Profiting from Black Swan Trades:
Evidence from S&P 500 puts.
Michael J. Naylor*, Jianguo Chen and Udomsak Wongchoti
School of Economics and Finance
Massey University
Private Bag 11 222
Palmerston North
New Zealand
* Corresponding author: M.J.Naylor@massey.ac.nz. T: +64-6-3599099 x 7720
Acknowledgements
Version 1.5: 15 June 2011
Profiting from Black Swan Trades:
Evidence from S&P 500 puts.
_________________________________________________________________________
Abstract
The seemingly increase in large security market swings due to unexpected extreme
(Black Swan) events and the associated increase in volatility raises the question as to
whether or not profits can be made from exploiting these tendencies. A common swan
trading hypothesis is that abnormal profits can be made by continually buying far-out-ofthe money puts in the expectation a periodic crisis will more than cover any short-term
losses. This paper examines the profitability of this hypothesis using a range of investment
strategies involving puts on the S&P 500 index over the 20 year period 1990-2009. We find
that swan strategies can be highly profitable over the long-term; though a high level of
trading expertise and holding capacity is required to capture these profits.
While the Swan hypothesis has been extensively covered by the financial press, and
Swan hedge funds established, this paper is the first academic test of the Swan trading
hypothesis. We also examine what the characteristics of extreme events, in terms of
frequency and size, have to be to make Swan strategies profitable.
The paper also provides evidence as to whether or not far out-of-the-money puts are
correctly priced, the volatility-smile puzzle, by indirectly examining whether market traders
do incorporate market extremes into prices.
_______________________________________________________________________
JEL classification: G12, G13, G14
Key words: Black swans, grey swans, crises, peso problem, rare disasters, rare event risk, option
pricing, volatility smile, investment strategies.
-1-
1.
Introduction
1.1 Swan Species
The seemingly increasing incidence of unexpected extreme events, (“Black Swans”), and the
associated increase in security markets volatility, raises the question as to whether or not profits can
be made from exploiting these tendencies. A common swan trading hypothesis is that abnormal
profits can be made by continually buying far-out-of-the money puts in the expectation a periodic
extreme crisis will deliver abnormal profits (Malliaras & Yan, 2008). This paper examines puts on the
S&P 500 index to ascertain the profitability of this hypothesis, by use of perfect trading as well as a
range of passive trading strategies. We also examine what the characteristics of extreme events in
terms of frequency and size have to be to make passive swan strategies profitable. This is an issue
which has not been discussed in the academic literature, though it has been well covered by the
financial press and swan hedge funds established 1. This paper is the first test of the swan trading
hypothesis. Our question is; what would the risk adjusted return have been if a swan strategy had been
applied to S&P 500 puts over the 20 year period 1990-2009?
Taleb (2007) defines a Black Swan2 as an event which has to meet three criteria; 1) it is an outlier,
lying outside the normal range of expectations because nothing in the past can convincingly point to
its occurrence (rarity); 2) it carries an extreme impact (extremeness); 3) despite being an outlier,
plausible explanations can be made ex-post, giving the appearance of it being explainable and
predicable (retrospective predictability). Black swans are inherently unpredictable and often
unforeseeable, e.g.; the turkey forecasts a rosy future on the day before Thanksgiving. Two other
swan species can be defined. “White Swans” can be defined as uncommon events which are within
the realm of what is foreseeable. They are random events that we know of, have a measureable
probability of occurring and create minimal impact in our lives; normal financial market volatility
which we know how to deal with. “Grey Swans” can be defined as intermediate to these; random
events that we can conceive of, which have a low probability of occurring but have significant
impacts. Grey swans can be seen as events which are outside the range of events defined as 95% of a
fat-tail distribution, but not so uncommon as to be unforeseeable3. They will have significant impact,
1
This coverage includes examples of successful Swan traders; Tiger's Julian Robertson roars again, CNN
Money.com, January 29 2008; In Beverly Hills, A Meltdown Mogul Is Living Large, Wall Street Journal,
January 15, 2008; Trader Made Billions on Subprime, Wall Street Journal, January 15, 2008, as well as a
popular book (Zuckerman, 2009).
2
The idea that all swans must be white was used for centuries in Europe as a metaphor of something that had to
be true, based on countless sightings of white swans. Europe was accordingly shocked at the discovery of
black swans in Australia in 1679. “For all x, if x is a swan, then x is white?”is used as the basic falsifiability
argument in Philosophy; no number of white swans can prove all swans are white.
3
An alternative definition would be that white swans are “known knowns”, grey swans are “known unknowns”
and black swans are “unknown unknowns”.
-1-
and we can somewhat take them into account but cannot predict accurately their occurrence or
quantitatively estimate their impacts. We can, however, create a ball-park estimate of their impact.
It has been argued that Oct/19/1987, Black Monday, was a Black Swan; prior to that the Dow had
only fallen twice in its entire history more than 10% in one day4. Yet on Oct 19th the Dow fell by
22.6% or 21 standard deviations5. The Asian Crisis of 1997/8 and the Sub-Prime Crisis of 2008/9
have been similarly labeled as Black Swan – high impact and predictable only in hindsight. It was
certainly unexpected, prior to the events, that problems with Thai banks, or with a small number of
sub-primes, would have such huge impacts, yet in both cases there has been an avalanche of research
showing why, ex-post, both were inevitable6. These crises illustrate the risks inherent in models which
ignore the possibility of unlikely extreme events.
Taleb (2007) argues that Black Swans are the basic generator of paradigm shift in finance due to
their high impact. However, they are very difficult to deal with because they are highly improbable
events and thus very unlikely to re-occur in the same form; the next financial crisis will always catch
markets by surprise because it will be generated by an equally unexpected event. Taleb’s basic
hypothesis is that extreme events are not priced into far-out-of-the-money put options and therefore
profitable long put strategies exist.
1.2
Crash-risk and Volatility Smiles
The academic issue with Taleb’s hypothesis is that theory and research argues that long put
strategies, being risk reducing, should earn expected returns below that of the underlying asset.
Bondarenko (2003), for example, finds that average at-the-money put returns on the S&P500 are 40% per month and deep out-of-the-money put returns are -95% per month. Driessen and Maenhout
(2004) and Santa-Clara and Saretto (2007) find that short put options provide large certainty
equivalent gains. These results seem to contradict any swan strategy, as these will only earn an
abnormal risk-adjusted profit if far out-of-the-money puts are under-priced.
Asset pricing theory argues that option risk is composed of two parts; leverage and curvature. The
existence of curvature in the implied volatility (strike price-time to maturity) surface (the volatility
smile) is seen as evidence for the need for more complex generating functions than geometric
Brownian motion. This curvature also tends to be more pronounced out-of-the-money than it is for inthe-money (the volatility skew). One reason for this is that realized volatility is higher for down-shifts
in the underlying than for up-shifts. This increases the gamma of lower strike price options more than
4
This was on Oct/28/1929 (12.8%) and Oct 29/1929 (11.7%).
The chance of this happening in a Gaussian framework is one in less than 1050. This is the same as a US man
growing to 9’8” without a growth hormone disorder.
6
Note that some analysts, including Taleb in some interviews, have argued that the sub-prime crisis was a grey
swan; uncommon but predictable. A small number of traders, like Paulson, did predict it.
5
-2-
higher strike price options, leading to a need for more frequent re-hedging. However, whilst the smile
and the skew imply far out-of-the-money puts are over-priced against Black-Scholes-Merton
predictions, they do not imply over-pricing against more realistic models. For example, a common
finding in empirical work is that index options embed large risk premia for jump/ volatility risk 7, e.g.;
Coval and Shumway (2001) show that S&P 500 puts, at levels close to the money, do earn negative
expected returns versus the index, this increases with out-of-moneyness, and that higher moments are
priced. Research also shows that a certain amount of crash-risk is priced into option markets, e.g.;
Jackwerth and Rubinstein (1996) argue that crash-risk is priced into S&P 500 options and show that
the market-assigned probability of a three-to-four standard deviation crash increased 10-fold
following the 1987 crash.
It can, however, be argued that adjustments required to align model-based prices to actual
distributions are complex8 and, as there is no agreement on the correct adjustments, models are
adjusted on an ad-hoc basis. This means that not all crisis events may be priced. Swan strategies
would argue that markets are not pricing in the chance of unknowable extreme events occurring;
primarily because it is very hard to model these events (the missing path problem). Jurek (2008), for
example, found strong evidence that far-out-of-the-money foreign exchange puts do not price in the
possibility of black swans. Note that in theory it is ambiguous as to whether swan strategies using farout-of-the-money puts will generate more profit than swan strategies using closer out-of-the-money
puts, as this is a empirical question based on the relative curvature of the volatility surface priced to
the market versus that which incorporates extreme events. This is illustrated in figure 4.
Our research question can be put another way; does the volatility surface of out-of-the-money puts
accurately reflect the chance of unknowable extreme events occurring?
1.3 Fractals and Models
It is well established that financial markets do not follow a Gaussian distribution, even with a wide
tail. Mandelbrot (1963), Fama (1965) and Aparicio and Estrada (2001) provide conclusive evidence
that a range of financial markets are characterized instead by a stable Paretian distribution with a
degree less than 2 and data independence. Mandelbrot and Hudson (2005) show large swings in
particular follow a power-law and are far more likely to be clustered than a Gaussian distribution
would predict. Fama (1965) argues that the above characteristics imply that in the long run stock
prices are dominated by the large changes which occur during the sporadic extreme periods. An
7
Ait-Sahalia (2002) and Anderson et al (2002) show the underlying index value is subject to stochastic
volatility, Buraschi & Jackwerth (2001) show this is priced, Coval & Shumway (2001) show the presence of a
negative volatility risk premium and Bates (2008) shows a positive jump risk premium.
8
Close OTM puts are typically corrected by reducing the price, and far OTM are corrected by increasing the
price, with this correction reducing as OTM-ness increases.
-3-
implication of a stable Paretian infinite variance process is that sample variance and standard
deviation are not meaningful measures of variability for option strategies.
Black swans provide an inherent challenge to builders of financial models as models can only
include the most important movements, not the extremes. It can also be argued that Black Swans
cannot be included as they are unforeseeable events which have never happened and they are not just
uncertain, but are likely to be inherently unknowable. Unknowable and improbable events are
impossible to encapsulate in models individually, as there are a large number of improbable events
yet to occur, and as these have not yet occurred they do not appear in data sets so are not quantifiable.
Even those Black Swans which have occurred cannot be included as they are outliers with a different
generating system, and are very unlikely to re-occur.
It can be argued, conversely, that models do not set prices, traders do, and traders use the models
only as guides; financial markets also incorporate traders’ intuitive feelings for where reality differ
from the models - if markets did not reflect reality there would be scope for profit. In this sense any
gaps between model prices and market prices can be regarded as the best guess by traders of the
impact of events which are not reflected in data. Thus whilst traders cannot price black swans into
markets on an individual basis, the possibility of an extreme event occurring from one of the wide
array of improbable events is quantifiable and can be factored into markets as a crash-risk premium.
The only way to test if market traders do incorporate possible extremes into prices is by tests of
trading strategies on actual market data.
1.4 Swans in the S&P 500 Composite Index
Our paper tests the swan hypothesis by implementing passive trading strategies using options on the
S&P 500 index. S&P 500 options are well suited to the task; the underlying index is a suitably broad
indicator, the options are the most heavily traded9 on the market with a wide range of strike prices,
and the underlying exhibits a fractal nature (Mandelbrot & Hudson, 2004). Estrada (2008) shows over
the period 1928-2006, for daily data, the S&P 500 composite index was dominated by the extreme
days – the black swans10. For example the lower three standard deviation was -3.35% and given
20,918 trading days, 0.27% or 28 days were expected to be below that. Estrada found 180 or 6.4 times
too many. The mean of the worst 10, 20 and 100 days was -10.40%, -8.79% and -5.87%. This
represents 9.3, 7.8 and 5.2 standard deviations. The worst day, Black Monday, was 18.14 standard
deviations below the mean. Figures 1 to 3 show the S&P 500 closing for our sample period and the
daily change in the S&P500 index, in both percentage terms, and in standard deviation. These clearly
show the fractional nature of the generating function. Mauboussin (2006) found similar results for the
9
The average daily volume in 2008 was 670,629 contracts, with a peak day of 2,170,870 (10/06/08).
Daily metrics; mean 0.03% (arithmetic), 0.02%; (geometric), standard deviation 1.13%, skewness -0.1,
kurtosis 18.4, maximum 16.6%, and minimum -20.47%. These show a clear departure from normality.
10
-4-
more recent period 01/3/1978-10/31/2005. Bogle (2008) notes the volatile distribution of the S&P 500
is a puzzle as the data on the underlying companies do not exhibit the same degree of volatility.
1.5 Nickel vs Swan Strategies
It is a common perception that index options are mispriced. However traders seem to have always
found it difficult to exploit this perception11, leading to the “option pricing puzzle”. Taleb (2001)
argues one reason for the inability of traders to exploit mispriced options is that financial markets are
dominated by “Taleb Distributions”, which he defines as heavily negatively skewed distributions in
which there is a high probability of a small gain, and a small probability of a very large loss. In these
situations the expected value is (very much) less than zero, but this fact is camouflaged by the
appearance of steady returns and low risk. In the short-term returns seem positive and volatility is
low; until the crisis reveals a far higher level of volatility than was evident in prior data.
Fung and Hsieh (1997) argue the dominant strategy in such markets involves earning the small but
steady income that can be made from following current trends and ignoring the possibility of
suddenly disaster (“blow-up” or “nickel” strategies12). Taleb (2004) argues the blow-up income
profile generated suits fund managers as it consists of long periods of steady income until large losses
are generated by rare crises13. While investors ultimately lose significant sums of money fund
managers enrich themselves in the short-term. The alternative Swan strategy is unviable, as while it
has a higher expected value, it is an unsustainable “bleed” strategy, involving unsustainably long
periods of steady losses before the sudden large gain. Malliaris & Yan (2008) argue reputational
reasons compounds this, as bleed fund managers cannot survive short-term performance comparison
to their temporarily successful blow-up peers. Taleb (2007) argues another reason is that traders tend
to overestimate what they know and underestimate the uncertainty that is derived from those things
they don't know; they ignore the possibility of Black Swans as these have not been experienced.
Given that these black swans are largely unpredictable and represent less than 0.1% of trading days
Estrada (2009) concludes that successful market timing is near impossible, and the key to good
investing is to hold assets over time, while trying to minimize negative black swans and maximize
exposure to positive black swans. The problem with this suggestion is that while it tries to gain from
the slow steady nickels it is still exposed to the unforeseen bust. Any extra gain would only come
from success at picking stocks exposed or not exposed to black swans; yet the unknowable nature of
future black swans ensure that this is a high risk activity. Can we do better by actively leveraging off
the extreme days through a swans trading strategy? Are the profits generated during Black Swans big
11
Broadie, et al (2009) argue this is because S&P 500 option prices are not inconsistent with Heston SV models
if jump risk variables are incorporated.
12
From “picking up nickels in front of a steamroller”.
13
Thus they “blow up”.
-5-
enough to compensate for the extreme return / risk profile? This is an unexplored research question,
despite the existence of swan events being widely debated on Wall St.
2.
Our Strategy
A swan strategy of buying far-out-of-the-money puts should generate a highly asymmetrical
distribution with a high probability of regular small losses, and a small probability of episodic very
large gains. Thus, while the expected long-run rate of return might be high, actual profit would be
highly variable, with profit coming in large amounts, at inherently uncertain times in the future when,
for unknowable reasons, the market’s collapse. Certainly a swan based trading strategy involves a lot
more uncertainty around the expected return profile than other strategies, as highly profitable days
could be expected to occur less than 0.3% of the time – once a decade. The time to actual
profitability would be unknowable. The traditional monthly, quarterly or even annual performance
review would be irrelevant.
The core problem with swan strategies is that there can be a long gap between those large gains;
meanwhile cash is continually running out and the next profit opportunity is an undetermined time
away. While Taleb himself made gains between 65% - 115% in 2007 via his Universa hedge fund, he
lost money on his initial Empirica Capital effort to exploit black swans when his bets against the
raising stock market in 2004 failed to bear fruit in time. The suggested strategy to avoid cash drain is
to keep 80-90% of assets in cash, and use 10-20% to buy far-out-of-the-money puts on stocks and
stock indexes. The cash is used to support the long periods where the fund makes regular small losses.
An important issue would seem to be – how do we define we define extreme events? It is standard
to define an outlier as more than three standard deviations. However from a trading viewpoint this
question is irrelevant. The important issue from a trading viewpoint is – how does varying the out-ofthe-moneyness affect profitability? The further (less) out-of-the-money we choose the cheaper (more
expensive) puts are so the lower (higher) are the regular losses, but the rarer (more common) are the
periods of profit. We thus used several different definitions of out-of-the-moneyness and assessed
their profitability. Intuitively swan trading is based on the argument that volatility is more complex
than the summation of daily price fluctuation and that long-term implied volatility is underpriced.
Our strategy was passive; we tested S&P 500 index puts rather those than on component stocks, as
we wanted to look at the return from a swan strategy, rather than our retrospective skill in stock
picking. One reason for this is that because we are back-testing on known events it would be invalid
to try to increase alpha because of unconscious bias. Another reason is that black swans are by their
-6-
nature very hard to specify, so we are likely to be wrong about whatever possible future event we
speculate on. It is useful to note that swan hedge funds are unlikely to follow this strategy; they will
instead sort through stocks to empathize those stocks more likely affected by black swan events,
positively or negatively, or choose stocks which have large betas. Since this will increase both their
return and risk levels our results can be regarded as the minimum achievable. The positions have very
low vega but potentially high delta and gamma for large price changes.
We run many test portfolios, each run involving $1,00014. Firstly; we looked at differing out-ofmoneyness by using strike prices set at increasing out-of-the-money ratios. We incorporated trading
costs using actual ask-bid quotations, buying at the closest available ask to the strike price required
and selling at the closest available bid. Buying prices were taken at the horizon period start date or the
next business day. We did not require that actual trades had been made, merely that an ask/bid
quotation was available. We tested increasing out-of-the-money puts using ten differing index prices
to strike (I/S) ratios, from 0.90 to 3.0 (out-of-moneyness 1.11 to 0.33). Because the higher ratios
represent extreme drops in the index there were a limited availability of quotes, especially prior to
1998. There is thus minimal difference in results once a I/S ratio of 2.0 is passed.
Secondly; we examined trading on puts alone, then as part of a put/cash portfolio. We use two
different cash weighting of 80% and 90%. A higher cash weighting will lower overall return but will
extend the length of time the fund can survive between crises. We examined both (i) a replenishing
fixed $1,000 investment each period, with profits/ losses accounted for separately and (ii) an initial
sinking $1,000 sum whose size fluctuated depending on the sequence of profits/ losses experienced.
The former enabled us to make a simple assessment of profitability as the portfolio size was constant,
whilst the latter enabled us to examine how long a swan strategy would be viable.
Thirdly; we examined differing investment horizons, using 3, 6, 12, 18, and 24 month ahead options
horizons, with automatic rollover at the end of the period, using both non-overlapping and
overlapping maturities.
Fourly; we examined differing sell strategies. Initially we assumed perfect hindsight, where we sold
at what turned out to be the best time during the horizon period. While unrealistic, this scenario gave
us base-line maximum profitability. We then examined the impact of technical charting, using three
rules; (i) we sold at the end of the horizon period, (ii) we sold when the price reached a set I/S level
was reached, (iii) we sold when a set level of index bounce back was reached.
14
Note that the use of a dollar figure per period, rather than a set number of puts, replicates industry practice. It
does have the implication that the number of puts purchased per period varies widely with the premium.
-7-
It is important to note that we use very passive automated rules, with no attempt to maximize profit;
we do not actively forecast the future of the index, we only buy one type of put at a time and hold it
for the required period even it turns out to be sub-optimal, we do not choice higher beta products.
Swan hedge funds are likely to devote considerable resources to selection and sell strategies, so our
results can be regarded as a minimum level of profitability achievable in practice.
This paper uses puts on the S&P 500 index over the 18 year period 1990 to 2008, which was
obtained from the MarketExpress division of CBOE. We use full-sized contracts for both short-dated
(SPX) and long-dated (SPL) options, both of which are European. Minis, weeklies and quarterlies
were eliminated. We use the last daily bid or ask on the CBOE as appropriate. While use of the last
bid/ask, rather than actual last trade, raises the possibility that our theoretical portfolio profits may not
be achievable in practice because of order size mismatch, we believe this issue is minor due to the
long trading horizon used and the high volume of trade in S&P 500 index options.
3.
Results
3.1 Perfect Timing
Our initial strategy was to assume perfect hindsight, by selling in-the-money options at the best
price achieved within the horizon period. Options which were always out-of-the-money during the
horizon period were allowed to expire. Our prior expectation was that as the strategy moved from
buying near-the-money puts to buying far out of the money puts, the percentage of portfolio periods
which expire with a profit opportunity would fall, whilst the profit made from each opportunity would
rise. The impact on overall return is ambiguous in theory. The Taleb hypothesis would be that, since
under-pricing of puts is more likely for extreme events than for more common events, overall period
profitability should rise as out-of-moneyness increases.
3.11 Put trading – perfect timing, six-month horizon, six-month puts.
We initially examined the profit potential of buying puts alone. We purchased $1000 of six month
put contracts, using a six month trading horizon, using six month non-overlapping options, for the
period 1990/06 – 2009/12. The horizon periods were set at 01/22-06/21 and 06/22-01/21 to match the
S&P 500 option cycle. The amount of options purchased was the amount invested divided by the
closing bid at the opening horizon day, or the next available trading day. The option was then sold at
the highest closing ask experienced during the period, with any profit or loss accumulated in a
separate account.
-8-
Detailed results are shown in Table 1for index-strike ratios of 1.0, 1.5 and 3.0. These could be
regarded as “normal”, “white/ grey swans” and “black swans”. The first sub-table, for 1.0 moneyness,
shows that in period 1 $1,000 is invested in puts. At a strike price of 355 puts are $11.5, so 86.9 puts
are purchased. These are sold at $56.75 for $4,934.80. Profit is thus $3,934. In the second period, at a
strike of 330, puts are $17.50 so 57.14 are purchased and sold at $27.88. Profit is $592.86. The
accumulated P/L is $4527 on capital of $2,000 so the return is 393% and 59% for each 6 month
period or 453% over the whole year. The second sub-table, for 1.5 moneyness, shows a profit of
362.60%, over the first period, with puts bought at $0.50 and sold at $2.81, for 6 month returns of
465% and 37% or 500% over the whole year. The third sub-table shows the same profit in the first
period as puts were not available at a higher level of moneyness prior to 1995.
For the whole period, the mean return for 1.0 moneyness was 161% p/a, the IRR was 308%, with
fourteen years positive and five years negative. There are no long periods of loss. For 1.5 moneyness
the mean return was 243% p/a, IRR was 369%, with seven positive years and twelve negative years.
For 3.0 moneyness the mean return was 304% p/a, IRR was 369%, with six positive years and
thirteen negative years. There is an eleven period stretch with no gain, 02/12 to 08/06. For all three
samples profits were made predominately during the 1990 decline, the 2000-02 tech crash and the
2007-08 sub-prime crisis.
Several results can be deducted from the comparison of six-monthly returns; firstly, substantial
profit is available ATM. While this is contrary to the idea of efficient markets, it needs to be
remembered this result assumes the ability to sell with perfect timing. A substantial proportion of the
profits were generated by changes in the moneyness of options at levels below that which would be
classified as large. Secondly, average profitability rises with out-of-the-moneyness, supporting the
Taleb Hypothesis. It needs to be noted, however, that while very substantial profits can were made
from buying far-out-of-the-money puts, 5,780% during the subprime crisis, regular losses during
normal market activity produce long periods of loss. Due to the asymmetrical nature of option trading
the return profile is heavily negatively skewed with regular small losses and occasional large gains.
This is more pronounced the more out-of-the-money is the strategy. A summary15 of all results are
shown in Table 2, and Figure 4 shows a comparison of the differing out-of-moneyness on 6-monthly
returns. It is useful to note that in each run we only use puts closest to the set level of moneyness, and
do not vary the choice of put based on forward estimates of volatility in the next six month period.
We also do not buy back into the market until the start of the next horizon period, thus losing possible
profit opportunities. Active portfolio management thus may produce better results than those
illustrated.
15
Detailed results are available from the corresponding author on request.
-9-
3.12 Portfolio trading - perfect timing, six-month horizon, six-month puts.
We then examined the purchase of put contracts within a portfolio. We initially used a 10/90
(puts/cash) fixed portfolio, with a six month portfolio horizon, using six month non-overlapping
options. The initial capital was a nominal $1,000, with $100 invested in 6 month S&P 500 index put
options on the CBOE, with a strike price closest to the required out-of-moneyness and $900 at the 6month US T-bill rate. Any profit (loss) from the puts and t-bills was added (deducted) from an
accumulating account.
Detailed results are shown in Table 3 for strike-index ratios of 1.0, 1.5 and 3.0. The first subtable, for 1.0 moneyness shows that in period 1 10% of $1,000, $100, is invested at $11.5 per put, so
8.69 puts are purchased. These are sold at $56.75 for $493.48. Since T-bills return $931.79 the
portfolio is worth $1,425.27 at the start of period 2. In the second period $100 is invested at $17.5 per
put, so 8.13 puts are purchased, which are sold at $27.88 for $59.29 profit. The return is 43% in the
first period and 8.6% in the second, with an annual return of 51%. The second sub-table, for 1.5
moneyness, shows a profit of 49.4% over the first period, 6.3% over the second, with an annual return
of 56%%. The third sub-table, for 3.0 moneyness, shows the same profit in the first period as puts
were not available at a higher level of moneyness for that period.
For the whole period, the mean return for 1.0 moneyness was 20%, the IRR was 8.5% and all
years but one returning a positive profit. For 1.5 moneyness the mean return was 28%, IRR was 7.7%,
with eight positive years and eleven negative years. For 3.0 moneyness the mean return was 34%,
IRR was 7.4%, with six positive years and thirteen negative years. There is an eleven period stretch
with no gain, 02/12 to 08/06. The pattern of results was, as expected, similar to those for put only
trading, with the scale of gains/losses reduced. A summary of all results are shown in Table 4 and
Figure 5 shows a comparison of the differing out-of-moneyness on 6-monthly returns.
We then examined the results using a 10/90 portfolio with a floating balance. Here any profit
(loss) from the puts and t-bills was added (deducted) from the trading balance. The process was then
repeated. Detailed results are shown in Table 5 for strike-index ratios of 1.0, 1.5 and 3.0. The first
period results are the same as for the fixed portfolio as the portfolio size is the same. After that, while
the percentage returns are the same, the profits experienced in first period then allow the size to grow
steadily. It needs to be noted that while profits are higher with a floating than a fixed balance this is
due to high profits in the initial years pushing up the portfolio size. One of the obvious dangers
reasons with the use of a rolling balance means that when the large profits do come for the Swan
trades the portfolio may be too small to take advantage of it. The effect of this is evident in the third
sub-table, for 3.0 moneyness, where the drain on the portfolio size caused by constant losses, impact
- 10 -
severely on the ability to benefit from the occasional large gains. The higher average return achieved
thus fails to translate into a higher effective end portfolio.
Tables 7 to 10 show the results using a 20/80 portfolio, for a fixed portfolio and for a floating
portfolio. For the fixed portfolio the whole period, the mean return for 1.0 moneyness was 35%, the
IRR was 18% and all years but one returning a positive profit. For 1.5 moneyness the mean return
was 52%, the IRR was 15%, with eight positive years and eleven negative years. For 3.0 moneyness
the mean return was 64%, the IRR was 13%, with seven positive years and twelve negative years. A
summary of all results are shown in Table 9 and Figure 5 shows a comparison of the differing out-ofmoneyness on 6-monthly returns.
Tables 12 to 14 show a comparison of dollar profits from trading at, respectively 1.0 moneyness,
1.5 moneyness and 3.0 moneyness for 6 year sub-periods. These tables further compare put only
trading with a 10/90 portfolio and a 20/80 portfolio. Neither of the 1.0 or 1.5 moneyness trading show
any sub-period with aggregate losses. However only the 06/96-12/01 and 12/02-06/08 sub-periods
exhibit returns comparable to other investments. The results for 3.0 moneyness, which represents
extreme movements, show aggregate losses for the 06/93-12/98 and 06/99-12/04 sub-periods. Note
that the latter period covers the tech crash. The only opportunities for extreme profits are 06/90-12/90
and 12/07-06/08 periods. These tables illustrate the rareness of profiting from black swans.
3.13 Portfolio trading - overlapping 6-mth horizons, perfect timing.
Results pending
3.14 Varied horizons - perfect timing, six-month puts, fixed portfolio
The results so far assume that we can capture the full profitability from a market crisis with the
use of fixed six month trading horizon. The put is purchased at the start of the horizon and sold at the
end of that six month horizon regardless of our expectation about the foreseeable future. We do not
time either choice. Note that maximum profit is made from a swan trade if the put is purchased during
a calm period when the market is not forecasting undue volatility. Once a crisis occurs or becomes
foreseeable then put prices will rise, which will substantially reduce any profit potential from further
trading periods. This leads to two issues, (i) a crisis may start towards the end of the horizon or (ii) a
crisis may take place over more than a six month period. In both cases any profit experienced may be
foreshortened by forced sale at the end of the investment horizon.
- 11 -
We thus examined the impact on our results of varying the investment horizon. There is, it needs
to be noted, a potential issue with longer term puts, as their relative illiquidity means that far-out-ofthe-money puts are less likely to be available. We examined the profitability of perfect timing at 3,
12, 18 and 24 month horizons.
Results pending
3.2 Fixed Timing
The major issue with the prior results is that the assumption of perfect timing is unlikely to be
achievable in practice. It is more common to calculate investment strategy profitability by selling at
end of the investment horizon. We thus examined the profitability of swan trading using fixed
horizons.
3.21 Put trading – end of horizon rule, six-month horizon, six-month puts, fixed portfolio
Results pending
3.22 Portfolio trading – end of horizon rule, six-month horizon, six-month puts, floating portfolio
3.23 Varied horizons – end of horizon rule, fixed portfolio
3.3 Non-Perfect Timing
Our third strategy was to examine the impact of imperfect timing by following two passive
technical analysis rules; (i) we sold when the price reached a set strike/ index level was reached, (ii)
we sold when a set level of index bounce back was reached. We used automated rules, with no
attempt to maximize profit. The results discussed so far are useful in that they illustrate the maximum
profitability of swan investing and shed light on the efficiency of pricing of out-of-the-money puts.
They are, however as discussed, unrealistic in practice.
- 12 -
It is also useful to note that the profitability of put trading strategies depends whether the future
volatility in the underlying priced at the time of purchase is realized. Swan strategies are thus most
successful when puts are purchased when market volatility expectations are low. This volatility only
correlates weakly to the level of the market index. Our use of the market index to decide sell timing is
thus likely to provide lower profits than would be achievable if a more direct indicator of volatility
was used. We also examined the profitability of a very passive form of swan trading, using a broad
index. Swan funds are likely to follow more active strategies in two ways. The first strategy would be
to devote a lot of resources to actively timing their selling. The second strategy would be to increase
their alpha, which essentially involves choosing stocks of markets with a high beta to possible market
crises. The resultant increase in profits will go some way to off-set their inability to time selling
perfectly.
3.31 Put trading – strike/price rule, six-month horizon, six-month puts, fixed portfolio
The difficulty with imposing a fixed technical analysis rule is that it ignores major issues of
market dynamics. Put strategies make profit when the market falls by a sufficient amount to cover
costs. As discussed earlier, downside market fluctuations tend to follow a power-law with a stable
Paretian distribution with a degree less than 2. This implies a large number of minor declines
interspersed with a number of episodic crises. Profits are only made from swan strategies which fully
capture the profits from the all types of swan species. The issue when setting strike/ price rules is that
a rule a low level will capture most of the profit from smaller grey swan declines but will cut short the
extremely large profits available from black swans. Conversely a rule set at the higher level will
capture more the profits available from crisis but will fail to capture the more frequent profits
available from the smaller white or grey swans.
An additional issue with dynamics is that the length of crisis can vary. The Oct/87 crisis was short
and thus could be captured within one trading period, with the buying price within the range of calm
markets. However the 2007 crisis was longer with trading horizons tending to end before full
advantage could be taken of the crisis, as buying prices at the start of the new trading period will
reflect the crisis. Differing technical selling rules will be required to take advantage the differing
varieties. Selling at the end of the trading horizon will only capture profits from the long slow
declines crisis variety, and then only for the first trading period.
Our first rule was to sell when the index fell by a set ratio below the strike. We varied this ratio
from 0.4 to 4.0. Our initial results are in Table 14, which shows the results of an imperfect selling
swan strategy, using only puts, and the two portfolios, using a six month trading horizon and nonoverlapping options. $1,000 is invested each period and P/L’s accumulated in a separate account. The
period was restricted to 1993-2009 due to restrictions on the availability of hedge fund data. The
- 13 -
results show both percentage return and accumulated profit and loss in the form of a matrix with the
strike to index moneyness buying on the vertical and the strike to index selling rule on the horizontal.
Put are bought at a strike price as close to the required S/I as possible and sold if the index falls to the
set moneyness level or in default at the option price on day before closing. Note that results were the
sale S/I ratio is less than the purchase S/I ratio are listed as n/a as this would involve the put being inthe-money.
These results clearly show that obtaining profits using a set strike/ index rule is challenging, as all
periods show very low accumulated gains. Detailed results show that purchase ratios above 1.25
profits are only made two or three times during the period. Note that the average return is positively
biased due to the extreme skewness of the profit profile.
3.32 Put trading – price recovery rule, six-month horizon, six-month puts, fixed portfolio
Our second rule was to sell when the strike price fell by a set ratio below the strike and then
recovered by a set percentage.
Results pending
3.3 Swan Portfolio Performance.
3.31 Performance metrics for options trading
Our paper is trying to assess the possibility of using Swan Trades to make a realistic profit. The
conventional metric to measure the performance of a trading strategy is the buy and hold abnormal
return (BHAR) between our chosen Swan Trade and its benchmark, the return on holding a
benchmark portfolio (possibly a 3 month T-bill or the S&P 500 index):
BHAR   1  rPortfolio,t    1  rBenchmark,t 
T
T
t 1
t 1
This is equivalent to the compounded return of the annual abnormal return, which is found by
regressing a portfolio’s excess return, rt, on a set of K return-generating factors, ft. BHAR is an
appropriate measure of the difference in investor’s terminal wealth between the two portfolios.
However Fama (1998) and Mitchell and Stafford (1997) argue that compounding issues mean that
BHAR is subject to serious distortion when used as a metric for the abnormal return impact of any
particular event. They argue that average abnormal return (AAR) should be used instead:
AAR 
1T
rPortfolio,t  rEBenchmark,t 


T  t 1

- 14 -
The major performance metric issue for swan strategies is that option returns are highly nonnormal (commonly hyperbolic with fat tails) and thus mean-variance based metrics which assume
normality and symmetry, such as CAPM alphas or Sharpe ratios, are inappropriate as they assume
either the investor's utility function is quadratic or the returns are normally distributed (Glosten &
Jagannathan, 1994, and Grinblatt & Titman, 1989). It is now agreed in the literature that performance
metrics for portfolios which exhibit extreme skewness and kurtosis, as displayed by most hedge
funds, need to take account of this asymmetry. The Sharpe ratio penalizes high volatility and ignores
correlations, and can produce a ranking which is drastically wrong (Lo, 2002).
An additional problem is that most hedge funds have a ‘blow up’ strategy whereby they earn
regular small income at the expense of exposure to bust under certain extreme market conditions.
Since these bust circumstances are rare outliers their impact is minimized when using symmetrical
mean-variance based performance or risk metrics, even if fat tailed. However, recently researchers
have argued that investors’ return-risk preferences are focused more on the possibility of major downside risks, rather than the more minor fluctuations in upside performance which are captured by
symmetrical metrics. Performance metrics should focus on downside risk.
A widely used asymmetrical metric which is often used to overcome these issues Sortino ratio
(Sortino 1994), which replaces the standard deviation in Sharpe by the downside deviation. This is the
excess return over the risk free rate over the downside semi-variance.
𝐸(𝑅𝑝) − 𝑀𝐴𝑅
√ 1 ∑𝑇 𝑡=0 (𝑅𝑝𝑡 − 𝑀𝐴𝑅)2
𝑇 𝑃𝑟<𝑀𝐴𝑅
where RPt is the return of the portfolio in the sub-period t, Rp is the average of the returns of the
portfolio over the whole period, MAR is the minimum acceptable return, and T is the number of subperiods. A higher value is better than a lower value. The problem with the Sortino ratio is that it does
not capture the problem of higher moments.
A metric which is regarded as better is “Omega”, developed by Keating and Shadwick (2002).
Focus is placed on size of possible downside risks by partitioning returns below a chosen threshold.
De Souza and Gokcan (2004) calculated Omega as follows:
b
( L ) 
 (1  F ( r ))dr
L
L
 F ( r )dr
a
where L is the required return threshold, a and b are the return intervals, and F(r) is the cumulative
distribution below the chosen threshold L. Omega involves petitioning returns into loss and gain
above and below a required threshold and then considering the probability-weighted ratio of returns
- 15 -
above and below the partition. It therefore employs all the information contained within the return
series. When the threshold is set to the mean of the distribution the Omega measure is equal to one. In
the discreet case, with equal frequency, De Souza and Gokcan (2004) provide:
b
( L ) 
 Max (0, R
a
b
 Max (0, R


)
)
a
where R+
(-)
is the return above (below) a threshold L. Our chosen threshold was return on the S&P
500. High Omegas are preferred to low Omegas at equal points of the threshold.
Another useful asymmetrical metric, which is a generalized downside risk-adjusted performance
measure, is “Kappa”, developed by Kaplan and Knowles (2004). This.
Kn𝜏 =
𝜇−𝜏
𝑛
√𝐿𝑃𝑀𝑛(𝜏)
where μ is the expected periodic return, τ is the investor’s minimum acceptable or threshold return
and LPM is the lower partial moment. The Sortino ratio is equal to K 2, and Omega to K1 +1. “n” is
strictly greater than 0. A higher value is better than a low value. Kappa has to be cautiously handled
as it is sensitive to skewness (Géhin 2004).
A more recently developed metric is the higher moment adjusted CAPM which can be used to
capture coskewness and cokurtoisis. Favre and Ranaldo (2003) first use the market model to for the
standard CAPM:
Ri,t – Rf,t = α + β1,iRm,t – Rf,t + εt
and add a relation to the third moment via:
β2,i(Rm,t – E(Rm))2
Favre and Ranaldo find the quadratic model to be particularly useful for option strategies. The higher
the β values the better and ideally α should be positive.
Costs of a typical hedge fund are around 1-2% p/a of the account value plus about 20% of any
profit.
3.32 Swan Portfolio Metrics
Table 15 shows the results of swan trades in terms of standard metrics (mean, standard deviation,
skewness and kurtosis) as well as Sharp, Omega, Kappa and higher moment CAPM. Table 15 then
compares a swan strategy on these metrics compared to the performance of a range of hedge fund
types. Note that metrics have not been provided for T-bills and the S&P 500 as these were used ti
calculate the other metrics.
- 16 -
The rankings show the effects of the extreme skewness of swan profits; based on Sharp the swan
strategies are low ranked, however based on Sortino, Omega, Kappa and higher-moment CAPM the
swan strategies are markedly superior. Swan trades have a potential for high long-term returns as well
as a useful return-risk profile.
It needs to be noted that the Sortino, Omega and Kappa metrics were created to capture the
performance of blow-up-profile hedge funds which have a large negative skewness, and a high risk of
bankruptcy in the long-run. Swan strategies, with a bleed profile, have a large positive skewness and a
very low risk of bankruptcy. These metrics are not designed for use with swan profiles, and thus the
comparison of performance metrics needs to be handled with care.
4.
Conclusion
The Taleb hypothesis is that option markets do not price in the occurrence of unlikely extreme
events, Black Swans. This paper examined whether or not profits can be made from exploiting the
increased occurrence of extreme downswings experienced by the US stock market. We used a swan
trading hypothesis of continually buying far-out-of-the money puts on the S&P 500 index to ascertain
the profitability of this hypothesis, by use of a range of passive trading strategies.
We show that swan trade strategies have a potential for very large long-run returns. While there
were long runs of losses all five year periods showed positive gains. We also showed that average
long-run returns rise as the swan strategy moves further out-of-the-money, and the return profile
becomes more skewed.
Our results fulfil the Taleb hypothesis that the put option market is not pricing in the risk of extreme
events. As outlined by Malliaris & Yan (2008) this is probably due to the difficulties hedge funds
reliant on outside funding which does not have the capacity to sustain three or four year runs of steady
losses. The markets are in general unable to arbitrage away gains available only to investors who can
sustain those losses.
What is pattern of implied volatility over last ten years?
If last ten years repeated what is profit potential?
How much of max profit potential do traders have to capture to get 15% return?
How big / frequent do BS have to be to give traders 15% return? 10% return?
Pending
- 17 -
- 18 -
References
Ait-Sahalia, Y. (2002) Telling from discrete data whether the underlying continuous-time model is a
diffusion, Journal of Finance, 57: 2075-2112.
Aït-Sahalia, Y. Wang,Y. and Yared, F. (2001) Do option markets correctly price the probabilities of
movement of the underlying asset? Journal of Econometrics, 102: 67-110
Anderson, T., Benzoni, L. and Lund, J. (2002) An Empirical Investigation of Continuous-time
Models for Equity Returns, Journal of Finance, 57: 1239-1284.
Aparicio, F. & Estrada, J. (2001). Empirical Distributions of Stock Returns: European Securities
Markets, 1990-95. European Journal of Finance, 7, 1-21.
Bates, D. (2008), The Market for Crash-Risk, Journal of Economic Dynamics and Control, 32 (7):
2291-2321
Board, J., Sutcliffe, C. and Patrinos, E. (2000), Performance of Covered Calls, European Journal of
Finance, vol. 6. No. 1, March, pp. 1-17.
Bogle, J.C. (2008), Black Monday and Black Swans, Financial Analysts Journal, Volume 64 (2)
Bondarenko, O. (2003) Why are Put Options So Expensive? (April 2003). University of Illinois at
Chicago Working Paper. SSRN: http://ssrn.com/abstract=375784
Broadie, M., Chernov, M. and Johannes, M. (2009) Understanding Index Option Returns, Review of
Financial Studies, 22 (11): 4493-4529
Buraschi, A. and Jackwerth, J. (2001) The price of a smile: Hedging and spanning in option markets,
Review of Financial Studies, 14: 495-527
Coval, J.D. and Shumway, T. (2001) Expected Option Returns, Journal of Finance, 56 (3): 983-1009.
De Souza, C. and Gokcan S., (2004) Hedge Fund Investing: A Quantitative Approach to Hedge Fund
Manager Selection and De-Selection, The Journal of Wealth Management, Spring
Driessen, J. and Maenhout, P. (2007) An Empirical Portfolio Perspective on Option Pricing
Anomalies, Review of Finance, 11: 561-603.
Estrada, J. (2008), Black Swans and Market Timing: How Not To Generate Alpha, Journal of
Investing, Fall, 20-34.
Estrada, J. (2009), Black Swans, Market Timing, and the Dow, Applied Financial Economics Letters,
16 (11): 1117 - 1121
Estrada, J. (2010), Black Swans, Beta, Risk, and Return, SSRN paper # 1692490
Farhi, E. and Gabaix, X. (2008), Rare Disasters and Exchange Rates, Working Paper 13805, National
Bureau of Economic Research, Cambridge, Ma
Fama, E. (1965) The Behavior of Stock Market Prices, Journal of Business, 38, 34-105.
Fama, E. (1998) Market Efficiency, Long-term Returns, and Behavioral Finance, Journal of Financial
Economics, 49: 283-306
Favre, L. and Ranaldo, A. (2003) How to price Hedge Funds: from Two to Four-Moment CAPM,
EDHAC Working paper.
Fung, W. and Hsieh, D. (1997), Empirical Characteristics of Dynamic Trading Strategies: The Case
of Hedge Funds, Review of Financial Studies 10 (2): 275-302.
Géhin, W. (2004) A Survey of Literature on Hedge Fund Performance, EDHEC working papper.
Glosten L. and Jagannathan, R. (1994) A Contingent Claim Approach to Performance Evaluation,
Journal of Empirical Finance, 35 (5): 61-76
Grinblatt, M and Titman S. (1989) Portfolio Performance Evaluation: Old Issues and New Insights,
Review of Financial Studies, 2: 393-421
Jackwerth, J. and Rubinstein M. (1996), Recovering Probability Distributions from Contemporary
Security Prices, Journal of Finance, 51: 1611-1631
- 19 -
Jegadeesh, N. and Titman, S. (1993), Returns to Buying Winners and Selling Losers: Implications for
Stock Market Efficiency, Journal of Finance, 48 (1): 65-91
Jegadeesh, N. and Titman, S. (2001), Profitability Of Momentum Strategies: An Evaluation Of
Alternative Explanations, Journal of Finance, 56 (2): 699-720
Jegadeesh, N. and Titman, S. (2002), Cross‐Sectional and Time‐Series Determinants of Momentum
Returns, Review of Financial Studies, 15 (1): 143-157
Jones, C. (2006), A Nonlinear Factor Analysis of S&P 500 Index Option Returns, Journal of Finance,
61(5): 2325-63
Jurek, J. W. (2008), Crash-neutral Currency Carry Trades, http://ssrn.com/abstract =1262934
Kaplan, P. and Knowles, J. (2004), Kappa, a Generalized Downside Risk-Adjusted Performance
Measure, Journal of Performance Measurement, 8 (3): 42−54
Keating, C. and Shadwick, W. (2002), A Universal Performance Measure, Journal of Performance
Measurement, 6 (3): 59-84.
Lo, A.W. (2002) The Statistics of the Sharpe Ratio, Financial Analysts Journal
Malliaris, S. and Yan, H. (2008), Nickels versus Black Swans: Reputation, Trading Strategies and
Asset Prices, Yale ICF Working Paper No. 08-26, Yale School of Management
Mandelbrot, B. (1963), The Variation of Certain Speculative Prices, Journal of Business, 36, 394-419.
Mandelbrot, B. and Hudson R. (2005). The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin
and Reward. Profile Books.
Mauboussin, M. (2006). More Than You Know: finding financial wisdom in unconventional places.
Columbia University Press.
Mitchell, M. and Stafford, E. (2000), Managerial Decisions and Long-Term Stock-Price Performance,
Journal of Business, 73: 287-329
McLean, R.D. (2009), Fooled by Compounding, University of Alberta working paper, SSRN:
http://ssrn.com/abstract =1485179
Peters, E. (1991), Chaos and Order in the Capital Markets. A New View of Cycles, Prices, and
Market Volatility. John Wiley and Sons.
Rabin, M. (2000), Inference by Believers in the Law of Small Numbers, Economics Department,
University of California, Berkeley, manuscript.
Santa-Clara, P. and Saretto A. (2009), Option strategies: Good deals and margin calls, Journal of
Financial Markets, 12(3): 391-417
Taleb, N. (2001), Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets,
Random House, USA.
Taleb, N. (2004), Bleed or Blowup? Why Do We Prefer Asymmetric Payoffs? Journal of Behavioral
Finance, Vol. 5 (1): 2-7
Taleb, N. (2007), The Black Swan: The Impact of the Highly Improbable. Random House.
Taylor, J. and Williams, J.(2009), A Black Swan in the Money Market, American Economics Journal,
1 (1)
Zuckerman, G. (2009), The Greatest Trade Ever, Broadway Books, USA: NY.
- 20 -
Tables and Diagrams
Figure 1
Figure 2
- 21 -
Implie d Vola tility
Figure 3
Figure 4 - Vola tility Smile
Any profits from swan trading have to derived from the
difference between the market price for put options, P,
1
and the correct extreme event adjusted pric e, P2, the
gap, AB. While Talib argues this gap is larger OTM than ATM,
there is no theoretic al reason to support this.
B
Extrem e-event adjusted pric e
A
Market pric e
0
P2
B-S Price
P1
OTM
ATM
- 22 -
Strike Price
- 23 -
- 24 -
- 25 -
- 26 -
- 27 -
- 28 -
- 29 -
- 30 -
- 31 -
- 32 -
- 33 -
- 34 -
- 35 -
- 36 -
- 37 -
- 38 -