day to trade around $98.25
at expiry. This downbet closes in-the-money, and is a winning bet
settling at 100.
1.5 Downbet Pricing
The expiry price profile of a downbet is illustrated in Fig 1.5.1. It is Fig
1.2.1 reflected through the vertical axis but with a strike of $99 as
opposed to $101.
1. In this case if the underlying is above the strike of $99 at expiry, the
downbet is out-of-the-money, has lost and is worth zero.
2. At $99 the downbet is at-the-money, is deemed a draw and worth 50.
3. While if the downbet expires with the underlying below the $99 strike,
the downbet is in-the-money, has won and is worth 100.
1.6 Downbet Profit & Loss Profiles
Trader A and Trader B now decide to trade a downbet with each other.
Trader A is no longer feeling bullish and wishes to buy a downbet (Fig
1.6.1) and since Trader B has conveniently turned bullish, he sells it to
him. This is not an aggressive trade that Trader A is putting on; since the
strike price is $101 and the underlying is $100 therefore the downbet is
already $1 in-the-money and has a better than an ‘evens money’ chance
of winning. The price of his downbet has to reflect this probability and the
price is agreed at 60, where they trade for $1/pt.
Trader A’s maximum loss since he bought the downbet is 60 × $1 = $60,
and this he will have to bear if the share price rises by over $1 from its
current level of $100. His maximum potential winnings have been
reduced to $40, which he will receive if the underlying either falls, stays
where it is at $100, or rises less then $1. In other words Trader A has
backed an ‘odds-on’ bet.
Fig 1.6.2 shows Trader B’s profile having sold the in-the-money downbet
to Trader A for 60. Trader B needs the share price to rise $1 in order to
win. If the underlying rises exactly $1 to $101, then the downbet will be
worth 50 and Trader B wins $10. A rise over $1 and the downbet expires
with the underlying above $101 and Trader B collects the full $60.
lose. The starting point is yet again $100 with twenty-five days to expiry,
except here the strike is $1 below at $99. In this instance the downbet is
‘out-of-the-money’ when the underlying is above the strike of $99 and ‘inthe-
money’ below the strike.
1. After day three RW1 falls to $99.01 and bounces up. This is the closest
RW1 gets to the strike and is trading at around $99.75 at the downbet
expiry. Consequently RW1 closes out-of-the-money and is a losing bet.
2. RW2 initially falls to the $99 level in tandem with RW1 but breaches the
strike. After ten days the underlying travels back up through the strike to
trade alongside RW1 at expiry. Therefore this too is a losing bet.
3. RW3 trades down to the $99 level with seven days left. With three
days to go RW3 trades back up to $99 from below the strike before
making a final downward move on the last day to trade around $98.25
at expiry. This downbet closes in-the-money, and is a winning bet
settling at 100.
1.5 Downbet Pricing
The expiry price profile of a downbet is illustrated in Fig 1.5.1. It is Fig
1.2.1 reflected through the vertical axis but with a strike of $99 as
opposed to $101.
1. In this case if the underlying is above the strike of $99 at expiry, the
downbet is out-of-the-money, has lost and is worth zero.
2. At $99 the downbet is at-the-money, is deemed a draw and worth 50.
3. While if the downbet expires with the underlying below the $99 strike,
the downbet is in-the-money, has won and is worth 100.
1.6 Downbet Profit & Loss Profiles
Trader A and Trader B now decide to trade a downbet with each other.
Trader A is no longer feeling bullish and wishes to buy a downbet (Fig
1.6.1) and since Trader B has conveniently turned bullish, he sells it to
him. This is not an aggressive trade that Trader A is putting on; since the
strike price is $101 and the underlying is $100 therefore the downbet is
already $1 in-the-money and has a better than an ‘evens money’ chance
of winning. The price of his downbet has to reflect this probability and the
price is agreed at 60, where they trade for $1/pt.
Trader A’s maximum loss since he bought the downbet is 60 × $1 = $60,
and this he will have to bear if the share price rises by over $1 from its
current level of $100. His maximum potential winnings have been
reduced to $40, which he will receive if the underlying either falls, stays
where it is at $100, or rises less then $1. In other words Trader A has
backed an ‘odds-on’ bet.
Fig 1.6.2 shows Trader B’s profile having sold the in-the-money downbet
to Trader A for 60. Trader B needs the share price to rise $1 in order to
win. If the underlying rises exactly $1 to $101, then the downbet will be
worth 50 and Trader B wins $10. A rise over $1 and the downbet expires
with the underlying above $101 and Trader B collects the full $60.
Upbet Pricing
Fig 1.2.1 illustrates the expiry price profile of an upbet. One of the
features of binaries is that at expiry, bets have a discontinuous distribution,
i.e. there is a gap between the winning and losing bet price. Bets don’t
‘almost’ win and settle at, say 99, but are ‘black and white’; they’ve either
(except in the case of a ‘dead heat’) won or lost and settle at either 100
or zero respectively.
1. If the upbet is in-the-money, i.e. in the above example of Fig 1.1.1 the
underlying is higher than $101, then the upbet has won and has a
value of 100.
2. Alternatively if the upbet is out-of-the-money, i.e. the underlying is lower
than $101, then the upbet has lost and therefore has a value of zero.
3. In the case of the underlying finishing exactly on the strike price of
$101, i.e. the upbet is at-the-money, then the bet may settle at 0, 50
or 100 depending on the rules or contract specification.
One issuer of binaries may stipulate that there are only two alternatives, a
winning bet whereby the underlying has to finish above $101, or a losing
bet whereby the underlying finishes below or exactly on $101. A second
company might issue exactly the same binary but with the contract
specification that if the underlying finishes exactly on the strike then the bet
wins. A third company may consider that the underlying finishing exactly
on the strike is a special case and call it a ‘draw’, ‘tie’ or ‘dead heat’,
whereby the upbet will settle at 50. This company’s rules therefore allow
three possible upbet settlement prices at the expiry of the bet.
N.B. Throughout the examples in this book the latter approach will be
adopted whereby in the event that a bet is a ‘dead heat’, or in other
words, where the underlying is exactly on the strike price at expiry, then
it is settled at 50.
1.3 Upbet Profit & Loss Profiles
The purchaser of a binary option, just like a conventional option, can
only lose the amount spent on the premium. If Trader A paid 40 for an
upbet at $1 per point then Trader A can lose a maximum of just 40 × $1
= $40. But with a binary not only the loss has a maximum limit but the
potential profit has a maximum limit also. So although Trader A’s loss is
limited to $40, his profit is limited to (100 – 40) × $1 = $60. As a general
rule the profit and loss of the buyer and seller of any binary must sum to
100 × $ per point.
In Figs 1.3.1 and 1.3.2 respectively Trader A’s and Trader B’s P&L profiles
are illustrated. Both traders are taking opposite views on whether a share
price will be above $101 at the expiry of the upbet.
In Fig 1.3.1 Trader A has bought the upbet at a price of 40 for $1 per point
($1/pt) so his three possible outcomes are:
1. Trader A loses $40 at any level of the underlying below $101.
2. At $101 the rules of this particular upbet determine a ‘dead heat’ has
taken place and the upbet settles at 50 with Trader A making a profit
of $10.
3. Above $101 Trader A wins outright and the upbet is settled at 100 to
generate a profit of $60.
Upbets
An upbet can only win or lose at the moment the bet expires and not at
any time leading up to the expiry of the bet.
Examples of upbets are:
1. Will the price of the CBOT US Sep 10 year Notes future be above
$114 at 1600hrs on the last trading day of August?
2. Will the Dow Jones Index be above 12,000 at 1600hrs on the last
trading day of the year?
3. Will the LIFFE Euribor Dec/Sep spread be above 10 ticks at settlement
on the last day of November?
4. Will a non-farm payroll number be above +150,000?
Examples 1, 2 & 3 enable the bettor to make a minute by minute
assessment of the probability of the bet winning. Example 4 is a number
(supposedly) cloaked in secrecy until the number is announced at 13.30
hrs on a Friday.
In all the above examples the bet always has a chance of winning or
losing right up to the expiry of the bet although the probability may be
less than 1% or greater than 99%. The Notes could be trading two full
points below the strike the day before expiry but it is possible, although
highly improbable, for them to rise enough during the final day to settle
above the strike. Conversely the Notes could be trading a full two points
higher than the strike the day prior to expiry and still lose although the
probability of losing may be considered negligible.
Ultimately the upbet is not concluded until the bet has officially expired
and until then no winners or losers can be determined.
Downbets too can only win or lose at expiry. Although in many circumstances
the downbet is simply the reverse of the upbet, the downbet has
been treated with the same methodology as the upbet in order that other
bets, e.g. the eachwaybet, can be analysed within a uniform structure.
Also a separate treatment of downbets will provide a firmer base on
which to analyse the sensitivity of downbets.
1.1 Upbet Specification
Fig 1.1.1 presents three different random walks that have been generated
in order to illustrate winning and losing bets. All the upbets start with an
underlying price of $100, have twenty-five days to expiry and a strike
price of $101.
1. RandomWalk 1 (RW1) flirts with the $101 level after five days, retreats
back to the $100 level, rises and passes through the $101 strike after
eighteen days and then drifts to settle at a price around $100. The
buyer of the upbet loses.
2. RW2 travels up to the $101 level after the eighth day where it moves
sideways until, with nine days to expiry, the underlying resumes its
upwards momentum. The underlying continues to rise and is around
the $102.75 level at expiry, well above the strike of $101, so is
consequently a winning bet with the seller ending the loser.
3. RW3 drifts sideways from day one and never looks like reaching the
strike. RW3 is a losing bet for the backer with the underlying settling
around $100.50 at expiry.
any time leading up to the expiry of the bet.
Examples of upbets are:
1. Will the price of the CBOT US Sep 10 year Notes future be above
$114 at 1600hrs on the last trading day of August?
2. Will the Dow Jones Index be above 12,000 at 1600hrs on the last
trading day of the year?
3. Will the LIFFE Euribor Dec/Sep spread be above 10 ticks at settlement
on the last day of November?
4. Will a non-farm payroll number be above +150,000?
Examples 1, 2 & 3 enable the bettor to make a minute by minute
assessment of the probability of the bet winning. Example 4 is a number
(supposedly) cloaked in secrecy until the number is announced at 13.30
hrs on a Friday.
In all the above examples the bet always has a chance of winning or
losing right up to the expiry of the bet although the probability may be
less than 1% or greater than 99%. The Notes could be trading two full
points below the strike the day before expiry but it is possible, although
highly improbable, for them to rise enough during the final day to settle
above the strike. Conversely the Notes could be trading a full two points
higher than the strike the day prior to expiry and still lose although the
probability of losing may be considered negligible.
Ultimately the upbet is not concluded until the bet has officially expired
and until then no winners or losers can be determined.
Downbets too can only win or lose at expiry. Although in many circumstances
the downbet is simply the reverse of the upbet, the downbet has
been treated with the same methodology as the upbet in order that other
bets, e.g. the eachwaybet, can be analysed within a uniform structure.
Also a separate treatment of downbets will provide a firmer base on
which to analyse the sensitivity of downbets.
1.1 Upbet Specification
Fig 1.1.1 presents three different random walks that have been generated
in order to illustrate winning and losing bets. All the upbets start with an
underlying price of $100, have twenty-five days to expiry and a strike
price of $101.
1. RandomWalk 1 (RW1) flirts with the $101 level after five days, retreats
back to the $100 level, rises and passes through the $101 strike after
eighteen days and then drifts to settle at a price around $100. The
buyer of the upbet loses.
2. RW2 travels up to the $101 level after the eighth day where it moves
sideways until, with nine days to expiry, the underlying resumes its
upwards momentum. The underlying continues to rise and is around
the $102.75 level at expiry, well above the strike of $101, so is
consequently a winning bet with the seller ending the loser.
3. RW3 drifts sideways from day one and never looks like reaching the
strike. RW3 is a losing bet for the backer with the underlying settling
around $100.50 at expiry.
Clearing and Settlement
A major constraint on starting and operating a derivatives exchange is the
necessary cost of engaging a clearing house. The Eurex and the Chicago
exchanges operate their own clearing houses, which require huge sums
of cash in order to operate their exchanges with financial integrity. Clearly
binary options alleviate this cost since the risk management is a more
exact methodology. This is likely to lead to a proliferation of binary/betting
exchanges globally.
Regulation
Regulatory authorities are placed between a rock and a hard place over
the regulation of binary options exchanges. The nature of the risk involved
means that these exchanges will distribute binary options via the internet
in much the same way as eBay offers everything but bets via the internet.
If regulation in any one jurisdiction becomes overburdensome then the
exchange will up sticks and go offshore.
It is clear that the ability to distribute the product cheaply over the internet
will be a major advantage to the exchange and the binary option
user/trader. The combination of a homogenous, limited-risk instrument
with zero credit facility ensures zero account defaults.
Leverage
The nature of binaries changes sharply as expiry nears. Shares and futures
have linear P&L profiles that lie at a 45º angle to the horizontal axis.
Conventional traded options have profiles that approach an angle of 45º.
A 45º angle means that if the share/future/option goes up by 1¢, then the
owner makes 1¢. A binary approaching expiry has a P&L profile that can
exceed 45º (indeed it will approach the vertical for an at-the-money
option) meaning that a 1¢ rise in the underlying share could translate into
a multiple (say 5¢) increase in value of the option. Clearly this feature is
Introduction
likely to attract the player who is looking for short-term gearing, for it is
safe to state that a binary option can provide greater gearing than any
other financial instrument in the marketplace.
Dexterity
At present financial fixed odds betting suffers from a paucity of available
strategies compared with conventional options. Spreadbetting companies’
binary offerings are usually restricted to the regular upbets, downbets and
rangebets in regular or one-touch/no-touch mode; but this is very plain
fare in comparison to what’s available on the high table of the OTC
market. Knock-Out bets, Knock-In bets, Onions and bets on two separate
assets are all available and over time are likely to, with one handle or
another, enter the trader’s vocabulary. Of course the trading community
will require educating in order that they may use these instruments
proficiently, but this is standard in all new markets. Once the trading
community has a clearer understanding of binaries, there will no doubt
be increased pressure on mainstream exchanges to issue binaries on their
current products – with the CBOT’s introduction of a binary on the Fed
Funds rate the process has already started.
Product Sets
When considering binary options, the usual product revolves around
foreign exchange bets and, more recently, bets on economic data and the
aforementioned Fed Funds rate. Earlier in this introduction sports bets
were proffered as alternative forms of binary options, but the product set
need not end there. Bets on political events are prevalent particularly at
the time of elections. Furthermore, the media is increasingly becoming a
sector where wagers may be placed. Reality TV events are now widely
accepted as a betting medium, but there is no reason why this should not
be extended to, for instance, the film industry. Binary options on weekend
cinema box office takings would no doubt be a welcome hedge for film
producers and nervous actors.
Binary Options
Summary
Although this book steers clear of complex mathematics, it systematically
analyses the anatomy of fixed odds bets. Hopefully this book will:
allow the part-time punter to learn enough to eradicate amateurish
mistakes;
open up the financial and commodity fixed odds market to the sports
betting enthusiast; and
provide enough material and new concepts for the professional binary
options trader to, at the very least, look at combining different forms
of financial instruments with binaries in order to maximise potential
profits and minimise unnecessary losses.
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