deviation
$2,109
Skewness
1.0
Excess kurtosis
−0.9
Median
−
$3,032
10th percentile
−
$3,032
Minimum
−
$3,032
Percent profitable
26%
Using a butterfly tames the extremely bad results of a straddle but
this comes at the expense of incurring the maximum possible loss
26% of the time.
Because a condor has less vega than a butterfly, we need to trade
more of them to get the same volatility exposure. This means that
the worst case is slightly worse than with the butterfly. And the
condor realizes this worst case 30% of the time. Also, the
maximum possible profit is lower than the butterfly's.
Now we look at the results where we sell implied volatility at 30%
and subsequent realized volatility is 70%. The returns of the
butterfly and condor in this scenario are shown in Figures 6.12
and 6.13, and statistics are summarized in Tables 6.11 and 6.12: We can see that the condor has a higher winning percentage, but
its wins are capped at the premium of $1,998. This occurred 15%
of the time. By having a higher initial premium the butterfly can
have larger wins. It wins 12% of the time more than $1,998. The
average of these “condor-beating” wins is $3,342.
In terms of risk and reward, the butterfly is to the condor what the
straddle is to the strangle: lower winning percentage but higher
upside and lower downside.
Aside: Broken Wing Butterflies and Condors
120
A broken wing fly or condor is one with only one long strike. For example, a broken wing (iron) butterfly might be long an out-of-the-money put and short a straddle. This has the same payoff as a
one- by two-call spread, and we will look at the risks of these in
more detail in Chapter Eight. But for now, I want to emphasize
that a common reason for implementing these strategies is wrong.
It is generally accepted that stock market down moves are more
severe than up moves. So, many traders are only concerned with
hedging short exposure on the downside. Be careful with this.
There isn't much of a difference between daily up moves and down
moves. For example, consider daily S&P 500 returns from 1990
through to the end of 2018. Summary statistics of absolute up and
down returns are shown in Table 6.13. None of the differences are significant at the 5% level.
The difference between the “speed” of breaks and rallies is real,
but it is due to correlation effects and is also far smaller than many
traders think. It is well known that before 1987, the index option
markets had almost no implied volatility skew. After the crash the
markets priced in the crash risk. But markets also massively
overcompensated. This is both why put options are usually the
most overpriced and why, if a trader wants to hedge extreme
moves, it is generally best to hedge both extreme drops and
extreme rallies.
TABLE 6.13 Summary Statistics of S&P 500 Returns from
1990 to 2018
Statistic
Positive Daily
Negative (Absolute) Daily
Returns
Returns
Average
0.0073
0.0076
Median
0.0051
0.0049
90th
percentile
0.0156
0.0177
99th
percentile
0.0387
0.0384
Maximum
0.1158
0.0903
Calendar Spread
121
The calendar spread is used to speculate on relative implied
volatility levels. Specifically, we are trying to capture the greater
variance premium in the shorter-dated options. For example, we
buy the 60-day ATM straddle at an implied volatility of 30% and
sell the 30-day ATM straddle at an implied volatility of 40%. We
hold the position until the front-month expires, and over this time
realized volatility is 30%. Because the edge in this trade comes
from the overpriced short-term implied volatility, we also simulate
a short 1-month straddle position, which is the most direct way to
capture this variance premium.
Summary statistics of the PL distributions from simulations of
10,000 paths are shown in Table 6.14.
The calendar spread has lower dispersion around the average P/L
(which is the same for both positions because it is entirely due to
the mispricing of the 1-month implied volatility). At an intuitive
level, this is because, at the expiration of the front-month options,
the spread's payoff diagram is very similar to the value of a
butterfly. This is shown in Figure 6.14, which shows the P/L of the
spread.
TABLE 6.14 Summary Statistics of the PL Distribution for the Straddle Spread and the Short Front-Month
Straddle
Statistic
Short
Straddle
Straddle
Spread
Average
$261
$265
Standard
deviation
$597
$210
Skewness
−1.4
0.0
Excess kurtosis
2.7
−1.4
Median
$411
$260
90th percentile
$873
$552
Maximum
$899
$578
10th percentile
−$574
−$18
Minimum
−$3,753
−$66
Percent profitable
72%
87%
122
FIGURE 6.14 The P/L of the straddle spread at expiry of the front-month options.
The cost of this risk reduction is that the spread is long vega. If
implied volatility drops, the spread will underperform the short
straddle. This risk can also be at odds with our volatility forecast.
We are projecting that the front-month volatility is too high. This
will usually also imply that the second-month volatility is also
overpriced (although not by as much). So, the straddle spread is a
long vega position that we put on when we are hoping volatility is
overpriced.
The spread still acts as a diversifier, but we also need to be correct
on our view of overall, rather than relative volatility. We
demonstrate this problem by simulating a situation in which the
realized volatility is 20%. Summary statistics of the spread and the
short straddle are shown in Table 6.15.
We can reduce this effect by weighting the spread components, so
that the overall position is flat vega. This will also reduce the
variance reducing properties of the spread.
Another weighting scheme is to calculate a volatility beta that
relates the change in the front-month volatility to that of the
second-month volatility.
The market regime in which the benefits of the calendar spread
are most obvious is when there is a large variance premium a
nd a
low implied volatility.
123
As always, the trader's decision will depend on exactly what risk he is most concerned about.
TABLE 6.15 Summary Statistics of the PL Distribution for the Straddle Spread and the Short Front-Month
Straddle (Front-month implied volatility was 40% and
second-month implied volatility was 30%. Realized
volatility was 20%.)
Statistic
Short
Straddle
Straddle
Spread
Average
$522
$223
Standard
deviation
$375
$147
Skewness
−1.2
−1.0
Excess kurtosis
1.0
0.4
Median
$630
$266
90th percentile
$877
$375
Maximum
$900
$380
10th percentile
−$16
$10
Minimum
−$2,141
−$370
Percent profitable
89%
91%
Including Implied Volatility Skew
The preceding analysis assumed that all strikes had the same
implied volatility. Of course, this is generally not the case. Usually,
the OTM puts will trade at a volatility premium to the ATM
options. Because ATM option-implied volatility is the most
predictive of future realized volatility (Bondarenko, 2003; Poon
and Granger, 2003, and further references from Chapter Four), by
selling OTM puts we can collect a volatility premium. This makes
selling strangles relatively more attractive than would be the case
in a world where all strikes had the same implied volatility.
The shape of the implied volatility curve is fairly persistent. A
given delta option's implied volatility will be a constant times the
ATM implied volatility. For example, in the SPX the 10 delta put
will have an implied volatility about 1.45 times the ATM volatility
(Sinclair, 2013). Similarly, the 25 delta call will usually have a
volatility of about 0.88 of the ATM volatility. Using this rule of
124
thumb we can surmise that the 70 strike put (which has a 9 delta)
will trade at an implied volatility of 43.5% if the ATM volatility is
30%. Similarly the 130 call (23 delta) will be expected to have an
implied volatility of 0.264.
FIGURE 6.15 The profit distribution of a strangle with an implied volatility skew and a fair value for realized volatility.
Again assume that future realized volatilities are 30%. If we run
the same simulations as previously but price the strangle with
more realistic implied volatilities of 43.5% for the put and 26.4%
for the call, we get the results shown in Figure 6.15 and Table 6.16.
These numbers are better than the strangle when all strikes are
priced at a 30% volatility (the results for both situations are shown
in Table 6.17 for ease of comparison).
TABLE 6.16 Summary Statistics for the Returns of a Strangle with an Implied Volatility Skew and a Fair Value
for Realized Volatility
Average
$278
Standard
deviation
$2,102
Skewness
−3.8
Excess kurtosis
18.0
Median
$1,123
10th percentile
−$1,765
125
Minimum
−
$37,372
Percent profitable
79%
TABLE 6.17 The Results for Both the Flat Skew Condor and the Skewed Case
Statistic Strangle with Constant Strangle with Implied
Strike Volatility
Volatility Skew
Average
−$6.12
$278
Standard
deviation
$2,040
$2,102
Skewnes
s
−4.8
−3.8
Excess
kurtosis
24.2
18.0
Median
$841
$1,123
10th
percentile
−$1,994
−$1,765
Minimum
−$24,683
−$37,372
Percent
profitable
78%
79%
Strike Choice
Although straddles are almost always struck at the ATM forward
price, the other combinations require us to choose strikes.
To do this, start with the short strikes. For straddles and
butterflies this will be the ATM strike but for strangles and
condors we want to choose strikes that maximize the volatility
premium due to the implied volatility curve.
The part of the implied volatility curve that is most predictive of
the future realized volatility is the ATM, so when we are looking
for a volatility edge we should sell a strike with the highest
volatility over this value. In pure volatility terms that will almost
always be the farthest out-of-the-money put strike. Consider the
SPY puts in Table 6.18.
The highest implied volatility is in the 210 strike. So let's imagine
we sell this option and also sell the 360 call (at an implied
126
volatility of 11.0%) to create a delta-neutral strangle. We assume that realized volatility was what the ATM predicted (14.2%). We
use Monte Carlo to simulate 10,000 instances of this strategy and
get the results shown in Table 6.19. We sell $1,000 of vega.
However, this combination has very little vega per option.
Contrast this with selling the put strike that has the greatest dollar
premium over what the option would be worth if it were priced at
the ATM implied volatility. Table 6.20 shows this choice.
TABLE 6.18 The Put Prices of the SPY June 2020
Expiration on July 30, 2019 (Down to the 5 Delta Strike)
(SPY was 300.48.)
Strik Market
Implied
e
Price
Volatility
210
1.68
0.253
215
1.95
0.248
220
2.25
0.243
225
2.58
0.237
230
2.95
0.232
235
3.35
0.226
240
3.82
0.221
245
4.33
0.215
250
4.91
0.209
255
5.55
0.203
260
6.28
0.197
265
7.07
0.192
270
7.96
0.186
275
8.95
0.179
280
10.03
0.173
285
11.26
0.167
290
12.57
0.160
295
14.10
0.153
300
15.78
0.
146
TABLE 6.19 The Summary Statistics from Selling $1000
Vega of the 210/360 SPY Strangle
127
Statistic
Maximum Implied Volatility
Strangle
Average
$6,600
Standard
deviation
$105,400
Skewness
−5.5
Excess kurtosis
32.1
Median
$29,400
10th percentile
$28,670
Minimum
−$1,463,400
Percent profitable
92%
TABLE 6.20 The Dollar Premium of Options Over Their Being Priced at the ATM Volatility (14.0% in this
Instance)
Strik Market Priced with ATM
Premium Due to
e
Price
Volatility
Volatility Skew
210
1.68
0.03
1.65
215
1.95
0.06
1.89
220
2.25
0.10
2.15
225
2.58
0.17
2.41
230
2.95
0.28
2.67
235
3.35
0.43
2.92
240
3.82
0.64
3.18
245
4.33
0.94
3.39
250
4.91
1.34
3.57
255
5.55
1.86
3.69
260
6.28
2.52
3.76
265
7.07
3.36
3.71
270
7.96
4.37
3.59
275
8.95
5.60
3.35
280
10.03
7.05
2.98
285
11.26
8.73
2.53
290
12.57
10.66
1.91
295
14.10
12.84
1.26
128
Strik Market Priced with ATM
Premium Due to
e
Price
Volatility
Volatility Skew
300
15.78
15.26
0.52
The greatest dollar premium is in the 260 strike. The market has
them worth 6.28, but if they were priced at the ATM volatility,
they would only be worth 2.52. Assume we sell 1.2 of the 260/335
strangles so we have the same vega exposure as in the previous
simulation. Results are summarized in Table 6.21.
The statistics that describe typical results (average, median,
percent profitable, and 10th percentile) all favor selling the highest
volatility premium. That is to be expected. The edge in selling
options comes from selling expensive implied volatility. However,
Positional Option Trading (Wiley Trading) Page 14
