Correlation Swaps 7

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First draft: 20 February 2007This version: 9 May 2007

Sebastien [email protected] Structuring ECD London

A New Approach For Modelling and PricingCorrelation Swaps


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Contents

Section Page:

Introduction 1

1. Fundamentals of index volatility, constituent volatility, correlation and dispersion 2

1.1. Definitions 2

1.2. Proxy formulas for realised and implied correlation 5

1.3. Variance dispersion trades 7

2. Toy model for derivatives on realised variance 10

2.1. Model framework 10

2.2. Fair value of derivatives on realised variance. Volatility claims. 11

2.3. Parameter estimation 12

2.4. Model limitations 12

3. Rational pricing of equity correlation swaps 13

3.1. Two-factor toy model 14

3.2. Fair value of the correlation claim 14

3.3. Parameter estimation 15

3.4. Hedging strategy 16

3.5. Model limitations 16

4. Further research 18

5. References 19

6. Appendices 20


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2

Fundamentals of index volatility, constituent volatility,correlation and dispersion

A New Approach For Modelling and Pricing Correlation Swaps

Introduction

This report carries forward an earlier work (2005) on the arbitrage pricing of correlation swaps on a stock index. Thetheoretical derivations have been made more rigorous, and we also include tentative parameter estimates based ona break-even historical analysis. Our aim here is to provide some elementary fundamental and practical results,which may serve as guiding principles and rules of thumbs for a new category of derivatives on realised volatility andcorrelation. These statistical derivatives have grown in popularity over the past few years, giving sophisticatedinvestors the opportunity to take advantage of specific market structural imbalances.

Correlation swaps are over-the-counter derivative instruments allowing to trade the observed correlation betweenthe returns of several assets, against a pre-agreed price. In the equity derivatives sphere, these contracts appearedin the early 2000s as a means to hedge the parametric risk exposure of exotic desks to changes in correlation.Exotic derivatives indeed frequently involve multiple assets, and their valuation requires a correlation matrix as inputparameter. Unlike volatility, whose implied levels have become observable due to the development of listed option

markets, implied correlation coefficients are unobservable, which makes the pricing of correlation swaps a perfectexample of chicken-egg problem. We show how a correlation swap on the constituent stocks of an index can beviewed as a simple derivative on two types of tradable variance the square of volatility , and derive an analyticalformula for its fair value relying upon dynamic trading of these instruments.

The report is organised as follows. Section 1 gives precise definitions of the concepts of realised and impliedvolatility of an index and its constituent stocks, realised and implied dispersion as well as realised and impliedcorrelation; some key mathematical properties and practical applications are then introduced. Section 2 proposes aone-factor toy model for derivatives on realised variance which is a straightforward modification of the Black-Scholes (1973) model; an analytical formula for the fair value of volatility (as opposed to variance) is then derivedand used for parameter estimation. Section 3 extends the toy model to two factors in order to derive an analyticalformula for the fair value of realised correlation; our numerical results suggest that the fair value of a correlationswap should be close to implied correlation; finally, a formal link between dispersion trading and the hedging

strategy for correlation is established.

1. Fundamentals of index volatility, constituent volatility, correlation and dispersion

1.1. Definitions

Consider a universe ofNstocks S = (Si)i=1..N, and a vector of positive real numbers w= (wi)i=1..Nsuch that1i=1..Nwi=

1. Denote Si(t) the price of stock Siat time t, with convention S(0) = 1, and define their geometric average as:

(110) ( )=

N

i

w

iitStI

1

)()(

From an econometric point of view, (S, w, I) is a simplified system2 for the calculation of a stock index I withconstituent stocks S and weights w.

We complete the quantitative setup by considering a probability space (, E, P) with a P-filtration F, and assumingthat the vectorSof stock prices is an F-adapted, positive Ito process.

1The positivity and unit sum conditions imply N 2.

2In practice, most stock indexes are defined as an arithmetic weighted average, with weights corresponding to

market capitalisations; as such, weights change continuously with stock prices. Additionally the constituent stocks

are typically reviewed on a quarterly or annual basis.


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3



Given a time period3and a positive Ito processX, define:

(111) | | ds

(112)1 2( ) ( ln )

X

sd X

(113) ( )2

1

( ) ( )iN

SS

i

i

w =

(114) ( )2

2

1

( ) ( )iN

S

i

i

w =

From an econometric point of view, (111) is the length of a time period, (112) is the continuously sampled realisedvolatility of any positive Ito processX(in particular that of the constituent stock prices S, and the index value I), (113)is the continuously sampled average realised volatility of the constituent stocks

4, and (114) corresponds to a

realised residual quantity useful below.

It is easy to see that ( ) ( )S I , and )()( >S . Define:

(115) ( ) ( )2 2

( ) ( ) ( )S Id

(116)( ) ( )

( ) ( )

2 2

2 2

( ) ( )( ) 1

( ) ( )

I

S

From an econometric point of view, (115) is the continuously sampled average realised dispersion5

betweenconstituent stocks, and (116) is their continuously sampled average realised correlation

6. (116) is consistent with

usual econometric and market practices (see for instance Skintzi-Refenes7, 2005), and we refer to it as canonical

realised correlation.

3Here a time period may be a segment such as [t0, t1], or a finite reunion of segments.

4Note that (113) corresponds to the canonical quadratic norm, whereas constituent volatility is more frequently

defined as the weighted arithmetic average of realised stock volatilities. Our choice is motivated by the economic

fact that only variance has a liquid market.

5To see this clearly rewrite:

2

12

1 1

( ) ln ( ) ln ( )N N

i i k k t

i k

d w d S t w d S t

= =

=

6To see this clearly observe that:

( )

,

2 2

22

1

( )

i j

i

S S

i jIi j

N Si j

ii i jw

=

=

, where iii w and

1, 1( ) ( ) ( ln )( ln )i j jiS S SS

i jd S d S

.

7 Note that Skintzi-Refenes implicitly define )( S as the weighted arithmetic average of constituent volatilities.


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4



We now consider a variance market on (S, w, I) where at any point in time tagents can buy and sell future realisedvariance

8

2over any given period , against payment at maturity

9of a pre-agreed price

10called implied variance

11

and denoted *2

. In the absence of arbitrage, this means that there exists a P-equivalent, F-adapted measure P*

such that forX= IorX= Si:

(117) ( )2

* *( ) ( )X Xt tE =

where Et*denotes conditional expectation under P

*with respect to Ft.

Additionally, define implied constituent volatility and implied residual as:

(118) ( ) ( )2 2

* * *

1

( ) ( ) ( )iN

SS S

t i t t

i

w E

=

=

(119) ( ) ( )2 2* 2 * *

1

( ) ( ) ( )iN

S

t i t t

i

w E =

=

From an economic point of view, (118) and (119) correspond to the unique no-arbitrage price of a portfolio of the Nfuture realised variances of the constituent stocks, with weights wand (wi

2)i=1..N, respectively.

No arbitrage considerations imply that )()( ** I

t

S

t , and )()(** t

S

t > . Define:

(120) ( ) ( ) ( )2 2 2* * * *( ) ( ) ( ) ( )S It t t t d E d =

(121) ( ) ( )( ) ( )

2 2* *

*

2 2* *

( ) ( )( ) 1( ) ( )

I

t t

tS

t t

We refer to (120) as implied dispersion and (121) as canonical implied correlation.

Here, we must emphasise that, while (120) corresponds to the no-arbitrage price of realised dispersion as defined in(115), (121) does not necessarily correspond to the fair value of realised correlation as defined in (116): in general,

* *( ) ( ( ))t tE . It is the aim of Section 3 to bridge the gap between implied correlation and the fair value offuture realised correlation.

8Throughout this report, variance means the square of volatility, and volatility means the square root of variance.

9Here maturity means sup .

10In this report, the prices or values of all derivatives are considered in their natural currency and as of maturity, i.e.

they are forward prices or values.

11Note that in the absence of a variance market, the price of variance can be determined using listed option prices

(see e.g. Derman et al., 1999).


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5



1.2. Proxy formulas for realised and implied correlation

In Appendix A, we derive proxy formulas for (116) and (121) using limit arguments. Subject to fairly reasonable

conditions on the weights and the pair-wise realised correlations between constituent stocks, we find that residual

terms ( ) and *( )t vanish as Ngoes to infinity:

)()(

)()(

+ S

I

N

)()(

)()( *

*

**

tS

t

I

t

Nt

+

We refer to )( and )( * t as realised and implied correlations, respectively.

In Exhibit 1.2.1, we compare three measures of the realised correlation of the Dow Jones EuroStoxx 50 index, over

1-month and 24-month rolling periods since 2000. We can see that the distance between the proxy and canonical

measures does not exceed a few correlation points, and also that the distance between the proxy and average pair-

wise measures can occasionally be significant (more than 10 correlation points), particularly for 24-month rolling

periods.

In Exhibit 1.2.2, we introduce an alternative measure which is based on a reconstitution of index values with

constant weights and constituent stocks at the start of each 24-month period, as well as a substitute calculation of

realised constituent volatility. We can see that the distance between this measure and the average pair-wise

measure does not exceed a few correlation points.


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6



Exhibit 1.2.1 1-month and 24-month realised correlation measures of the Dow Jones EuroStoxx 50 index

(20002007)In the two charts below, we report on each monthly listed expiry date the correlation level realised over the following (a) month or (b) 24 months,

using formulas below. On each monthly start date, we retrieved the constituent stocks and their weights, and held them constant over the 1-

month or 24-month time period. In all cases, realised volatilities were calculated as the annualised zero-mean standard deviation of daily log-

returns, using closing levels. For index volatility, we used actual index closing levels as disseminated by the calculation agent.

Proxy:

2

S

I

, Canonical:

( )( ) 22

22

S

I

, Average pair-wise:ji SS

j

ji

ij

ji

i wwww,

1

. Our numerical results in Appendix G tend to indicate that, for> 80% and typical values

for other model parameters, the probability of the terminalrealised correlation cTbeing above 1 would be less than5%. This is also confirmed in Exhibit 3.5.1 below where we used the theoretical volatility of volatility estimates found

in Appendices D and E and a value of 50% for the initial implied correlation*

0 .

Here, we seem to have a trade-off between model simplicity and accuracy. Because index and constituentvariances are traded assets, we cannot introduce mean reversion or any other type of constraint on their drift underthe forward-neutral measure P*. One solution could be to make the correlation of volatilities parameternon-constant, so as to further reduce the probability of an arbitrage. Another, more ambitious solution might be todevelop a large-factor model consistent with option and variance prices on the index and each constituent stock

21.

Another limitation of the toy model is that it assumes static weights and constituent stocks. While this assumptionseems reasonable for short time periods, it is likely to affect results for longer time periods.

21A step in this direction can be found in Driessen et al. (2005), who derive endogenous dynamics for index

variance based on the dynamics of constituent variances and an instantaneous correlation of stock returns processof the Wright-Fisher type. However, the model of Driessen et al. is for variances and correlations of constant rollingmaturity, which are non-tradable assets. It is unclear to us whether their results can be extended to the fixed

maturity case, with forward neutral drifts.


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17

Rational pricing of equity correlation swaps


Exhibit 3.5.1 Probability ofcT > 1 in function of maturity T, for various values of the correlation between

index and constituent volatilities parameter, and initial implied correlation *0 0.5 = In the chart below, we report for each maturity the implied probability of the terminal realised correlation cTbeing above 1 assuming an initial

implied correlation of 50%. To construct each curve, we applied the analytical formula found in Appendix G with a given value forand the

theoretical estimates found in Appendices D and E fors.

0%

2%

4%

6%

8%10%

12%

14%

16%

0 3 6 9 12 15 18 21 24Maturity (months)

= 0.5 = 0.55 = 0.6 = 0.65 = 0.7 = 0.75 = 0.8 = 0.85 = 0.9 = 0.95 = 1.0

1 year

Source: Dresdner Kleinwort


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18

Further research


4. Further research

Despite its limitations, our approach is, to our knowledge, the first of its kind to establish that correlation swaps onthe constituents of a stock index can be replicated by dynamically trading variance dispersions, and that their fairvalue is straightforwardly related to implied correlation. In fact, using a parameter estimation methodology whichrelies on few historical factors, we obtain numerical results supporting the intuitive idea that the fair value of acorrelation swap should be close to implied correlation. Dynamic arbitrage opportunities may therefore existwhenever the market price of correlation swaps substantially differs from implied correlation.

Further research is now needed:

On the fundamental side, the toy model needs to be refined to be made entirely arbitrage-free.

On the practical side, the toy model needs to be extended in order to calculate the fair value of other correlation

measures, for instance the canonical or average pair-wise measures. Additionally, allowing for free-floatweights and changes in index composition would render the model closer to index calculation practices.

On the numerical side, more sophisticated parameter estimations, over longer historical periods and in othermarkets, would be extremely valuable.

We believe our approach constitutes a first step towards a consistent pricing theory of statistical derivatives (i.e.derivatives on realised variance and correlation). The toy model provides us with elementary analytical formulasand rules of thumbs. We also see two other research areas which may benefit from our results: the pricing andhedging of exotic derivatives on multiple stocks (in particular the modelling inclusion of correlation skew), and thestochastic modelling of volatility and correlation.


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19

References


5. References22

Buehler, Hans (2006). Consistent Variance Curve Models. Finance and Stochastics, Vol. 10, No 2, April 2006.Available at: http://www.math.tu-berlin.de/~buehler/dl/VarSwapCurves3622.pdf

Black, Fischer and Scholes, Myron S. (1973). The Pricing of Options and Corporate Liabilities. Journal of PoliticalEconomy, Vol. 81, Issue 3, pp. 647-654.

Bossu, Sebastien and Gu, Yi (2004). Fundamental relationship between an indexs volatility and the averagevolatility and correlation of its components. JPMorgan Equity Derivatives report, Working paper, May 2005.

Bossu, Sebastien (2005). Arbitrage pricing of equity correlation swaps. JPMorgan Equity Derivatives, Workingpaper, July 2005.

Carr, Peter and Sun, Jian (2005). A New Approach for Option Pricing Under Stochastic Volatility. Bloomberg LP,Working paper, March 2005.

Carr, Peter and Lee, Roger (2005). Robust Replication of Volatility Derivatives. Bloomberg LP and University ofChicago, Working paper, April 2007.

Derman, Emanuel, Demeterfi, Kresimir, Kamal, Michael, and Zou, Joseph (1999). More Than You Ever Wanted ToKnow About Volatility Swaps. Journal of Derivatives, Vol. 6, Issue 4, pp. 9-32. Also available as Goldman SachsQuantitative Research Notes, March 1999, at: http://www.ederman.com/new/docs/gs-volatility_swaps.pdf

Driessen, Joost, Maenhout, Pascal J. and Vilkov, Grigory (2005). "Option-Implied Correlations and the Price ofCorrelation Risk", EFA 2005 Moscow Meetings, December 2006. Available at SSRN: http://ssrn.com/abstract=673425

Duanmu, Zhenyu (2004). Rational Pricing of Options on Realized Volatility The Black-Scholes Way, Global

Derivatives and Risk Management Conference, Madrid, May 2004.

Dupire, Bruno (1992). Arbitrage Pricing with Stochastic Volatility, Proceedings of AFFI Conference in Paris, June1992. Also available in: Derivatives Pricing, pp.197-215, Risk 2004.

Friz, Peter and Gatheral, Jim (2005). Valuation of volatility derivatives as an inverse problem. Quantitative Finance,Vol. 5, Issue 6, pp. 531-542.

Potter, Chris W. (2004). Complete Stochastic Volatility Models With Variance Swaps, Oxford University, Workingpaper. Available at http://gemini.econ.umd.edu/cgi-bin/conference/download.cgi?db_name=QMF2004&paper_id=71

Skintzi, Vasiliki D. and Refenes, Apostolos N. (2005). "Implied Correlation Index: A New Measure of Diversification".Journal of Futures Markets, Vol. 25, pp. 171-197, February 2005. Available at SSRN: http://ssrn.com/abstract=460080

or DOI: 10.2139/ssrn.460080

22When referring to a working paper, we indicate the year of the first known version after the authors name,

followed by the title and date of the latest known version.


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20

Appendices


6. Appendices

A Proxy formulas for realised and implied correlations

In this appendix, we render the dependence on implicit and the dependence on Nexplicit in the notationsintroduced in Section 1.1. Define the pair-wise correlation coefficient between stocks Siand Sjas:

( )

)ln)(ln(11,

ji

SjSSS SdSdiji

Write:

( ) ( )2 2,

( ) ( ) j i jiS S SSI

i j

i j

N w w N

= +

Define:i

i

N

ii

NS

i

NS

i

NSS

ji

Nwwwwiiji max,min,max,min,min )(max

)(

min

)(

max

)(

min

,

,

)(

min

Assuming( )

min 0N > and

( )

min0

N > , we have:

(A1)( )

( )

( )

( )

( )

( )

( )

( )

2 2 22 2 ( ) ( ) ( ) ( )max max max max

2 2 2 2 ( )( ) ( )2 ( ) ( ) ( )minmin min minmin min

( ) ( ) 10

( ) ( )

N N N N

NN NS I N N N

N w wN N

N N N N w w

=

If we further assume that (a) all realised volatilities never degenerate towards zero nor explode towards infinity, and(b) all pair-wise correlations never degenerate towards zero, we obtain the reduced convergence condition:

( )( )max

( )

min

N

Nw o Nw

=

Combined with inequality (A1), this condition ensures:

( ) ( )( )( ) ( )( )

22

22

( ) ( )

( ) ( )

I

S

N o N

N o N

= =

whence:

)(

)(

)()(

+ S

I

N

This condition is also sufficient to ensure the convergence of implied correlation:

)()(

)()( *

*

**

tS

t

I

t

Nt

+

Note that we can drop assumption (b) and obtain the relaxed convergence condition:

( )

( )

2( )

max

2( ) ( )

minmin

( )

N

N N

wo N

w=

It is clear that this condition ensures the convergence of realised correlation. We do not know if it is sufficient for

implied correlation without making the additional market assumption that one can trade( )

min

N .


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21

Appendices


B Analytical formula for the volatility claim within the toy model

Applying the Ito-Doeblin theorem, we can write the diffusion equation for ln v:2

* 2 *ln( ) 2 2t tT t T t

d v dz T T

= +

Equivalently:2

* * 2 *

2exp 2 ( ) 2 ( )

T T

T t st t

v v T s ds T s dz T T

= +

Calculating the first integral explicitly:

(B1)

3

* * 2 *2exp 2 ( )3

TT t s

tT tv v T T s dz

T T

= +

Taking the square root and then the conditional expectation of both sides of (B1), we can write that the price of thevolatility claim is given as:

(B2)

3

* 2 * *1exp exp ( )3

T

t t t st

T tv T E T s dz

T T

=

V

The stochastic integral*( )

T

st

T s dz has a conditional normal distribution with respect to F t, with zero mean and

standard deviation

2/3

3

1

)( tT . Thus:

=

=

3

22

2

2**

6

1exp)(

2exp)(exp

T

tTTdssT

TdzsT

TE

T

t

T

tst

Substituting this result in (B2) we obtain:3

* 21exp6

t t

T tv T

T

=

V


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22

Appendices


C Analytical formula forct within the 2-factor toy model

Applying the Ito-Doeblin theorem onS

I

v

v*

*

ln , we find:

( ) ( ) +

= *2*

2

22

*

*

1222ln tSI

tSIISS

t

I

t dzT

tTdz

T

tT

T

tT

v

vd

Equivalently:

(C1) ( )

+

=

T

ts

ST

t

I

sSI

ISS

t

I

tT dzsT

TdzsT

TT

tTT

v

vc

*

2

*

3

22

*

*

)(1

2)(23

2exp

Taking conditional expectations under P* yields:

( )

++=

3

22222

*

*

)1()(3

2exp

T

tTT

v

vc SSIISS

t

I

tt

Expanding the squares and simplifying, we obtain:

( )

=

3

2

*

*

3

4exp

T

tTT

v

vc ISSS

t

I

tt

D Estimation of the volatility of index volatility parameter

When historical prices for implied variance are available, as is the case with the major stock indexes such as theDow Jones EuroStoxx50, several estimators can be constructed for the volatility of volatility parameter. Theproblem here is that does not correspond to the volatility of realised volatility, nor the volatility of implied volatility;

corresponds to the volatility of the variance price process*

tv over its lifetime. To get around this problem, we

propose to estimate by isolating the quadratic effect of variance versus volatility.

Inverting (221), we can write:

= ln6

T

where is the ratio of fair variance to fair volatility, respectively denoted*

0v and 0V in (221). Note that if time

series of fair variance and volatility were available, we could back out an implied volatility of volatility parameter foreach historical date and analyse its statistics. However, in practice, volatility swaps are much less liquid instrumentscompared to variance swaps, precisely due to the absence of market consensus on their fair value. Ourmethodology is to determine such that historical spread trades between variance and volatility break even onaverage.

Given a sequence ofMtime periods Mmm ..1)( = , define:

1

1

2

2

11

=

M

m m

mm

K

KR

M

where )(m

I

m

R denotes realised index volatility overm, and*

inf

( )m

I

m m

K

denotes fair variance at the

start of time period m.


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23

Appendices


From an economic point of view, corresponds to the historical quadratic adjustment to be used so that anarbitrageur repeating normalised spread trades between variance and volatility would break even on average. Tosee this clearly, assume that for each historical time period ordinal m, future realised volatility overm trades at fair

variance Km divided by a constant quadratic adjustment factor. Buying 22

1

mKunits of variance and selling

mK

1units of volatility, and repeating the trade for all historical dates, the total profit or loss is:

==

=

=

M

m m

mmM

m m

m

m

m

K

KR

K

R

K

Rlp

1

2

12

21

12

11

2

1

2/

Assuming p/l = 0 and solving for, we find=

.

We applied this methodology on a monthly basis, for 1-, 2-, 3-, 6-, 12- and 24-month listed expiries, using at-the-money implied volatility levels as a proxy for fair variance. Exhibit D1 below shows the results obtained on the DowJones EuroStoxx 50 for the period 20002005 (64 data points). We can see that the break-even quadraticadjustment increases with the length of the time period up to 6 months and then remains stable, while thecorresponding theoretical volatility of volatility decreases.

We must emphasise that this analysis is fairly coarse due to the small number of data points, and the significantchanges in market conditions within the historical period

23. As such, the figures obtained should not be seen as any

form of recommended parameter values for trading purposes; they might, however, form the basis for sensiblevalues in a risk management context.

For completeness, we mention two other possible methodologies which require the reconstitution of the price

processmtmt

v

)(* for each time periodm:

Realised volatility of volatility: ( ) mt mtm vdT

2* )(ln4

3

Break-even volatility of volatility: m is the solution to the break-even delta-hedging p/l equation:

22*2

* * 2 2

2 *

( ) sup ( , ( ), , ) ( ) 4 0

( )mt m m

t m m m t m mt

t m m

dv tt v v dt

v v

=

V

where:

3

2 sup

6

1exp),,,(

tvvtV .

23Internet bubble in 2000, September 2001 terror attacks, bear stock markets in 2002, war in Iraq in 2003, stock

markets recovery since 2004.


26/30

24

Appendices


Exhibit D1 Break-even quadratic adjustment and corresponding theoretical volatility of volatility for the

Dow Jones EuroStoxx 50 index (20002005)In the chart below, we report for each maturity the values of and observed on the Dow Jones EuroStoxx 50 index between 2000 and2005, corresponding to 64 periods starting on a monthly listed expiry date. The calculation methodology of realised and implied volatilities is

identical to Exhibit 1.2.1.

1.029

1.0641.063

1.059

1.043

1.051

42%

61%

87%

123%

145%

109%

1.02

1.03

1.04

1.05

1.06

1.07

0 3 6 9 12 15 18 21 24

Maturity (months)

20%

50%

80%

110%

140%

170%

Index break-even quadratic adjustment (lhs) Index theoretical vol of vol (rhs)

1 year


E Estimation of the volatility of constituent volatility parameter

We apply the same methodology as in Appendix D to estimate the volatility of constituent volatility parameter S .

For an index such as the Dow Jones EuroStoxx 50, the difficulty here is to obtain reliable implied volatility surfacesfor each of the 50 constituents. The amount of data mining involved in such an operation can be considerable; welimited ourselves to estimating historical at-the-money implied volatility levels on a monthly basis for 1-, 2-, 3-, 6-, 12-and 24-month listed expiries.

Exhibit E1 below shows the results we obtained for the period 20002005, based on 64 monthly data points. We

can see that the break-even quadratic adjustment increases with the maturity, while the corresponding theoreticalvolatility of constituent volatility decreases. Compared with Exhibit D1, both figures are systematically lower than forthe index. This should not be surprising: in any framework where constituent volatility and correlation have positive

covariance, we have: ( ) ( )1

ln ln ln ln2

I S S Var Var Var

= +

.


27/30

25

Appendices


Exhibit E1 Break-even quadratic adjustment and volatility of volatility of the constituents of the Dow

Jones EuroStoxx 50 index for the period 20002005In the chart below, we report for each maturity the values of and observed on the constituent stocks of the Dow Jones EuroStoxx 50 indexbetween 2000 and 2005, corresponding to 64 periods starting on a monthly listed expiry date. The calculation methodology of realised and

implied volatilities is identical to Exhibit 1.2.1.

1.021

1.042

1.050

1.051

1.033

1.029

89%

123%

101%

70%

54%

39%

1.02

1.03

1.04

1.05

1.06

1.07

0 3 6 9 12 15 18 21 24

Maturity (months)

20%

50%

80%

110%

140%

170%

Constituent break-even quadratic adjustment (lhs) Constituent theoretical vol of vol (rhs)

1 year


F Estimation of the correlation parameter between index and constituent volatilities

Inverting (321), we have:

SI

ST

+=

ln4

32

where denotes the ratio of implied correlation to fair correlation, respectively denoted*

0 and c0 in (321).

Similarly to the situation described in Appendix D, if time series of fair correlation and implied correlation wereavailable, we could back out an implied correlation of volatilities parameterfor each historical date and analyse itsstatistics. In practice, correlation swaps are illiquid instruments and interbank quotes are infrequent.

Our methodology is to estimate using a break-even analysis of the fair to implied correlation adjustment. Given a

sequence ofMtime periods Mmm ..1)( = , define:

1

11

2

1

==

+

M

m

m

M

m m

m

m

m

m

mm IC

K

R

IC

RC

IC

RCIC

where )( mS

mR is the realised constituent volatility overm, )(*

inf m

S

m mK is the corresponding implied

volatility at the start of time period m, )( mmRC is the realised correlation overm, and )(

*

inf mm mIC isthe corresponding implied volatility at the start of time period m.


28/30

26

Appendices


From an economic point of view, corresponds to the historical adjustment to be used so that an arbitrageurrepeating normalised spread trades between variance dispersion and correlation swaps would break even onaverage. To see this clearly, assume that for each historical time period ordinal m, future realised correlation overm

trades at implied correlation ICm divided by a constant adjustment factor. Selling 2

1

mKunits of a vega-neutral

dispersion24

and selling 1 unit of correlation, and repeating the trade for all historical dates, the total profit or loss is:

( ) ==

=

=

M

m m

mm

m

mm

M

m

mmmm

m

m

K

RIC

K

RRC

ICRCICRC

K

Rlp

1

22

1

2

11/

Assuming p/l = 0 and solving for, we find = .

We applied this methodology on a monthly basis, for 1-, 2-, 3-, 6-, 12- and 24-month listed expiries, using at-the-money implied volatilities to calculate implied correlation. Exhibit F1 below show the results obtained on the DowJones EuroStoxx 50 for the period 20002005 (64 data points). We can see that the break-even adjustment factor

takes values between 0.991 and 1.041 while the corresponding theoretical correlation parameter increasesfrom 80.7% to 98.0%. We can also notice that, contrary to results in Appendices D and E for the break-even

quadratic factor, the term structure of the break-even adjustment factor is non-increasing.

Exhibit F1 Break-even adjustment factor and theoretical correlation between index and constituentvolatilities, for Dow Jones EuroStoxx 50 (20002005)

In table (a) below, we report for each maturity the values of , s and observed on the Dow Jones EuroStoxx 50 index and its constituents

between 2000 and 2005, corresponding to 64 periods starting on a monthly listed expiry date. Chart (b) plots the term structure of , and only.

(a)

Maturity

Break-evenadjustment factor

Index theoreticalvolatility ofvolatility

Constituenttheoreticalvolatility ofvolatility

Theoreticalcorrelation ofvolatilities

1m 0.991 144.7% 123.4% 80.7%

2m 1.013 122.6% 101.2% 87.2%

3m 1.027 109.2% 88.9% 89.8%

6m 1.041 86.5% 69.9% 90.8%

12m 1.018 60.5% 54.1% 93.5%

24m 1.022 41.5% 38.6% 98.0%

24 Recall that a short variance dispersion position is long correlation.

I

S


29/30

27

Appendices


(b)

1.018 1.022

1.041

0.991

1.027

1.013

90%

81%

87%

91%

94%

98%

0.98

0.99

1.00

1.01

1.02

1.03

1.04

1.05

0 3 6 9 12 15 18 21 24

Maturity (months)

72%

76%

80%

84%

88%

92%

96%

100%Break-even adjustment factor Theoretical correlation of volatilities

1 year


G Probability ofcT > 1 within the two-factor toy model

From the particularisation of (C1) at t= 0, we can write:

{ }( ) ( )

+ TdzsT

TdzsT

TPcP IS

T

s

ST

I

sSI

T

22*

00

*

2

0

***

3

2ln)(

12)(21

The two stochastic integrals in the above expression being independent normals with zero mean and variance T3

4,

we obtain after simplifying terms:

{ }( )( )

+

+=>

22

22*

0*

23

2

3

2ln

1

SSII

IS

T

T

T

NcP

where N(.) denotes the cumulative distribution of a standard normal.

We calculated this value for 1-month and 12-month maturities, using the theoretical volatility of volatility parameters

found in Appendices D and E, for between -1 and 1 and*

0 between 50% and 100%. Results are shown in

Exhibit G1. We can see that the probability increases with*

0 and decreases with .


30/30

Appendices


Exhibit G1 Probability ofcT > 1 forT= 1/12, in function of the instantaneous correlation between index

and constituent volatilities , for various values of the initial implied correlation *0 .In the charts below, we report the value ofP

*({cT> 1}) for (a) 1-month and (b) 12-month maturity, in function of , using the values ofs found in

Appendices D and E. Each curve corresponds to a given value of*

0 .

(a)

0%

10%

20%

30%

40%

50%

-100%

-85%

-70%

-55%

-40%

-25%

-10%

5%

20%

35%

50%

65%

80%

95%

Instantaneous correlation of volatilities

= 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 1Prob cT > 1

(b)

0%

10%

20%

30%

40%

50%

-100%

-85%

-70%

-55%

-40%

-25%

-10%

5%

20%

35%

50%

65%

80%

95%

Instantaneous correlation of volatilities

= 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 1Prob cT > 1


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