straddle or a strangle. It is easy to work out the expected profit of
an option position. It is just the position value at the volatility we
sold at, minus the position value evaluated at the realized
volatility. Or in terms of vega:
(6.1)
But this doesn't tell the whole story. Unhedged option positions
are not exposed to path dependency in the underlying, but their
profitability is still tied to the returns of the underlying. Returns
are not relevant when pricing options, but the final option return
is enormously dependent on returns. And even a truly driftless
process will have periods when it appears to trend.
Further, when selling options our upside is capped by the collected
premium, but our downside is infinite. This means that all short
option positions will have significant negative skew in their P/L.
Because of all this I evaluated the strategies by running
simulations. I sold a 1-year ATM straddle on a $100 stock at 30%
implied volatility and then simulated 10,000 paths of the stock
where the realized volatility was also 30%. The drift and interest
rates were both zero.
There are two ways to quantify these results. We could either track
returns on the margin required (for the short straddle, strategy-
based margin would be $2,000) or in terms of dollars. Most
professional option traders think in terms of dollars so that is
what we will do, but the overall conclusions would be very similar
if we considered returns.
106
FIGURE 6.1 The profit distribution of the short straddle.
The distribution of profits (in dollars) is shown in Figure 6.1.
The dollar value of the straddle was $2,385 (assuming the
convention of the options being on 100 shares). So, the maximum
profit is $2,385. As the implied and subsequent realized
volatilities were the same, the average profit should have been
zero. The simulation confirms both of these figures. Statistics are
summarized in Table 6.1. (The minimum is included as an
indicative number and should not be relied on as the sampling
error is huge for extrema. The 10th percentile is a better measure
of downside risk.)
Next we run the same simulation for a short 70/130 strangle
(corresponding to shorting the 9-delta put and the 23-delta call).
To get the same vega exposure as we had with the short straddle,
we need to sell 1.68 strangles. This position has an initial value of
$841. The distribution of profits (in dollars) is shown in Figure
6.2.
TABLE 6.1 Summary Statistics for the Returns of a Fairly Priced Short Straddle
Average
$8.04
Standard
deviation
$1,882
Skewness
−1.83
Excess kurtosis
6.41
107
Median
$384
10th percentile
−$2,274
Minimum
−
$15,321
Percent profitable
57%
FIGURE 6.2 The profits of the short strangle.
The dollar value of the position was $841. So, the maximum profit
is $841. Again, the average return should have been zero. As
before, the simulation confirms both of these figures. But the
results are also much more negatively skewed than for the
straddle. And even if you don't look at the numbers, a quick glance
at the histograms tells us a lot. The strangle hits its maximum
profit about 60% of the time, but, because the total premium we
took in is lower than with the straddle, the losses can be much
greater. Statistics are summarized in Table 6.2.
Because there are many situations in which selling volatility has a
positive expected value, when we enter short volatility positions,
we should focus on controlling our risk. If we can keep plugging
along, the profits should eventually take care of themselves. So
instead of just comparing the straddle and strangle in the case
where we had no edge, we now look at the results where we are
completely wrong. Specifically, realized volatility was 70%. The
straddle returns for this scenario are shown in Figure 6.3, and statistics are summarized in Table 6.3.
108
TABLE 6.2 Summary Statistics for the Returns of a Fairly-Priced Short Strangle
Average
−$6.12
Standard
deviation
$2,140
Skewness
−4.8
Excess kurtosis
24.2
Median
$841
10th percentile
−$1,994
Minimum
−
$27,683
Percent profitable
78%
FIGURE 6.3 The returns of the short straddle when our forecast was poor.
Not a good set of results. But we were very wrong in our volatility
forecast, so we can't really expect great results.
But now look at the returns of the strangle in Figure 6.4 and Table
6.4.
Again, we can see that the extreme results (minimum and 10th
percentile) were worse than the corresponding straddle returns.
However, generally speaking, the poor volatility forecast had
similar effects on both the straddle and strangle results.
109
TABLE 6.3 Summary Statistics for the Returns of a Mispriced Short Straddle
Average
−$3,071
Standard
deviation
$5,425
Skewness
−4.31
Excess kurtosis
29.01
Median
−$2,143
10th percentile
−$7,069
Minimum
−
$94,084
Percent profitable
25%
FIGURE 6.4 The returns of the short strangle when our forecast was poor.
TABLE 6.4 Summary Statistics for the Returns of a Poorly Priced Short Strangle
Average
−$3,230
Standard
deviation
$9,190
Skewness
−4.2
Excess kurtosis
21.7
Median
−$1,650
110
10th percentile
−$10,085
Minimum
−
$197,526
Percent profitable
36%
Next, we look at the case where our volatility forecast was neutral
(i.e., 30%) but there was also an unanticipated 20% drift in the
underlying. The returns of the two option structures in this
scenario are summarized in Tables 6.5 and 6.6.
Again, the strangle is profitable more often than the straddle, but
it can go more badly wrong. The difference between the average
profits is largely due to the initial delta of the positions. The
straddle had a delta of −12, and the strangle delta was −15. When
scaled by the size of each position, this leads to an expected PL
difference of $264.
Differences between straddle and strangle results are not
dependent on the actual process that generates the underlying
returns. Skew
ed and fat-tailed distributions will create more
dramatic results, but the straddle will still have fewer disasters
than the strangle.
TABLE 6.5 Summary Statistics for the Returns of a Short Straddle When Our Directional Forecast Was Poor
Average
−$800
Standard
deviation
$2,914
Skewness
−1.62
Excess kurtosis
6.14
Median
$12
10th percentile
−$4,712
Minimum
−
$28,143
Percent profitable
50%
TABLE 6.6 Summary Statistics for the Returns of a Short Strangle When Our Directional Forecast Was Poor
Average
−$1106
Standard
$3,925
111
deviation
Skewness
−4.72
Excess kurtosis
25.74
Median
$841
10th percentile
−$6,127
Minimum
−
$38,830
Percent profitable
65%
Instead of showing this by using another postulated distribution,
we will look at the returns of the S&P 500 from January 1990 to
December 2018. Over this entire period, volatility has been 17.6%,
skewness of daily returns has been −0.08, and the excess kurtosis
has been 8.9. We sample 252 returns from this population to find
the returns of a 1-year straddle and strangle on an imaginary $100
stock. Options are priced at a 17.6% volatility (this approach
ignores autocorrelation in the returns so it isn't exactly what
would have happened in the real index). To give each strategy the
same vega we sell one straddle and 1.68 strangles. The simulation
was run 1,000 times. Results are shown in Figures 6.5 and 6.6.
FIGURE 6.5 The returns of the short 100 straddle when the underlying has the S&P 500 return distribution.
112
FIGURE 6.6 The returns of the short 85/134 strangle (10-delta call and put) when the underlying has the S&P 500 return
distribution.
Again, the straddle has less downside. This is summarized in Table
6.7.
In practice selling a strangle will often collect an implied skewness
premium. Because the implied skew overstates the actual
skewness of returns, this effect will raise the average profit of the
strangle relative to the straddle, but it won't affect the relative risk
conclusions.
The strangle's win percentage is a very powerful piece of feedback
that can trick us into doing trades like this even when they have
negative expectation. The straddle has a better correspondence
between correctness of forecast and profits. Hence you will be far
less likely to fool yourself into thinking you have a volatility edge
than you would with a strangle.
TABLE 6.7 Comparing Results for Straddles and Strangles if the Underlying Has the Same Historical
Returns
Result
Straddl Strangl
e
e
Skewness
−1.32
−3.27
Excess
kurtosis
1.36
9.93
113
Result
Straddl Strangl
e
e
Worst case
−$9,428 −$16,522
Worst decile
−$3,356 −$4,211
This is where many people who sell options “for income” go
wrong. There is no magic in selling strangles, even if they are
struck a long way out of the money. If you don't have an edge in
volatility, you will lose eventually.
The straddle has a payoff that is less sensitive either to extreme
moves or to making a poor forecast. It won't be profitable as often
as the strangle, nor will it practically ever make its theoretical
maximum, but it also won't go as badly wrong as a strangle can.
By choosing to sell a strangle instead of a straddle, a trader is
gaining an increased median return in exchange for greater
extreme risks. It is impossible for someone else to say that a
choice like this is wrong as it depends on individual risk
preferences, but by most risk metrics the straddle might initially
appear to be the riskier position, but it really isn't.
Aside: Delta-Hedged Positions
This book has focused on trading options from the perspective of
speculators and end users. These investors tend not to delta hedge
much or at all. However, it is worth repeating the straddle and
strangle comparison assuming we hedge daily. Both implied and
realized volatilities are 30%, so we again expect an average profit
of zero. The results of 10,000 GBM simulations are shown in
Figures 6.7 and 6.8 and Table 6.8.
It is clear that delta hedging does exactly what it is supposed to do:
reduces risk. Extreme results are far more palatable than for
unhedged positions. We also see that the differences between
straddles and strangles are greatly minimized. The positions are
not exactly the same because the strangle will have a more
concentrated vega profile (shown in Figure 6.9). This means that when things are going very badly (i.e., the underlying has moved a
lot) the straddle risk will start to decrease as we move away from
the option strike. This need not happen for strangles. Also note
that although the positions were scaled to have the same vega at
114
initiation, if the stock rallies, the strangle picks up more vega. This
increases risk during adverse events.
FIGURE 6.7 The returns of the short straddle when hedging daily.
FIGURE 6.8 The returns of the short 70/130 strangle when hedging daily
TABLE 6.8 Comparing Results for Straddles and Strangles When Hedging Daily
Result
Straddl Strangl
e
e
115
Result
Straddl Strangl
e
e
Average
$7
$22
Median
$12
$126
Skewness
−0.2
−2.71
Excess
kurtosis
1.28
13.32
Worst case
−$856 −$2,170
Worst decile
−$242
−$450
FIGURE 6.9 Vega as a function of underlying price for the straddle (solid line) and 70/130 strangle (dashed line).
Hedging has the effect of reducing the variance of returns, but
hedging drastically increases transactions costs and introduces
operational issues associated with position monitoring. Most
people who are not broker-dealers should probably not be
dynamically hedging. For those who are interested, refer to
Sinclair (2013) for the theory and practicalities of actively delta
hedging.
Butterflies and Condors
These are the more conservative versions of straddles and
strangles respectively. In each case, exposure to adverse moves is
capped. Technically, these strategies are constructed from eit
her
116
all calls or all puts but in practice traders use the equivalent iron-
butterfly and iron-condor structures. These are constructed from
out-of-the-money options. Put-call parity means these positions
are exactly the same.
We repeat the straddle/strangle analysis to compare butterflies to
condors. Results are shown in Figures 6.10 through 6.13 and
Tables 6.9 through 6.12. (As before, we weight the positions so they have the same vega as a short straddle.)
FIGURE 6.10 The profit distribution of the fairly priced butterfly (long the 70/130 strangle and short the 100 straddle).
FIGURE 6.11 The profit distribution of the fairly priced condor
(long the 70/130 strangle and short the 80/120 strangle).
117
FIGURE 6.12 The profit distribution of the poorly priced butterfly (long the 70/130 strangle and short the 100 straddle).
FIGURE 6.13 The profit distribution of the poorly priced condor (long the 70/130 strangle and short the 80/120 strangle).
TABLE 6.9 Summary Statistics for the Returns of a Fairly-Priced Butterfly
Average
−$25
Standard
deviation
$2,460
Skewness
0.4
118
Excess kurtosis
−1.4
Median
−$321
10th percentile
−
$2,756
Minimum
−
$2,756
Percent profitable
46%
TABLE 6.10 Summary Statistics for the Returns of a
Fairly Priced Condor
Average
−$9
Standard
deviation
$2,252
Skewness
−0.4
Excess kurtosis
−1.7
Median
$1,524
10th percentile
−
$3,032
Minimum
−
$3,032
Percent profitable
58%
TABLE 6.11 Summary Statistics for the Returns of a Badly Priced Butterfly
Average
−
$1,530
Standard
deviation
$2,109
Skewness
1.5
Excess kurtosis
0.8
Median
−
$2,756
10th percentile
−
$2,756
Minimum
−
$2,756
Percent profitable
20%
119
TABLE 6.12 Summary Statistics for the Returns of a Badly Priced Condor
Average
−$1643
Standard
Positional Option Trading (Wiley Trading) Page 13