Profiting from Black Swan

Trades: Evidence from S&P 500 puts.

Michael J. Naylor*, Jianguo Chen and Udomsak WongchotiSchool of Economics and FinanceMassey UniversityPrivate Bag 11 222Palmerston NorthNew 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-of-the 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.

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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,

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.

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

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 in-

the-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

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

4 This was on Oct/28/1929 (12.8%) and Oct 29/1929 (11.7%). 5 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.

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finding in empirical work is that index options embed large risk premia for jump/ volatility risk7, 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 far-

out-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

implication of a stable Paretian infinite variance process is that sample variance and standard

deviation are not meaningful measures of variability for option strategies.

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

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

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.

9 The average daily volume in 2008 was 670,629 contracts, with a peak day of 2,170,870 (10/06/08).10 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.

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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

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.

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”.

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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-of-

the-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

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

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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-of-

moneyness 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.

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.

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

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.

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

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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.

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 6-

month US T-bill rate. Any profit (loss) from the puts and t-bills was added (deducted) from an

accumulating account.

15 Detailed results are available from the corresponding author on request.

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Detailed results are shown in Table 3 for strike-index ratios of 1.0, 1.5 and 3.0. The first sub-

table, 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

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-of-

moneyness on 6-monthly returns.

10

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.

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-of-

the-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

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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.

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.

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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 non-

overlapping 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

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 in-

the-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.

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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=∑t=1

T

(1+r Portfolio ,t )−∑t=1

T

(1+r Benchmark , t )

This is equivalent to the compounded return of the annual abnormal return, which is found by

regressing a portfolio’s excess return, r t, on a set of K return-generating factors, f t. 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= 1T [∑t=1

T

(rPortfolio , t−r EBenchmark , t )]The major performance metric issue for swan strategies is that option returns are highly non-

normal (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.

14

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 down-

side 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

√ 1T ∑

t=0Pr <MAR

T

(Rpt−MAR )2

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 sub-

periods. 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:

Ω( L )=∫L

b

(1−F (r ) )dr

∫a

L

F (r )dr

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

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:

Ω( L )=∑

a

b

Max (0 , R+ )

∑a

b

Max (0 ,|R−|)

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.

15

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 K2, 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.

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

16

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

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Tables and DiagramsFigure 1

Figure 2

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Figure 3

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