−2.03
−2.02
−2.72
Maximum
28.90
26.48
27.28
Minimum
−49.42
−53.79
−46.08
Median
4.22
3.91
4.05
90th
percentile
9.46
10.37
9.10
10th
percentile
−2.60
−3.11
−2.81
The variance premium varies considerably with the market volatility. Table 4.3
shows the variance premium statistics for the five VIX quintiles. Here, as in
Figure 4.6, the premium is expressed in volatility points.
So, broadly speaking, the typical size of the variance premium (measured by
either mean or median) increases with volatility levels. The variability of the
variance premium also increases with volatility. When trading the variance
premiums with options, a mid-level VIX is probably the best environment. Low-
volatility regimes give lower returns as a percentage of margin and require more frequent hedging due to the higher gamma. High-volatility regimes have good
average returns but a bad risk profile.
TABLE 4.3 Summary Statistics for the VIX Sorted by Quintiles
VIX
13.02
15.89
19.42
VIX
<13.0
2
9
2
5
5
Mean
2.61
3.37
4.35
4.19
5.87
Standard
deviation
3.54
3.60
4.58
6.53
9.06
Skewness
−2.36
−1.67
−2.23
−2.21
−2.46
Maximum
8.33
10.17
13.84
14.82
31.21
Minimum
−17.54
−16.84
−27.95
−37.66
−53.34
Median
3.25
4.05
5.25
5.44
7.23
90th percentile
5.16
7.21
8.73
10.45
13.81
10th percentile
−0.81
−0.80
−0.74
−3.15
−2.80
There is also more direct evidence of the premium. Short positions in delta-
neutral equity index option positions (straddles, strangles, butterflies, and
condors) have been profitable, as have short positions in variance swaps. Sharpe ratios vary between 0.4 and 1.0 depending on details of the implementation.
66
The effect has been studied in many countries and time periods (see, for
example, Driessen and Maenhout, 2006; Londono, 2011).
The CBOE publishes two separate short option volatility indices: CNDR and
BFLY. CNDR tracks the performance of a strategy that sells a 1-month SPX iron
condor (short the 20-delta strangle and long the 5-delta strangle). BFLY tracks a short iron butterfly. The hypothetical performance of these strategies is shown
in Figures 4.7 and 4.8.
It is worth discussing why neither of these indices has been particularly
successful since 2008.
The variance premium was virtually the same before the end of 2008 (median
of 4.63 points) and since (median of 4.61). However, the median VIX level
dropped from 18.3 to 16.3, and after the chaos of 2009, the median VIX
dropped further to only 15.4. The lower volatility level gives options higher
gamma, which drastically increases the effects of path dependency and drift.
The second reason is the effect of skew. Although lower ATM volatility usually
means all other options will also trade at lower volatilities, the relative implied volatility of teeny puts tends to increase. So as volatility decreased, the prices of the 5-delta puts increased relative to the premium from selling the options
closer to the ATM. This effect is illustrated by the CBOE SKEW index having an
average value of 116.2 in the earlier period and 125.7 afterwards. These
strategies were paying more for their put hedges. And these teeny puts are
always the most options with the highest variance premium. Unless buying
them for a hedge they are the ultimate sucker bets. Refer to Hodges et al. (2003) for a study of this fact.
FIGURE 4.7 Performance of the CNDR index.
67
FIGURE 4.8 Performance of the BFLY index.
So, although the variance premium is a strong and consistent phenomenon,
attention needs to be paid to the capture strategy. This will be discussed further in Chapter Six.
The Implied Skewness Premium
A large part of the index variance premium is due to the OTM (out-of-the-
money) puts being overpriced. Intuitively this should be the case, because much
of the value of a variance swap is driven by the value of the OTM puts. And it is well known that index put options are overpriced (see, for example,
Bondarenko, 2003).
The implied volatility curve predicts very high negative skewness in realized
returns. Although many pricing models can reproduce such a curve, their
parameters are not consistent with the absence of substantial negative skewness
in stock index returns. To misquote Samuelson, option markets predict nine out
of the past five market corrections. Also, the predictions are reactive, with the implied skew being steepest after crashes.
Kozhan et al. (2011) examined the profitability of selling skew swaps. The skew swap is a model-free skewness contract whose payoff is equal to the difference
between realized skew and implied skew. As with variance swaps, the skew swap
can be replicated from a portfolio of vanilla options. Using this contract enables the researchers to investigate the risk premium implicit in implied skew without having to worry about the misspecification of any specific option pricing model.
They show that for S&P 500 options (from 1996 to 2009) about half of the
excess return from selling out-of-the-money puts is due to the correlation
between returns and volatility: the realized skewness.
The Implied Correlation Premium
Selling index volatility and buying component volatility is a short correlation
position. All the index components dropping is the same as correlations rising.
68
Short correlation is also short implied volatility.
This strategy has Sharpe ratios comparable to selling index variance swaps.
Although the returns are lower, the variance is also considerably reduced
(Driessen et al., 2009). Dispersion trading suffers during market turmoil when the correlations increase, that is, the same periods when selling index variance suffers.
Commodities
The variance premium probably exists in commodities. Prokopczuk and Simen
(2014) used option prices to construct synthetic variance swaps and found significantly negative variance risk premia in nearly all commodity markets.
They examined 21 commodity markets between 1989 and 2011 and found that
18 of the commodities had statistically significant returns to short variance
swaps. Table 4.4 summ
arizes the results for the 60-day swaps (a premium of
10% would mean the 60-day implied variance was 10% above the 60-day
realized variance).
The correlation of variance premia for commodities in the same sector was
positive but small. And the correlation across sectors was also small enough to
make harvesting the premium in commodities a good diversifier. The
correlations are shown in Tables 4.5 and 4.6.
TABLE 4.4 The Size and Significance of the Variance Premium in Commodity Options
Commodit Variance Premium
T-
y
(%)
Score
Crude oil
3.4
6
Heating oil
3
7
Natural gas
10.2
9
Corn
2.3
8
Cotton
−0.6
−11
Beans
0.8
2
Bean meal
0
0
Bean oil
1
4
Sugar
2.6
6
Wheat
0.7
3
Hogs
1.2
3
Cattle
1
11
Copper
2.4
4
Gold
1
4
Silver
0.2
1
Cocoa
3
8
Coffee
1.7
1
Oats
6.2
8
OJ
2.3
3
69
Commodit Variance Premium
T-
y
(%)
Score
Rice
3
8
Lumber
3.5
10
Trading options on commodities requires some fundamental knowledge. An
equity option trader can trade any equity, often without even knowing more
than the ticker symbol. This is largely because the movement of equities is
almost random and the effect of any real fundamental knowledge is small
(indices are even more ignorance driven: the best situation for a statistically
driven trader). But people actually know things about commodities. There are
different crops, pipeline bottlenecks, storage squeezes, and weather effects.
There is no guarantee that a good wheat trader can become a good corn trader.
Different things affect different commodities.
TABLE 4.5 The Correlation of the Variance Premium Within Commodity Sectors
Sector
Correlation of 60-day
VP
Energy
33.4%
Grains
24.2%
Livestoc
k
31.4%
Metals
30.1%
Tropical
s
5.7%
Wood
N/A
TABLE 4.6 The Correlation of the Variance Premium Between Commodity Sectors
Energ Grain Livestoc Metal
S&P
T-
Tropical Woo
y
s
k
s
500
bonds
s
d
Energy
100%
Grains
9.0% 100%
Livestoc
k
13.2% 14.9% 100%
Metals
22.1% 14.5% 6.5%
100%
S&P 500 26.5%
2.0%
16.4%
30.7% 100%
70
Energ Grain Livestoc Metal
S&P
T-
Tropical Woo
y
s
k
s
500
bonds
s
d
T-bonds 20.2% 16.0% 5.3% 23.1% 39.3% 100%
Tropical
s
8.9% 24.7%
11.3%
11.6% 5.9%
0.3% 100%
Wood
7.3%
0.9%
7.8% 3.9% 9.5% 6.5%
11.7%
100%
We can see this in the variance premia measured by Prokopczuk and Simen
(2104). Traders who have fundamental insight might know why the corn
premium is so much greater than that of wheat and why the premium in silver
options is so different from that in gold options.
Bonds
Choi et. al (2017) looked at bond options from 1990 to 2012. They found that 1-
month variance swaps on US 5-year notes, 10-year bonds, and 30-year bonds
are overpriced by about 20% (18.7% for the 5-year, 27.6% for the 10-year, and
21.2% for the 30-year). The Sharpe ratio for a strategy that sells all these swaps is about 2. Selling 1-month ATM straddles was also profitable, although less so.
Most bond uncertainty (and hence most of the variance premium) is clustered
around the release of macroeconomic data. Jones et al. (1998) showed that from 1979 to 1995 about 90% of excess returns to Treasury bonds accrued on days
with either an employment or PPI announcement. Interestingly, Andersson et
al. (2009) showed that German government bonds reacted far more strongly to
US data releases than German ones. German unemployment releases had
almost no effect at all, leading them to conclude that the number was widely
leaked.
The VIX
There is also a variance premium in VIX options. Hogan (2011) used VIX
options to construct synthetic VIX variance swaps and showed that these had a
premium to the realized variance of the VIX futures that was similar in size to
the premium in equity indices. Kaeck (2017) found similar results using data from 2006 to 2014.
Currencies
Lo and Zhang (2005) found direct evidence of a variance premium in OTC
currency options. They found that a strategy of selling straddles was profitable for options on USD versus GBP, YEN, CHF, and the euro for terms between 1
month and 1 year. They also found that the variance premium increased as
options got closer to expiration.
Londono and Zhou (2017) reported significant positive variance premia for
variance swaps on USD/GBP, USD/Yen, and USD/euro at 1-, 3-, and 6-month
durations. However, the variance premia for some currencies, notably NZD and
71
AUD, was large and negative. As the antipodean currencies are generally seen as
“safe havens,” this suggests some linkage between the priced variance risk and
real-world political and macroeconomic risk perceptions.
Equities
The results of selling options on equities are not clear-cut. First, because we
know that implied correlation accounts for a portion of the index variance
premium, we would expect the returns to selling stock volatility to be lower than selling index volatility. This is true. Also, as with commodities, there are specific fundamental factors that affect the variance premia in stock options. This is
examined in more depth in Chapter Five.
TABLE 4.7 The Average Return to a Short 1-Month Variance Swap for Stock Options in Different Industries
Industry
Average
T-
Return
statistic
Utilit
ies
31.25%
10.39
Nondurable consumer
goods
17.71%
7.89
Other
9.83%
3.71
Durable consumer goods
8.56%
3.36
Energy
8.56%
3.55
Manufacturing
8.05%
3.29
Retail
4.68%
1.93
Health care
3.27%
1.39
Telecommunications
−1.40%
−0.48
Technology
−7.66%
−2.97
Di Pietro and Vainberg (2006) show how size and book-to-market firm characteristics are linked to the expensiveness of equity options. They find that options on small stocks are more expensive than options on large stocks and
that options on value stocks are more expensive than options on growth stocks.
This was partially corroborated by Vilkov (2008). He did not find a robust size
effect but found that value stocks had more of a variance premium than growth
stocks. He also found that illiquid stock had high variance premia (this won't be practically useful because these stocks will also have illiquid options). He also looked at the variance premia of various industry sectors (as defined by French).
These results are shown in Table 4.7.
Anyone thinking of trading a portfolio of equity options should probably study
the interrelationships among these various factor exposures. I'm not aware of
any published work on this.
Reasons for the Variance Premium
There are many reasons for the existence of the variance premium. This means
it is likely to persist, because it is doubtful that all of these reasons would
72
disappear at the same time, even if they all vary across different market and economic regimes.
Insurance
The most compelling reason for a variance premium is that people are willing to
pay for insurance. The most obvious insurance buyers are those who are long
stocks and buy puts for downside protection. This demand is a major driver of
the implied skew and the resulting skew premium. However, there are also
investors who will buy calls to insure against missing out on large rallies.
Obviously, for every option buyer there is also a seller, but the customer
demand sets the price. Hedgers are prepared to pay for insurance. The sellers
take on the risk and also demand an extra premium. The variance premium is
reflective of a risk premium, but it is also a mispriced risk premium.
Jump Risk
If underlying prices were continuous, options could be perfectly replicated. This would make their existence redundant. However, underlying prices can jump
Positional Option Trading (Wiley Trading) Page 8