MEMOIRE

En vue de lobtention du Diplme de Sup de Co Reims

REIMS MANAGEMENT SCHOOL

CYCLE MASTER

2010-2012

VOLATILITY AS AN ASSET CLASS: STUDY OF THE RISK AND RETURN IMPACTS ON A PORTFOLIO

MEMOIRE ACADEMIQUE

PAR: Fernando PRIETO

JURY: Yves RAKOTONDRATSIMBA

MAY 2013

2

I would like to thank:

M. Bucchiccio Oliver, Head of Institutional Investor Sales at BNP Paribas, for the

inspiration

M. Lleo Sebastien, Finance Teacher/Researcher at RMS, for all the references

M. Sarter Stephan, Responsible for Master Thesis at RMS, for the patience and

assistance with paperwork.

And special greetings to M. Rakotondratsimba Yves, Teacher at ECE, for his tutoring.

3

Table of Contents

Asset classes ...................................................................................................................... 8 Volatility: Standard deviation, Variance and Covariance .................................................. 10 Correlation and diversification effect ............................................................................... 12 Volatility: Historical and Implied volatility ........................................................................ 14 Volatility Index: example of the VIX ................................................................................. 18 Volatility: from the Smile to the Skew .............................................................................. 21 Exposure to the VIX via swaps, futures, options ............................................................... 24 Risk adjusted return: Sharpe ratio and some others ......................................................... 29 Hypothesis 1: Volatility is a new asset class ...................................................................... 36 Hypothesis 2: Volatility improves risk-adjusted returns .................................................... 42 Hypothesis 3: Volatility protects from Black-Swans .......................................................... 49 Our conclusions ................................................................................................................ 55

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Abbreviations and acronyms

c. Circa CAC40 Cotation Assiste Continue 40 CAGR Compounded Average Growth Rate CAPM Capital Asset Pricing Model CBOE Chicago Board Options Exchange DAX Deutsche Borse Ag German Stock Index e.g For example ETF Exchange Traded Fund ETN Exchange Traded Note i.e That is OTC Over the counter PF Portfolio Pp Percentage point S&P Standard & Poors SPX S&P 500 SPXTR S&P 500 Total Return VIX S&P 500 Volatility Index Y Years (e.g. 10Y means ten years)

5

I Introduction

During my first internship at BNP CIB as equity and commodity derivatives sales

assistant, I was given the opportunity to discover a new category of products the bank wanted

to push: volatility products. A volatility product is defined as a security that provides

exposure to volatility alone without being affected by directional movements of the

underlying asset. This was a major surprise for me because as a Junior (and it was also the

case of not so junior people) volatility was a measure of risk.

Moreover, to simplify, some people spoke indifferently of volatility and variance. But

at that time the variance, for me, was not even a finance concept but a mathematical one. And

when I expressed my surprise to a sales I was working with, he answered: if you young

undergraduate, working for us on complex products in a daily basis and supposed to

understand, explain and sell these products are so lost, can you imagine the situation of the

clients? . That is why I decided to write my thesis on this new approach of a mathematical

and portfolio management concept. The goal of this study is to recreate the context and

constraints given to the banks that logically leaded to the creation of these products and

expansion of the market.

During the previous decades, liquidity has been the central concept of research in

finance. But the exhibit 1, a Google Ngram graphic (a Google Books tool analysing a

significant amount of books over time and giving as an output a graph showing how a phrase

have occurred in history) illustrates how, since the 90s, Volatility is the more commented

concept. We will see later that this date is in line with Variance Swaps creation!

Exhibit 1: Google Ngram graph for Liquidity and Volatility between 1930 and 2008

6

If some institutional investors and quants have started to work on this kind of project

in the 2000s, the intense development and growth of the volatility product market boomed

after the recent financial crisis and especially the collapse of Lehman Brothers in September

2008. Most of the asset classes declined sharply - a decline rarely explainable by the

fundamental methods of asset valuation. For example, in 2009, G. Subleys article 1demonstrated that a 3000pts CAC40, the average level of 2009 (note that it reached its

lowest close level in March 2nd 2009, 2534.45pts), means that half of the underlying

companies are valued to a Price to Book-Value ratio of less than 1. He also demonstrates that

the multiples low levels are the result of an increasing uncertainty environment. In parallel of

that phenomenon, the correlation levels of the different asset classes increased dramatically.

In average from 1983 to 2007, the Equity and Commodities (considering the S&P500 and the

S&P GSCI) correlation level was 0,57: it increased to 0,91 during the 08-09 crisis.

What was the impact on for the portfolio managers? The Modern portfolio theory is

based on Markowitz approach of risk: it is measured by the standard deviation (or the

variance, its square). Then Markowitz derived the general formula to measure the standard

deviation of the whole portfolio of risky assets. Thanks to this formula he clarified the

relationship between the risk, the return, and the different assets correlation. Thus he was able

to demonstrate the impact of the correlation levels on the risk diversification: the global risk

level of a portfolio is less than the risk of carrying each asset independently if the correlation

coefficients are less than 1.

The challenge for the portfolio managers is to find an efficient way to diversify the

risk of their portfolios and optimize the return for each unit of risk. For that reason the asset

managers use different asset classes as they are exposed to different economical phenomenon

(e.g. when equities drop, the investors are supposed to sell it and invest in bonds and then

bonds will be negatively correlated to the equities). So what happen when the correlation

levels of the asset classes converge to 1? The level of undiversified risk carried by a portfolio

composed of these assets increases.

1 Analysed in (Surbled, 2009)

7

Our idea here is to consider the volatility as an asset class. By volatility we mean the

implied volatility, the level implied by the market as it is derived from the mark-to-market of

the options (calls and puts). To be more precise, it is the volatility level matching the Black

and Scholes formula and the prices of the options on the market (in other words the implied

volatility is the future volatility level estimated by the market). Structurally it is uncorrelated

to the other asset classes (especially on bear markets). In 2008 after Lehman collapse, the

implied volatility levels reached records high levels across all the markets (for example the

VIX has reached its highest level 80,86 2 months after the crash of Lehman) when, as

mentioned before, the other assets where declining.

So the first hypothesis is about the volatility, in other terms the products allowing an

exposure to the volatility of assets, can we consider this set of securities as a new asset class?

To test this hypothesis, we will design a framework to define what is an asset class. Then we

will study the different solutions offered by investment banks to buy volatility or variance

with different wrappers (like swaps, options or shares in funds Voledge-like) and see if they

meet the definition of an asset class.

Then it will be interesting to study the impact of diversifying a portfolio of risky assets

with these securities. While volatility has a given return and risk profile in a given market, the

question then is to know how the portfolio will react to it in different situations and over time.

Do these assets allow a statically significant improvement of the risk to return profile of our

portfolio? The second hypothesis will be then is that the integration of volatility to a portfolio

of risky assets does improve the risk to return profile.

Finally, my third hypothesis will derive from the precedent: as the implied volatility

takes advantage of the market uncertainty, then we will test the efficiency of these products to

hedge a portfolio against what Nassim N. Taleb so-called a Black Swan - those huge, not

anticipatable, and highly improbable market movements. For example, during the last

financial crisis, the incredible surge in volatilities of most markets and especially the

observation of a rare phenomenon - the recorrelation of the asset classes in a bearish market -

caused a "Black Swan" out of the scope of usual Risk Management Scenarios. We will try to

test the efficiency of these solutions in extreme scenarios like these.

8

II State of the Art

Volatility products are highly technical; approximations and abuses of language are

very common in the industry about it. Thus we have decided to start with a global overview of

the required concepts. Moreover as the final goal of this paper is to help novice people to be

comfortable with volatility products, it appears relevant to start with the basics. As a

consequence, for that part of the study, I will particularly insist on the terminology and

definition of the concepts the lector could need to understand the analysis realised in the

second part of this research paper.

Asset classes

Asset classes are an asset allocation concept. The more frequent definition is: an

asset class is a set of assets that bear some fundamental economic similarities to each other,

and that have characteristics that make them distinct from other assets that are not part of that

class 2. The generally accepted traditional asset classes are: equities, bonds and cash. With

time, it is now widely recognised that there is some alternative asset classes: hard

commodities, soft commodities, real estate, and credits...

The idea of considering different asset classes is useful to diversify our exposure to

different economic factors or phases of a cycle. As this is out of the scope of this study we

invite the readers interested by the subject to consult the paper of Pim Van Vliet and David

Blitz Dynamic Strategic Asset Allocation: Risk and Return Across Economic Regimes to

know more about the asset allocation driven by the economic cycles. Our interest for that

theory here is limited to the proof it makes of the usefulness of the concept.

In this study we will use the following framework to define what an asset class is:

Does this asset class make sense as an economic factor? To answer that question

we can try to determine if the securities composing it have a homogeneous return

in a given economical context but it is also necessary to question the philosophy

2 Definition in (Greer, 1997)

9

of the product. In other words, it leads to answer a simple question: do these

securities give an exposure to the same economic factor?

Are the securities tradable? This criterion is in line with my intension of writing a

guide useful for the portfolio managers (and consequently close to the reality of the

market). This tradability criteria involve different questions:

The question of the general agreement on the measure of the return. We

will see that if it is easy to agree on the concept of the return, the way to

measure it is quiet different from a party to another

The existence of the securities and the market access. Neuberger and

Hodges (Neuberger & Hodges, 1989) published the variance swap

replication portfolio method (with a portfolio of options) in 1990 and the

first var-swap was traded in 1993. At that time the var-swaps was a very

small market with very restricted market access for years. It proves you

need to test the availability of the securities to define them as an asset class

More in depth the question of the securities liquidity still pending. If the

liquidity of an asset class condition, it is clearly a parameter and by

consequence need to be defined

Do these securities have a homogeneous risk/return profile? If this question is

close to the previous one, we will focus here more on the payoff structure than on

the timing or on the economic (or financial) background resulting in this return

for the security

My researches on how to define an asset class in the academic literature was not much

successful and it appears that for now the asset classes are more the result some arguments of

authority or empirical evidences than on a precise framework resulting in a classification. If

the framework allowing an asset class classification is not in the core of this study, I decided

to draft one and to use it in order to test my first hypothesis: can we consider volatility as a

new asset class?

10

Volatility: Standard deviation, Variance and Covariance

In 1954, Harry Markowitz wrote the efficient portfolio theory and in 1990 he received

the Sveriges Risksbank Prize in Economics Sciences in memory of Alfred Nobel for it. In his

theory, Markowitz defines the returns of an asset as a random variable. Consequently the

expected return is the average return and the risk is defined as the variance of returns or the

standard deviation (the square root of the variance) of the random variable.

Since then, the economists have highlighted the limits of this measure of risk of which

the more important is considering both up and down side equally risky, an irrelevant

hypothesis in most investment positions. Imagine a pharmaco stock with a given volatility.

The CEO of the company announce a vaccine against AIDS, the stock knows unusual positive

jumps in the stock exchange in prediction of the future positive expectations for the company.

As a consequence, the volatility (supposed to measure the risk) will rise. Does it make sense?

The market is only pricing the value of the future earnings! Another limit of the measure is

the high sensitivity to extreme figures (as we square the distances to the mean, the impact of

the extreme values is not linear).

But how is the volatility measured? The standard deviation is a 19th century

mathematical concept; it measures the variation (or dispersion) to the mean of a random

variable. The symbol used in mathematics is and the standard deviation formula (1) is:

(1) X = E X E X[ ]( )2"

#$%

Where X is a random variable representing the returns of the asset

Then, the variance formula (2) is:

(2)V X[ ] = 2X = E X E X[ ]( )2"

#$%

In other words, the variance is the average of the squared distances to the mean and

the standard deviation is the square root of it. The advantage of the standard deviation is that

it has the same dimension than the mean.

11

Now it is necessary to define the variance of a portfolio. When Markowitz defined the

standard deviation as a measure of risk, then he derived the general formula for the variance

of a portfolio of risky assets (3):

(3) 2P = w2i 2i + wiwj i ji, jji

i

i

Where wi and wj are the respective weights of the assets i and j in the portfolio P

Where 2i the variance of the asset i

Where i is the standard deviation of the asset i

Where i, j the correlation coefficient between the asset i and j

As this formula shows the dispersion is neutral to the mean. The sign of the distances to

the mean are completely neutral as they are squared. Again, this limit of the standard

deviation is important to note, as we will see later, this is not the case with tradable volatility

(cf the concept Volatility: from the Smile to the Skew).

If the variance is used to measure the variations of a variable with respect to itself, the

covariance will allow studying the simultaneous variation of variables from their respective

means. It can be noted indistinctly cov(X,Y) or X,Y . The covariance formula (4) is as

following:

(4) Cov(X,Y ) = E (X E X[ ])(Y E Y[ ])"# $% From this presentation of the covariance formula we can easily deduce a simple result: cov(X,X) = 2X . Please note this second presentation of the equation: (5) Cov(X,Y ) = 1N (Xi E X[ ])(Yi E Y[ ])

Where Xi is the value of the asset X at time i

Where Yi is the value of the asset Y at time i

The lower the obtained result is, the more the series are independent and a move from one

will not impact the second. Conversely the higher it is, the more the series are linked. Two

independent variables have a zero-covariance.

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Correlation and diversification effect

From the previous formula of the portfolio variance (3) we can understand the origin

of the diversification effect and the role of the correlation between the assets (and by

extension of the asset classes in it) in the volatility of a portfolio. Lets analyse it: as mentioned

before, the general formula of a portfolios variance is:

(3) 2P = w2i 2i + wiwj i ji, jji

i

i

But what does the correlation coefficient i, j represent? The formula of it is:

(6) i, j =cov(X,Y ) XY

The formula (6) is easy enough to understand, the essential point is to understand the

relation between the correlation coefficient and the covariance for a given level of volatility of

the assets. Finally we can deduct the consequences on (2), and the relationship between assets

impact the volatility of the portfolio:

The more correlation we have between assets, the bigger the second term is. The

consequence is obviously a reduced impact on the non-systematic risk. At the

opposite, when we have low or negative correlation between the assets, then the

second term will be small or negative and we will reduce quicker the non-

systematic risk

The more assets we have, the smaller is the second term as the weights will be

reduced and the correlations range from -1 to 1 increase the effect. This lead us to

the exhibit 2 resuming the relation between the number of securities and the

portfolio variance

Exhibit 2: Portfolio volatility and Risk diversification

Number of securities Portfolio

variance (risk

)

Systematic risk Non-Systematic risk

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Now that the concept of correlation between assets and asset classes is clear, it is

interesting to have a look on the real situation of the markets on the previous years. How

assets are correlated? In the exhibits 3 and 3bis, we will present US and HK Equity as

different asset classes because if both are equity, they give an exposure to very different

economic factors (cf. the presentation of the Asset Classes concept).

Exhibit 3: Performance evolution between different assets since 2000

Exhibit 3bis: Zoom on the increasing correlation of assets since 2005

Source: BNP CIB - Bloomberg, Data from 22nd March 2005 to 25th August 2010.

These two exhibits perfectly illustrate the recent constraints on the portfolio managers:

as we can see, the diversification of assets during the normal periods and the recorrelation

of the assets when they really need diversification: during the crisis periods. In this context, a

structurally negatively correlated asset, like the volatility, should be appealing for the asset

managers: here started the adventure of the volatility products

Perfo

rman

ce (%

)

High diversification during quiet times (relatively low

volatility regimes), when it is less needed

During the recent crisis, traditionally de-correlated asset

classes moved in the same direction, hence reducing

diversification benefits

0

50

100

150

200

250

300

Jan-00 Aug-00 Mar-01 Oct-01 May-02 Dec-02 Jul-03 Feb-04 Sep-04 Apr-05 Nov-05 Jun-06 Jan-07 Aug-07 Mar-08 Oct-08 May-09 Dec-09 Jul-10

US Equities European Equities Hong Kong Equities US Real Estate US Bonds Commodities Diversified

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2005 2006 2007 2008 2009 2010

European Equities Hong Kong Equities US Real Estate Credit Commodities

14

Volatility: Historical and Implied volatility

The historical volatility, also called realized volatility, is calculated as the variance of

the assets return time series. As it is easy to calculate and based on the hypothesis that assets

return follow a normal distribution, in the past years portfolio managers have supposed that

the historical volatility can be used as a proxy of the future volatility. However academic

research have proved that this indicator is not relevant of the future assets volatility

(historical volatility can be a good proxy in bull markets but not in bear markets where the

global volatility levels will increase more and faster), that is why they then used local

volatilities in their models and now the trend is to use stochastic processes to simulate future

volatilities.

As mentioned earlier, we not tend to use local volatility and usually the realized

volatility is expressed in 30 or 60 days volatility: the annualized standard deviation (in the

coming lines we will consider the standard deviation and the variance as equivalent but it is

important not to forget to take the square root to change the variance in volatility) of the daily

returns during the last 30 or 60 days - it is important to use the log returns in order to be time

consistent.

Concerning the formula we commented it in the previous section Volatility: Standard

deviation and Variance - Formula (1). It is important to precise how we average the distances.

We have seen that the historical volatility is usually expressed as an unweighted average of

the last 30 or 60 days using this variance formula (7):

(7) 2n = 1N 1 (Xn1 X)2i=1n Where Xi is the value of the asset X at time i

Where N is the number of studied periods and then the number of distances

You could have expected an N as a denominator and not N-1 but as we only use some

distances (a sample of distances) then we are not averaging the whole population and

then it is necessary to use N-1 as un unbiased estimator

In practice, you can simplify the formula (7) by taking two simple assumptions: the

daily mean return of a stock is small enough to be replaced by 0 and you can use m

15

instead of m-1, then you are not using an unbiased estimator but a maximum

likelihood one. Then we have (8):

(8) 2n =1m U

2n1

i=1

n

There is much more to say concerning the implied volatility. Here we need to do a

small digression to the Black and Scholes PDE (Partial Differential Equation), derived from

the Merton mathematical model. The B&S PDE is a very easy and commonly used way to

price options.

We will start with the pricing formula of a Call option (9). The put formula is easily

deductible from the call formula and the object here is not to present the B&S model:

(9) C(S, t) = N(d1)S N(d2 )Ker(Tt )

Where d1 =ln( SK )+ (r +

2

2 )(T t) T t

Where d2 = d1 T t Where N(.) is the cumulative distribution function of a standard normal distribution

Where T t is the time to maturity

Where S is the spot price of the underlying

Where K is the strike price of the option

Where r is the risk free rate (expressed in annual continuous compounding)

Where is the expected volatility (plus a premium?)

The implied volatility level is the volatility allowing the match between the actual

price of the option on the market (so-called the Mark-to-Market, or MtM) and the formula

result (3). Market operators use the implied volatility level (calculated as described above) as

a proxy of the expected volatility, which is the best traders forecast of the volatility during

the rest of the options life (the consistency of this hypothesis over time is function of the

hypothesis used but B&S model is based on flat volatility hypothesis).

For an illustrative purpose we present in exhibit 4 a analogy between the bonds and

the options in order to give a more empirical understanding of what the volatilities

(historical and implied) are when it comes to deal with derivatives instruments. The exhibit is

extracted from (Derman, 2003).

16

Exhibit 4: Analogy between Bonds and Options, Yields and Volatilities 3

Bonds Options

Interest rates are the parameters people use

to quote bond prices.

Volatilities are the parameters people use to

quote options prices

Realized daily interest rates: actual short-

term interest rates

Realized daily volatility: the actual volatility

of an index

Yield to maturity of a bond: the average of

the future realized rates that make that bond

price fair. Its the implied yield based on

price.

Implied volatility of an option: the average

of future realized volatilities that make the

options price fair, based on Black-Scholes.

Forward rates: the future realized rates,

moment by moment, that must come to pass

to make current yields of all liquid bonds

fair.

Local (forward) volatilities: the future

realized index volatilities, index level by

index level and moment by moment, that

must come to pass to make current implied

volatilities fair.

It is interesting to study the relation between the historical and implied volatility.

Indeed if implied volatility is a good proxy of expected volatility (we use here an if because

it is easy to question the B&S model hypothesis) it is thought that if implied volatility does,

indeed, contain information in forecasting future realized volatility, then implied volatility

may be useful in predicting stock market returns. Consequently the relation between historical

and implied volatility is the key to forecast the latter and define a rewarding trading strategy.

Christina Chui4 demonstrates by statistical inference:

The correlation between the historical and the implied volatility for the S&P100 (using

the VIX for the implied volatility and the HVOEX for the historical volatility) and

Nasdaq100 (using the VXN for the implied volatility and the HVNDX for the

historical volatility) at the 1% level of significance

3 In (Derman, 2003) 4 In (Chiu, May 2002)

17

The non-significant difference between the historical and implied volatility means for

the two index at the 5% level of significance

She also observe in the data some widely accepted empirical regularities relating to

the time series behaviour of volatility: persistence effect, mean reversion, asymmetric

reaction of volatility to the market changes, global increase in volatility levels over

time

However if the means are not significantly different, the data compiled by Chui

present a implied volatility higher than the historical one. This empirical observation could

lead to the hypothesis of an overpricing of the options (due to the positive relation between

implied volatility and price). If Chui did not investigate the question due to the high level of

risk implied by naked sell of straddles (volatility sell), some more recent studies (Wallmeier

& Hafner, 2007) reveal a strong negative volatility risk premium: strong enough to generate a

positive return.

18

Volatility Index: example of the VIX

In this study we will focus on one major implied volatility index: the Chicago Board

Option Exchange (CBOE) Volatility S&P 500 Index (symbol: VIX). Introduced in 1993, it

measures the expected 30-days volatility implied by the SPX listed option prices. This choice

has been made based on the abundant academic research available on the index and for more

material reasons: it is the most traded volatility index avoiding to deal with major liquidity

limits in our study and due to the variety of derivative instruments available to get an

exposure to it. Actually the closest and more sensitive exposure we can have to the VIX is

through the 1-month forward.

Indeed if there are financial instruments giving an exposure to the VIX, by its

calculation methodology the index is not directly investable. The VIX is calculated by

averaging the weighted prices (by 1/K2, where K is the strike of the option) of SPX puts and

calls over a wide range of strike prices. Due to this calculation methodology (in particular the

weighting of each option price based on the strike), the puts contribute more to the index than

the calls: the smaller is K, the highest is 1/K2 and then the weight of this option in the index

calculation). Based on the same rational we can deduct that the more the put is OTM, the

more it will impact the index. The call/put impact split is approximately 1/3 for calls and 2/3

for puts (this result is obtained by measuring the sensitivity of the VIX to a 10% change in

volatility for the calls only and the puts only: the result is asymmetric and is in favour of the

puts).

If the VIX is not directly investable, the derivatives on the index know an exponential

increase in traded volumes as illustrated in the exhibit 5. The exhibit not only shows the

significant increase in traded volumes but also the timing of this boom: 2010 and 2011, just

after the last financial crisis. However the approach given to the VIX is different today than it

was in 2010. Then, the VIX was sold as a efficient hedge in case of market turmoil, but today

we know that this not true and the VIX is sold as a good diversifier. For example, in (BNP

Paribas, 2010) the VIX is presented with an embedded Delta to the S&P500 due to the

volatility skew of the options on the SPX. However there is no mention of the variability of

this delta (necessary in case of hedging).

19

Exhibit 5: Average daily volumes of traded VIX Futures and Options

Source: CFE and CBOE

In the Finance jargon the VIX index is known as the investor fear index and is a

contrarian indicator (cf. exhibit 6). As discussed previously, the implied volatility is a proxy

of the uncertainty and the calculation methodology makes it more sensitive to bear markets. It

leads the VIX to perform when the markets expect important losses. At this point the core

characteristic of the VIX we are looking for is its negative correlation, its inner negative delta

to the SPX: this negative correlation will provide us with the diversification effect.

The next question is obviously about the level of negative correlation in case of

important market change. The exhibit 6 shows that the global level of negative correlation in

crisis years (2008, 2010, 2011) has globally increased and then proves the efficiency of the

VIX in turmoil market. If we refer to the relation between asset correlation and diversification

effect studied in the section Correlation coefficient and diversification effect, it means that a

portfolio with an exposure to VIX will be well diversified even if the 2009 asset class

recorrelation happen again.

Previous studies (Dash and Moran in 2005 or Black in 2006) illustrate the potential

diversification benefits of adding spot VIX exposure to hedge fund portfolios. In addition to

the diversification benefits of spot VIX, they suggest that the skew and excess kurtosis of

many hedge fund strategies can be eliminated by a small long exposure to spot VIX (using the

VIX to reduce the fat tail effect). However we can regret that none of these studies indicate a

good entry point optimisation methodology. Moreover it is important to keep in mind the

important difference between the historical and implied volatility mentioned in Volatility:

historical and implied volatility due to the potential strong negative risk premium on

volatility (Wallmeier & Hafner, 2007) meaning a important cost of carry of VIX in portfolio.

20

Exhibit 6: The 10-Y daily returns and correlations of the S&P500 and VIX

Source: Bloomberg & CBOE Documentation

To conclude, for empirical reasons, the VIX will be our main source of volatility

levels and the underlying of the considered derivatives. Obviously the use of the VIX as

reference of implied volatility index involves, for our model, to use US assets in order to be

consistent: the S&P500 for the equity and US Bonds for the fixed income side. However the

readers interested in other markets, there is implied volatility index in most of the major

financial markets and they are investigated also like (Wallmeier & Hafner, 2007) with there

study on the DAX and the Eurostoxx500. The impossibility of investing directly the VIX will

also involve complication due to the instruments used to get exposure to it but we assume that

it is necessary to make this study useful for portfolio managers.

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21

Volatility: from the Smile to the Skew

One of the limits of the Black & Scholes model is commented here. Actually B&S

predicts that the implied volatility is a constant function of the strike or the time to maturity. If

this hypothesis was almost true before the 1987s crash, it is absolutely no longer true. The

so-called volatility smile is the characteristic variation of implied volatility in relation to the

strike and time to maturity. It appeared after the 1987 crash and is inconsistent with the

Black-Scholes model - in this model we should have a flat implied volatility.

Volatility and rate of return

There are widely accepted reasons of this volatility skew. Mark Rubinstein studied the

volatility skew first (and probably discovered this irregularity in the smile). He suggested a

crash-o-phobia phenomenon5. As the skew is observed since the crash of 87, he suggests

that the markets fear another crash and price this event by charging more implied volatility on

low strikes. The empirical analysis confirms this hypothesis as a bearish market makes this

skew steeper and a bullish market less steep. Because the traders have to sell when the stocks

go down to stay hedged and because at those levels the stop loss orders fast the market drop,

this empirical observation makes sense.

Another explanation is related to trading strategies used by market agents and there

risk aversion (a development to the crash-o-phobia hypothesis). When a stock fall, the

demand, and thus the price for it will decline but demand for puts on the stock will increase

(resulting in short-sales for delta-hedge by market operators). This rational suggests that

investors indirectly set the level of implied volatility, as investors demand for calls and puts

set prices, and these prices, in turn, are used to imply the level of the implied volatility

indicators.

Finally a more fundamental explanation involves the leverage of the equity. When the

equity value declines, the leverage ratio of the companies increases and therefore makes the

equity riskier and sensitive to the market moves (cf. levered beta concept). The consequence

is obviously a higher implied volatility. This observation makes the implied volatility a

contrarian indicator, as it will spike more frequently on bearish markets than in bullish ones.

5 Mentioned in (Rubinstein, 1994)

22

Volatility and Strike

Exhibit 7: From the volatility smile to the skew

In the case of an equity underlying, we do not have a volatility smile (first graph of the

exhibit 7) but a downward sloping curve also called volatility skew (second graph of the

exhibit). This asymmetric smile of volatility is the result of a non-lognormal probability

distribution of the underlyings return (remember that the Black and Scholes pricing model

involves a lognormal distribution of the assets prices). More precisely the volatility skew

presented in the second graph of the exhibit 7 is the result of a positively skewed distribution

of the underlyings return: this skewness will involve a heavier left tail and a less heavy right

tail, in other words a higher probability of loss.

The consequence of this probability distribution is that a deep-out-of-the-money call

will have a lower price than with a lognormal distribution (as the surface under the curve after

the strike is smaller than with the lognormal bell). The price difference will result in a reduced

implied volatility. This rational also works with the puts. If you consider a deep-out-of-the-

money put, then the option pays off if the underlying price is below the strike. As this surface

is bigger than with the lognormal distribution then the price will be higher and consequently

the implied volatility also.

If we make the assumption that the previously mentioned positive risk adjusted return

of shorting implied volatility, it means that the skewness of returns is too high in comparison

to the actual volatility of markets (do not forget that the implied volatility is used as a proxy

of the expected volatility). Nassim N. Thaleb could answer with its barbell strategy: pay the

VIX premium can appear as a loss but in case of Black Swan in the equity market? Protection

is a cost until it protects you, then it is an asset.

Implied(vo

la,lity

(

Strike(

Vola,lity(skew(Equity'op*ons'

Implied(vo

la,lity

(

Strike(

Vola,lity(smile((FX#op&ons#

23

Volatility and Time to Maturity

Exhibit 8: The mean-reversion of the volatility

Short-term volatilities are higher than long-term ones. This is true for both historical

and implied volatility (which is normal due to the relation between the two) and can be

observed in our long term VIX spot graph (cf. exhibit 6). It suggests the empirically observed

mean-reverting property of the volatility. In other words, the instantaneous volatility (the limit

of the implied volatility of an at-the-money option as its time to maturity approaches zero)

tends to be flat.

Samuelson did a similar hypothesis related to the futures, the so-called the Samuelson

Hypothesis. In 1965, he said: it is a well known rule of thumb that nearness to expiration

date involves greater variability or riskiness per hour of per day or per month than does

farness. This hypothesis involves:

The mean reversion of volatility suggests that the value of the volatility in the long run

is equal to the average/historical volatility level. In markets where Samuelson

Hypothesis holds, the historical volatility is good proxy for the long-term expected

volatility level but it does not say anything about the short-term volatility

It has a impact on asset pricing as the futures will be affected via the margin calls

(positive function of volatility), the more volatility we have the more the traders will

have to post collateral to hedge there positions. This collateral is obviously costly. The

options will be affected via the hedging costs, if the traders price higher volatilities

near the maturity date, then the options will be more expensive.

Implied(vo

la,lity

(

Time(to(Maturity(

Vola,lity(and(Time(to(maturity(All#op&ons#

24

Exposure to the VIX via swaps, futures, options

In this chapter we will present different wrappers allowing an exposure to the

volatility (via variance swaps) and to the VIX index (via VIX futures and options). We will

start with the var-swap because it is considered as the purer instrument to get an exposure to

the volatility. Then we will study the VIX Futures and Options because they are easily

tradable in the CBOE. If these products are not the only ones to give an exposure to the

volatility, they are the more widely used (cf. Volatility Index: example of the VIX) and will

allow us to get factual information and market data.

Variance Swaps (so-called Var-swaps)

A volatility swap allows the investor to bid on the increase (by taking the long

position) or decrease (by taking the short position) of the volatility, more precisely the

variance of the underlying. We consider K, the strike (solved in order to make the present

value of the contract equal to 0), the buyer will receive the Vega notional (because it is

expressed in USD) for each Vega (percentage point of the volatility) above the strike and pay

the Vega notional for each point below. The payoff formula is:

(10) Payoff = Nvar * ( 2realized K 2 )

Where Nvar is the notional allowing the contract to pay approximately the Vega amount for each volatility point

Nvar =Nvega2K and Nvega is the Vega notional. To demonstrate

this we can consider realized = K +1 (as we want to study the impact of 1 percentage point, 100bp, of the realized volatility on

the payoff) and solve the equation:

Payoff = Nvar * ( 2realized K 2 )

=Nvega2K * (K +1)

2 K 2"# $%

=Nvega2K * 2K +1[ ]

= Nvega +12K

And 12K is negligible in front of Nvega

25

Where 2realized is the realized variance of returns, or the squared volatility Where K 2 is the squared strike (because the strike is expressed in volatility) A var-swap also needs to define in the term sheet the following parameters:

The source of information (as mentioned earlier in the chapter

Volatility: Historical and implied volatility, there is different ways to

calculate the realized volatilities)

Observations frequency (we can have no move between to observation

points but important movements between the two, cf. Volatility and

time to maturity relation)

Annualization factor

Formula used for the standard deviation

Usual assumptions with var-swaps:

252 Business days per annum as annualization factor (explaining the

famous 1% daily vol = 16 on the trading floors, this is the result of

252 Daily mean return = 0 (in order to allow additivity of the contract, 3

month var-swap + 9 months var-swap in 3 months = 1Y var swap).

This is typically a point to consider as this assumption will impact the

estimated realized volatility

Convexity of the payoff:

Exhibit 9: Contrary to volatility, variance offers convexity

!3#

!2#

!1#

0#

1#

2#

3#

4#

5#

0# 10# 20# 30# 40# 50#

Payo

&

Vola)lity&

Convexity&of&var3swaps&in&vola)lity&(with&strike,&K=24)&

Vola-lity#payo# Var!swap#payo#

26

The exhibit 9 illustrates how the payoff of a variance swap is convex in

volatility. This means that an investor who is long a variance swap (i.e.

receiving realized variance and paying strike at maturity) will benefit

from boosted gains and discounted losses. This bias has a cost reflected

in a slightly higher strike than the fair volatility, a phenomenon that is

amplified when volatility skew is steep. Thus, the fair strike of a

variance swap is often in line with the implied volatility of the 90% put

VIX Futures

VIX products are a second way to get an exposure to implied volatility. Indeed, if the

var-swaps are replicated via a portfolio of options on the studied asset (e.g. a var-swap on the

S&P 500 is replicable via a portfolio of options on the index), the VIX products work

differently. They allow an exposure to the VIX, which is a measure of the implied volatility of

the S&P 500. Basically, if we could invest on the spot VIX, the performance at maturity of

the VIX and the squared root performance of the var-swap could be the same.

In this context, the first products are the VIX futures; they started to trade in March

2004 at the Chicago Futures Exchange (ticker: VX).

Multiplier: USD1000 and USD100 for the mini-VIX futures

Underlying: VIX SOQ (i.e. Special Opening Quotation). This

procedure returns the VIX forward level for the 30 days after the

settlement (based on the options available on the SPX on the settlement

date). Please note that this procedure does not reflect only the expected

volatility in 30 days but the whole term structure during the period.

Pricing: futures are quoted in points with a two decimals precision and

each point is valued at the multiplier

Price intervals: min. 0.05 points (corresponding to USD50 for the

futures and USD5 for the mini-VIX futures)

Settlement style: contrary to other futures underlying, VIX futures can

only be settlement by a cash payment

27

Settlement: The amount of this payment will be equal to the final

settlement VIX future value times the multiplier and has to be done the

following day (the clearing house system is used). Please not that the

futures trade on the expected level of the VIX and consequently there is

no strict push-to-spot effect but only a convergence to the spot. It

means that when the future settle, the level will probably not worth the

spot VIX due to the mean-reverting properties of it.

When an investor wants to maintain its exposure, he can roll his position (close its

present position and buy a new one in a later term). The problem with a long volatility

exposure is the frequent steep contango on the VIX term structure and the consequent cost of

this contango for the investor.

VIX Options (ticker: VRO)

The second wapper offered by CBOE are the options in 2006. You can find quotation

and buy VIX calls and puts on the market. The options on VIX have a particular pricing and

settlement in comparison to other options, there is no nominal but a multiplier, like for

indexes, the premiums and payoffs are function of this multiplier. The fundamentals

characteristics of these options are:

Multiplier: USD100

Strike intervals: min. 2.5 points

Available strikes: in/at/out of the money strikes are available with a

dynamic range (new strike can be issued if necessary)

Premium quotation: premium are in points and each point worth the

multiplier

Exercise style: European (can only be exercised at expiration date)

Settlement: The exercise-settlement value for VIX options shall be a

Special Opening Quotation (SOQ) of VIX calculated from the sequence

of opening prices of the options used to calculate the index on the

settlement date. Exercise will result in delivery of cash on the business

day following expiration. The exercise-settlement amount is equal to

the difference between the exercise-settlement value and the exercise

price of the option times the multiplier

28

The strength of VIX options are the leverage it allows in our exposition to the VIX

and the easy way it works. However if the value at expiration date is easy to understand, the

value of the option between the inception date and the expiration date is not equal to the

payoff value. The value at anytime is the result of different impacts and the global impact of

the different sensitivities can even engender a drop of the option price on the secondary

market when the VIX is in our favour but this is also true for puts protection so portfolio

managers should be able to deal with it.

29

Risk adjusted return: Sharpe ratio and some others

Sharpe ratio

The Nobel laureate William F. Sharpe gave its name to this ratio allowing a measure

of risk-adjusted performance. Please note that the Sharpe index refers to the same indicator.

The idea here is to measure how much extra-return a security or portfolio gives me for each

unit of risk. The Sharpe ratio formula is:

(11) Sharpe.ratio = r p rf p

Where r p is the expected portfolio return

Where rf is the risk-free rate

Where p is the standard deviation of the portfolio

The strength of this ratio was the possibility to compare portfolios easily (as all the

information we need is directly and easily computable from the observed series of return).

Indeed, two portfolios can have different returns but the real question is how much risk (and

we all know risk is money) the extra-return will cost? Obviously the best investment is the

one with the higher Sharpe ratio, meaning that for each unit of risk the portfolio provide a

maximum extra-return to the risk-free rate, and a negative result will mean that risk-free

securities (or portfolio) is the best.

However a major weakness of this indicator is it requires normally distributed returns

(otherwise the standard deviation could be a highly inefficient measure of risk). Indeed we

have presented the standard deviation we commented that a major weakness of this measure is

the insensitivity to the sign. Therefore applied to a distribution of returns, the standard

deviation gives us accurate information only if the distribution is symmetrical to the mean.

30

Lelands Alpha and Beta

Lelands indicator where presented in an article with a very meaningful title Beyond

Mean-Variance: Performance measurement in a non-symmetrical world, (Leland, 1999),

based on an interesting assumption: market returns are normal but not securities or portfolios

returns. Leland presents an enhancement of the Jensen Alpha (as the Jensen Alpha is based on

the CAPM and this model suppose that investors only care about mean and standard deviation

and by that rejects the eventual preference for positive skew and kurtosis) allowing the

preference for low kurtosis and positively skewed distributions.

To determine the Lelands Alpha we first need to compute the market price of risk

(noted b here) which, if market returns are normal, worth:

(14) b = log E(1+ rmkt )[ ] log(1+ rf )var log(1+ rmkt )[ ]

Where rmkt is the return of the market (used as benchmark)

Where rf is the risk-free rate From that point we can compute the Lelands Beta: (15) Leland = cov rp,(1+ rmkt )b"# $%cov rmkt,(1+ rmkt )b"# $% Thus, the Lelands Alpha: (16) Leland = E rp!" #$Leland E(rmkt ) rf!" #$ rf Leland concludes as following: for assets or portfolios whose returns are jointly lognormal with the market, the differences between the correct beta and the CAPM beta are small and the mismeasurement of alpha is similarly small. For portfolio or asset returns that are highly skewed, the correct beta differs substantially from the CAPM beta. Thus, using the correct beta is critical for correct performance measurement of investment strategies that use options, market timing, or other dynamic strategies.

31

Stutzer ratio

Michael Stutzer in (Stutzer, 2000) presented also a new measure of risk-adjusted

returns robust to non-normal distributions of returns. For him, a portfolio manager motivation

is to exceed the returns of a particular benchmark (which is confirmed by the remuneration

policy of asset managers and in particular hedge-fund managers). Provided that his portfolio

expected return is greater than that of the benchmark, the probability to underperform the

benchmark decays over time to zero. Provided that we would like to be sure of that, we can

use the rate at which the probability of underperforming decays to zero as a measure of the

portfolio manager performance. In other words, the faster the portfolio manager reduces our

risk to underperform the benchmark the better he is. Stutzer captures the information statistic

via the following formula:

(12) Where is the excess return of the portfolio on the benchmark over time (e.g. we

measure the outperformance of the portfolio on the market)

Where is chosen to maximize . In this study we have defined this factor via an

Excel macro computing 1000 iterations. The calculation of this indicator is highly

calculation power consuming (e.g. 10 minutes on a Intel core i5 1.7Ghz). Which

probably explain why this indicator is not much popular.

(13) Where is the mean excess return

Where is the absolute value of the mean excess return Actually the strength another strength of the Stutzer index is that if the distribution is

normal, and then it will worth the Sharpe ratio! Otherwise the Stutzer index will penalize high

kurtosis and negative skewness.

Indeed, regarding the Sharpe ratio we will have:

and consequently Where is the Sharpe ratio of the portfolio

I p =Max log1T e

rt

t=1

T

#

$%

&

'(

)

*+

,

-.

rt

I p

StutzerIndex =rr 2I p

r

r

I p =12 p

2 StutzerIndex = p

p

32

III Methodology

The goal of this research paper is to try to determine whether or not the volatility can

be considered as a new asset class and if an investment strategy using it to diversify a

portfolio of risky assets will allow a better risk-adjusted return (in particular in crisis periods).

The underlying hypothesis is that the negative correlation of volatility to the other asset

classes (observed in exhibit 3 and 6) will allow a better diversification but also an efficient

protection against Black Swans. This research will require an inductive approach: we will

design different portfolios and analyse them thanks to risk-adjusted indicators in order to

determine what investment strategy has the better risk adjusted return profile.

Description of data

Equity exposure:

The Standard and Poors 500 Index is widely regarded as the best single gauge of the

U.S. equities market, this world-renowned index includes 500 leading companies in leading

industries of the U.S. economy. Although the S&P 500 focuses on the large cap segment of

the market, with approximately 75% coverage of U.S. equities, it is also an ideal proxy for the

total market. Then, we have decided to consider an exposure to the SPX for the equity pocket

of our portfolio.

At this stage we faced a difficulty to manage the dividends paid by the index as the

SPX is adjusted to reflect the payment of the dividends to the investors (originating an

underestimation of the returns as our portfolio strategy suppose to reinvest them). Please note

that we passed through this difficulty by using the S&P 500 Total Return index. The Total

Return version of the SPX simplifies the management of the dividends by making the

hypothesis that the investor reinvests the dividends (after tax, based on the hypothesis of an

US investor) in the SPX. This hypothesis fits better to our model and do not impact

significantly our monthly rebalancing strategy. Our SPXTR time series are based on the daily-

close returns extracted from the CRSP (Center for Research in Security Prices) via the

Wharton Research Data Service.

33

Volatility exposure:

As mentioned earlier, this study will focus on the VIX as implied volatility indicator.

This is to be consistent with the equity pocket of our portfolio but also because the VIX

presents major strengths: it is investable, has a large span of securities using it and presents a

good liquidity. Please remember that if the spot VIX is not directly investable, we have seen

in Exposure to VIX via swaps, futures and options what the more frequent VIX instruments

are.

We have decided here to use VIX futures: in particular we have used the S&P 500

VIX Mid-Term Futures Index. Indeed this index offers to investors directional exposure to

volatility through publicly traded futures markets, and seeks to model the outcome of holding

a long position in VIX futures contracts. This index allowed us to integrate a large part of the

transaction costs for the volatility pocket and to integrate the rolling strategy which is out of

our scope but represent a real issue with VIX futures: as commented earlier, the steepness of

the volatility term-structure generates an important rolling cost. For the same reason we have

decided to use the VIX mid-term futures instead of the VIX short-term futures (the rolling

cost is less important as the steepness of the slope is decreasing for the farer terms).

The liquidity is important for our study because it is measured via the bid/ask spread

and we have not integrated it in our model. The more liquid is the index, the smaller is the

bid/ask spread, the smaller is the over performance implied by this hypothesis. Consequently,

as the VIX presents a good liquidity (as shown in exhibit 5), it avoids us the difficulties and

transactions costs that a liquidity management could imply for the investors (this good

liquidity parameters have allowed us to suppose a reduced bias in the study by not including

the bid/ask spread in the returns). Note that we did not include it because we have not found

the appropriate time series. Our VIX time series are the daily-close levels extracted from

Bloomberg.

Bonds exposure:

With the pari passu rational about the coherence between the asset classes, we

needed to use US bonds for the fixed income exposure. The idea was to avoid a debtor

picking bias and consequently we have decided to choose US Government 10Y Bonds to

bench the bond asset class performance (which is a classic choice for this asset class). The

10 years maturity is due to our investment horizon, if we are analysing a long-term period, it

makes sense to use long-term maturities. Moreover, it allows us to maintain an important

34

duration in the portfolio without a complex bond investment strategy. The 10 years bond

yields are extracted from the U.S. Department of the Treasury database.

To finish, all the levels, prices and yields are based on daily-close time-series covering

an almost eight years period starting in January the 1st 2004 and ending in August the 31st

2012. The time horizon was limited in the past by the creation of the firsts derivatives

instruments on the VIX.

Analysis methodology

If the academic and theoretical aspects are part of our investigations, this study targets

to be useful for a finance industry professional with no experience in volatility products. The

thesis should be an answer to:

How volatility, if considered as a new asset class, will impact in terms of return

and risk a portfolio of risky assets?

The first hypothesis involved in this paper is that volatility is an asset class. But

volatility is most widely known as a measure of risk, that is why the first step is explain why

we can consider that the volatility, whatever the underlying is, can be considered as an asset

class meanwhile there are products allowing exposure to it. We tried in the previous Asset

classes section of this paper to define a framework allowing us to define if a set of securities

is an asset class or not. The framework we have defined takes into account:

The economic sense of the supposed asset class

The tradability of the securities (this is a very empirical approach but

really making sense to us. Moreover this approach raise other

interesting questions:

The question of returns measure

The existence of the securities and the market access

The liquidity question

The homogeneous risk and return profile of the considered securities

This hypothesis is more an academic/qualitative question. Answer to it will not have

an impact on the financial market industry and the implied strategy. Some

authors/professionals will consider this hidden asset class like a sub-asset class some others

like an investment strategy etc. The debate is open and our goal is not to close it but to design

a tool allowing the synthesis of what we have found on the topic.

35

The second hypothesis is that volatility is a good portfolio diversifier and improves the

return of a portfolio for a given level of risk. Our approach here is to simulate different

portfolios (with different assets exposure and different weights for each asset classes). We

have used 3 asset classes: volatility, equity and bonds. First, we will analyse two traditional

portfolios and then test different volatility exposure levels. The portfolio will be rebalanced to

maintain the chosen asset class weights the first working day of each month.

To resume the properties of our portfolios will be:

Traditional portfolio 1: 100% invested in 10Y bonds

Traditional portfolio 2: 100% invested in equity (in the SPXTR)

Traditional portfolio 3: 60% equity in SPXTR and 40% in 10Y bonds

For the volatility exposure, we will try a 5% and 10% long asset

allocation to the VIX trough the S&P Mid-term Futures Index

The transaction costs are not captured in the study due to the absence of

information but could be easily captured via a simple bid/ask spread

(not computed here due to the lack of information)

The weights in the portfolio are arbitrary and based on different studies we have used

for this paper. The reason is that during crisis periods the only performing asset are the

volatility products and the optimisation gives us a 100% of VIX products, which is a non-

sense and/or out of the scope of our study. Then we will compare the risk and return profiles

of the different portfolios over the long run.

The third hypothesis is about the efficiency of using variance to hedge a portfolio

against a Black Swan. Here we will suppose a dynamic exposure to the VIX in order to avoid

paying the negative risk adjusted premium of the volatility. The gearing optimization is, in

our opinion, out of the scope of this study and consequently we will consider a flat 10%

exposure to the VIX based on the distance between a moving-average of the VIX and the spot

VIX to trigger the exposure. Then we will implement the same analysis of the risk and return

than for the second hypothesis.

36

IV - Results

Hypothesis 1: Volatility is a new asset class

Hypothesis origin

Asset class and diversification are two portfolio management concepts useful when it

comes to control the risk carried by a portfolio (please find more details on these two concepts

in the previous sections Asset classes and Correlation and diversification effect). For years,

markets have split the two concepts and many industry professionals have considered, for

example, a smart stock picking were enough to protect a portfolio against systematic risk (by

opposition to the idiosyncratic risk which is not diversifiable). However if this diversification

scenario, based on a single asset class, may be rewarding in terms of return, it also bears

hidden risks like a potential increase in assets correlation or a change in the risk-adjusted

return.

They have based their diversification on a simplistic interpretation of the

diversification as described on the exhibit 2: if I have enough different securities in my

portfolio, with a good allocation between smid and large caps, with a mix between defensive

and pro-cyclical stocks and an exposure to international equity, then Im diversified and I

receive the equity required rate of return. However in a single asset class, it has been proven

that it exists some hidden correlations between securities and these correlations became more

visible in bear markets (when you need the most the diversification benefits!). Moreover the

following exhibit 10 illustrates the same phenomenon between many asset classes during the

financial crisis.

Exhibit 10: Asset class correlation in normal and crisis periods

Exhibit'10'*'Asset'class'correlations'of'2004'to'2006'(2007'to'2008)Equity Bonds High/yield/bonds Hedge/funds Commodities Private/equity Real/Estate

FY04/to/FY06

FY07/to/FY08

FY04/to/FY06

FY07/to/FY08

FY04/to/FY06

FY07/to/FY08

FY04/to/FY06

FY07/to/FY08

FY04/to/FY06

FY07/to/FY08

FY04/to/FY06

FY07/to/FY08

FY04/to/FY06

FY07/to/FY08

Equity 1.00//// 1.00////Bonds 0.02//// 0.22//// 1.00//// 1.00////High/yield/bonds 0.51//// 0.87//// 0.39//// 0.40//// 1.00//// 1.00////Hedge/funds 0.77//// 0.80//// (0.01)/// 0.18//// 0.54//// 0.74//// 1.00//// 1.00////Commodities (0.22)/// 0.52//// (0.06)/// 0.00 (0.07)/// 0.55//// 0.06//// 0.80//// 1.00//// 1.00////Private/equity 0.77//// 0.84//// 0.28//// (0.01)/// 0.61//// 0.80//// 0.77//// 0.75//// (0.23)/// 0.65//// 1.00//// 1.00////Real/Estate 0.56//// 0.85//// 0.46//// 0.31//// 0.44//// 0.91//// 0.47//// 0.63//// (0.19)/// 0.44//// 0.61//// 0.85//// 1.00//// 1.00////

37

The investigated hypotheses explaining the increasing correlation levels are in

particular:

Between international stock exchanges: the globalization and

increasing integration of economies are favouring the multi-national

companies and this phenomenon linked to the free movement of capital

is correlating equities around the world. The stock exchanges mergers

and concentration process to ease the international market access to

investors is probably in cause as well

Equities and bonds: if these two asset classes were traditionally the

mainstay of asset class diversification, it is possible that the increasing

relationship between investment banking and structured financing

could be the cause. Another hypothesis is based on the development of

the hedge fund industry and the growing success of the cross-asset

financial products. A loss in a position could generate a cash pulling via

the margin/collateral calls and force to close other positions and affect

the related asset classes

The research on these hidden risks has allowed the creation of securities giving an

exposure to them. Here comes the divergences: do we simply have to consider these sets of

securities as derivatives because their value is derived from other securities or we have to

try take into account the specific economical sense of these securities and finally split the

derivative family into different asset classes. In other words, the hypothesis we are doing

here is the prominence of the substance over the form.

Demonstration of the hypothesis

As mentioned in the methodology, in order to test this hypothesis, we have thought in

the section Asset Classes a framework defining a set of criteria we have found during our

research. The framework has been described earlier and we will not present it again, we will

focus here in applying it to the implied volatility in order to test whether or not it can be

considered as an asset class, a hidden asset class where the derivatives only allow an exposure

to it.

38

We have decided to consider the implied volatility and not the historical volatility for

two main reasons:

In our opinion financial asset have to be oriented to the future (as most

of them are valued to the actualized value of future cash flows).

Consequently, we consider the implied volatility (which, as we have

demonstrated in Volatility: Historical and Implied volatility, is a good

proxy of the expected volatility) is the more appropriate volatility for

this hypothesis

We also do it in order to be consistent with the two following

hypothesis of this research paper: analysing the impact of adding VIX

exposure in different portfolios (and we have seen that the VIX is a

measure of the implied volatility of the SPX)

Criterion 1: The economic sense of volatility

If we have to analyse the philosophy of volatility products and in particular the

philosophy of the VIX, then we have to remind the relation between the VIX and SPX. We

have seen that the VIX, a measure of the SPX implied volatility, is also called the market fear

gauge. This nickname of the VIX resumes the economical sense of volatility: it measures

the expected short-term level of risk on the market. The exhibit 6 presented in the previous

pages illustrates the negative relation between the two indexes: the VIX has a negative inner

delta to the SPX. Moreover the way the VIX is calculated makes the index not symmetric to

the SPX returns, it is more sensitive to market downturns: where the risk really is.

The underlying question then is the economical sense of the SPX. As commented

earlier, the S&P 500 Index measures the changes of the Top500 market caps in the US taking

into account the different sectors of the economy. Today, the index covers more than 75% of

US equities and is the most important equity index in the country (more than the DJIA which

presented the defect of being equiweighted and only presenting the Top30 market caps). By

giving an economical sense to the VIX (and the implied volatility in general), as a meaure of

risk of the underlying, we are assuming that the SPX does reflect the economical health of the

country. At this point we have a debate between people saying that the markets do not

measure the economic health and the opponents saying it does.

39

In our opinion this debate is out of the scope of this research paper and in particular in

this part of the paper and we will only argue with a Warren Buffet quotation6 the market

value of all publicly traded securities as a percentage of the countrys business that is, as a

percentage of GNP has certain limitations in telling you what you need to know. Still, it is

probably the best single measure of where valuations stand at any given moment. And as you

can see, nearly two years ago the ratio rose to an unprecedented level. That should have been

a very strong warning signal.

At the end of the day, the market value of a company is supposed to be equal to the

present value of the future cash flows it will generate and an equity index like the SPX

measure the change of this sum of present values. If the market can be irrational and/or

manipulated, in the long run, they do reflect the health of economy. As a consequence, if the

SPX has a strong economical sense, then necessarily the implied risk of it, the VIX, has

also an economical sense: it measures the vision of the market on the market/economy

health. However it is important to note that there is a potential limitation to the hypothesis

here: the underlying must have a strong economical sense to make this hypothesis consistent

based on our framework.

Criterion 2: The tradability of the securities including:

The question of returns measure

Here we have to distinguish two kinds of instruments: the publicly traded securities

and the securities traded OTC. In the first category we will have the VIX futures, options and

ETN. For this category of securities, there is no problem of returns measure. The VIX is an

index with a public calculation methodology and the instruments mentioned have a value

based on the VIX times a multiplier. The returns of that category of securities are not subject

to any calculation difficulty.

For the OTC securities, the securities are made and priced on demand. In this case the

bank and the client can agree on a particular return calculation. Moreover, for the variance

swaps in particular, the calculation of the realized volatility can be subject to discussion.

Different formulas can apply as commented in Volatility: Historical and Implied volatility

(e.g. the difference between the formula 7 and 8). The banks also smooth them margins

engendering a difference between the actual price of the product and the expected price. 6 In the Fortunes, an interview of Carol Loomies

40

In the mean time this differences have a limited impact on the payoffs of the securities

and are more subject to impact the rounding and the price of the products on the secondary

market than the approach of the return measure. As a consequence, if there are some

discrepancies in return calculations between securities, the return calculation is feasible and

globally homogeneous.

The existence of the securities and the market access

The principal products allowing an exposure to volatility have been presented in the

previous section Exposure to the VIX via swaps, futures, options. The question of the

existence and tradability of the securities is then answered in the state of the art of this

research paper. However we have to note that the complexity of the product reduces the

span of the potential clients and market regulations may require an accreditation of the

investors.

The remaining question is about the market access: this question is closely related to

the previously mentioned point of investor accreditation. We are here talking about very

specific products and consequently they are not as easy to access than equities or even bonds

but the CBOE is a large stock exchange with large coverage and lot of brokers allow at good

price a market access. Consequently we can then consider that the market access is good

and not difficult for market operators allowed to trade such products.

The liquidity question.

As mentioned earlier we have decided to do not consider the question of the liquidity

in this study. However the figures of the exhibit 5 shows a large and growing volumes for

volatility products.

Criterion 3: The homogeneous risk and return profile of the considered securities

The first question here is to know if the securities we are considering for an exposure

to the implied volatility. As mentioned earlier we do not have data series for options and

swaps but it is interesting to study as an example the different terms strategies for futures. The

exhibit 11 illustrates how different are the returns over the long term depending of the chosen

term however the profile (i.e. the directionality) of the series are similar. The difference

between the series are essentially due to the rolling cost.

41

Exhibit 11: Returns of different VIX futures term

If we have seen with the previous exhibit that we can have different returns but a

common directional exposure to the VIX, do these securities have a homogeneous risk/return

profile? If this question is close to the previous one, we will focus here more on the payoff.

Indeed what make the shares and bonds common families of asset; it is also the common

philosophy of the products (and not only of the underlying). Part of it is the payoff. At this

point we are touching a limit of our hypothesis: as we have different derivatives products

allowing an exposure to the implied volatility with extremely different philosophies (the VIX

futures pays the expected volatility when the var-swaps at maturity pays the different between

the implied and realized volatility). This point avoids considering all volatility products as

a homogeneous asset class but this result, in our opinion, only weakens our results and

not invalidates them.

As a conclusion, volatility (for the purpose of this study the SPX volatility) satisfies

most of the defined criteria required by our framework and then we can conclude that the

volatility of the major indexes can be considered as a de facto new asset class. We are

convinced that if the wrappers can weakens our conclusions, at the end of the day the driver

of the products returns (and underlying risk) is the level of risk measured by the market:

which is a common base to all these products. Then we cannot reject this first hypothesis

and it drives us to our second hypothesis: what is the impact of this new asset class on a

traditional portfolio?

!"!!!!

!100.00!!

!200.00!!

!300.00!!

!400.00!!

!500.00!!

!600.00!!

!700.00!!

!800.00!!

1/3/06! 1/3/07! 1/3/08! 1/3/09! 1/3/10! 1/3/11! 1/3/12! 1/3/13!

VIX$Returns$

S&P!500!VIX!ST!Futures!Index! S&P!500!VIX!MT!Futures!Index!

VIX!Spot!

42

Hypothesis 2: Volatility improves risk-adjusted returns

Hypothesis origin

As commented earlier, the volatility presents a negative inner delta to its underlying.

This negative correlation between the two could allow a portfolio manager to consider the

volatility as an equity hedge. Unfortunately, in the exhibit 6, we have highlighted a varying

level of correlation between the VIX and the SPX. This particular relation was also observed

in other indexes as demonstrated in (Wallmeier & Hafner, 2007) with the Deutscher

Aktienindex (DAX) and EuroStoxx50 index (ESX). Based on this observation, using the

volatility as an equity hedge is challenging, as the correlations between the implied volatility

index and the underlying index are conditional and not stable over time (cf. exhibit 6), but a

selectively applied long volatility position may provide significant diversification benefits,

particularly in times when the diversification benefits of other assets break down, such as in

the last two quarters of 2008.7

The question here is to measure this gain, if there actually is a gain, to invest part of

the portfolio in VIX instruments? But in finance industry we know (or should know) that

gains come with risks and then it is important for us to determine the impact on the

distribution of the portfolio return.

Demonstration of the hypothesis

Step 1: Determine a benchmark

As presented in the methodology, at this stage we have decided to model classic

portfolios to get an initial benchmark to compare in a second time the VIX impact.

Full bond portfolio: we have invested the portfolio nominal in a US

10Y bond with a total return assumption: we reinvest in bonds the

coupons received and simulate the impact of the yield change over

time. Please note that we do not have considered the management of

the portfolio duration here

7 Quotation from (Szado, 2009)

43

Full equity portfolio: we have invested the portfolio nominal in the total

return SPX. This allows getting an equity-diversified portfolio

Mixed portfolio: supposing 60% of equity and 40% of bonds, which is based on the same assumptions than the two previously presented

portfolios

Exhibit 12: Classic portfolio returns and distributions

(8) Note: The exhibit returns the following ranking (based on the Sharpe ratio and skew): 1st Mix, 2nd Full

Bonds, and 3rd Full Equity

To begin please note that these results come from a sample of 1,698 observations of

returns on a daily basis between January, 3rd 2006 and September, 28th 2012. This large

sample allows us to consider it as significant in regard to the analysed underlying. This is

particularly true when we know that the VIX futures started to trade in 2004.

8 The risk-free rate is the average 3 months US Treasuries rate over the period.

Portfolios)modelling)/)Distribution)analysis

ReturnSt.devCAGRSt.dev.p.aKurtosisSkewnessSharpe.ratio

Portfolios)modelling)/)Distribution)analysisMix0.02%0.79%5.56%

12.54%7.10...........2.74...........0.41...........

Portfolios)modelling)/)Distribution)analysisFull.Equity Full.Bonds

0.03% 0.02%1.48% 0.65%7.11% 4.62%

23.51% 10.36%3.68........... 6.95...........2.07........... 2.65...........0.29........... 0.41...........

600000#

1000000#

1400000#

1/3/06

#

1/3/07

#

1/3/08

#

1/3/09

#

1/3/10

#

1/3/11

#

1/3/12

#

Por$olios(performance(

Mix# Full#Equity# Full#Bonds#

0"0.02"0.04"0.06"0.08"0.1"

0.12"

)5.4%"

)4.8%"

)4.2%"

)3.6%"

)3.0%"

)2.4%"

)1.8%"

)1.2%"

)0.6%"

0.0%

"0.6%

"1.2%

"1.8%

"2.4%

"3.0%

"3.6%

"4.2%

"4.8%

"5.4%

"

Returns')distribu-on)

Mix" Full"equity" Full"bonds"

44

Another important note is about the rebalancing date. At this stage we have tried to

analyse the impact of different dates on the mixed portfolio (trying the 2nd and 3rd Monday of

the month) without any substantial difference. Indeed the idea was to be sure we are not

biasing our model by choosing an arbitrary rebalancing date (the 1st Monday of each month).

The bigger impact we have found is on September, 16th 2011 with a difference of USD27,243

for a portfolio value ranging from USD1,189,707 and USD 1,216,950 (c. 2.3% of the nominal). We have decided to mention this spread for each portfolio but not to analyse each

underlying portfolio as we consider this spread negligible (it is 2.3% in 6 years, the risk free

rate).

The first observation we can do based on the exhibit 12 is the non-normal distribution

of the returns. We all know that for years the finance literature has considered asset returns

normally distributed. However, nowadays this hypothesis is known as unrealistic but still in

use through widely used models and indicators based on it. For example the Sharpe ratio (and

the derived indicators like the Roy Safety ratio, etc., only works for normally distributed (or at

least symmetric distributions). Here we have a limited but real skewness and a high kurtosis

(ranging from around 4 for the full equity portfolio to more than 7 for the others). With such

returns we can consider the Sharpe ratio biased: that is why we have introduced the Stutzer

index and the Leland alpha.

The other observations we can do based on these usual indicators are mentioned

here following:

The CAGR of the full equity portfolio is the highest and the bonds have

a more limited return. This result is not surprising when we consider

the potential risks of each investment position. The mixed portfolio has

presented a return in the resulting range (which is normal as the

portfolio is a mix of the two)

The same observations can be done for the annualized standard

deviations

The interesting result here is the outperformance of the mixed portfolio

in terms of risk-adjusted return perfectly illustrating the diversification

gain! Indeed the Sharpe ratio of the mixed portfolio illustrates a

maximized return per each unit of risk carried by the investor

45

Step 2: The impact of an exposure to the VIX

First we have determined and analysed the benchmark portfolios. We can now

simulate the impact of an exposure to the VIX. F