Abstract

Several trading institutions are actively engagedin ‘volatility dispersion’ strategies. These involveselling volatility on the index and buyingvolatility on the components. This trade wastraditionally done using at the money (ATM)straddles. An important practical problem withthis approach is that market prices move andcause the original ATM options to become out ofthe money (OTM) and lose their vega exposure.Even if volatility moved as expected by thetrader, the profit potential of the trade would begreatly diminished as the options lost their vega.To correct this problem, a new style of tradinghas emerged in which some practitioners are

trading this strategy using a ‘variance swap’approach. This has the advantage that both legsof the trade have relatively constant vegaexposure, regardless of stock marketmovements.

INTRODUCTION

This paper reviews the volatility dispersiontrade and compares the two approaches. Itgives a heuristic motivation to the varianceswap style of the trade. It also providessome empirical evidence that seems toindicate that the variance swap approach ismore malleable to trading.

334 Derivatives Use, Trading & Regulation Volume Eleven Number Four 2006

Variance swap volatility dispersion

Izzy Nelken

Super Computer Consulting, Inc., 3943 Bordeaux Drive, Northbrook, IL 60062, USATel: �1 847 562 0600, e-mail: izzy@supercc.com, websites: www.supercc.com andwww.optionsprofessor.comReceived: 24th March, 2005

Izzy Nelken is President of Super Computer Consulting, Inc which specialises in complex derivatives,structured products, risk management and hedge funds. He holds a PhD in computer science from RutgersUniversity and now lectures in the Department of Mathematics at the University of Chicago. Super ComputerConsulting clients include several regulatory bodies, major broker-dealers, large and medium-sized banks aswell as hedge funds. Dr Nelken has published a number of books, including ‘Volatility in the Capital Markets’.

Practical applications

Volatility dispersion strategies involve selling volatility on the index and buying volatility onthe components, traditionally using at the money (ATM) straddles. A problem with thisapproach, however, is that as the orginal ATM options move out-of-the-money, they losetheir vega exposure. To correct this problem, a new style of trading has emerged the‘variance swap’ approach, which eliminates the constant rebalancing requirement associatedwith ATM options. Moreover, under variance swap methodology, the volatility differenceappears more tame.

Derivatives Use,

Trading & Regulation,

Vol. 11 No. 4, 2006,

pp. 334–344

� Palgrave MacmillanLtd1747–4426/06 $30.00

historical back tests but are notguaranteed to work in the future.

Somewhere between these two extremetypes of trading lies the field of statisticaltrading that is based on a formula. Theformula is not exact and depends on severalunobservable variables. This type of tradingcan be characterised by medium risk andmedium returns. There is anecdotalevidence from the markets that this type oftrading is becoming more and morecommonplace. Examples of this type oftrading include:

— Convertible bond ‘arbitrage’ Here, thetrader will purchase a convertible bondand short shares against it. Obviously,the price of the bond depends on theprice of the underlying share. This isnot an exact formula however, since italso depends on many unobservableparameters: implied volatility, creditspreads, recovery values etc.

— Volatility dispersion.

WHAT IS VOLATILITY DISPERSION?

Assume that one sells index options.Obviously, one can hedge the exposureto the index price by trading theunderlying index (‘dynamic deltahedging’). But one is still exposed to therisk that the index implied volatility willdrop and the price of the sold optionswill also decline. One method of hedgingthe exposure to implied volatility is topurchase options on the shares thatcompose the same index.

Obviously, there is a strong relationship

STYLES OF PROPRIETARY TRADING

It is useful to distinguish between two‘extreme’ types of trading.

(1) Simple relationships: these seek toexploit slight price differences betweenidentical products on differentexchanges or products whose prices aretied to one another by an exact,pre-defined formula. These tradestypically have very low risk and lowexpected profits. They rely onextremely efficient execution. Owing tothe proliferation of fast electronictrading tools in the last several years,the opportunities for these trades havebeen rapidly diminishing. Examples are:(a) The existence of several different

option exchanges in the USA,which, at times, allows one to findzero cost ‘butterflies’ or othercombinations of options that have asmall chance of a positive payoff;

(b) Trading an index (eg the NDX)against its component stocks.

(2) Historical relationships — these types oftrades rely on historical pricecorrelations that have been observed inthe past but have no inherent reason.There is no formula that dictates therelationship between the products. Suchtrading is quite risky as ‘pastperformance is no guarantee of futureresults’. Examples of this are:(a) Long short hedge funds that

purchase one share and sell anotherbased on historical, observed pricerelationships.

(b) All kinds of technical tradingsystems. These have worked well in

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between the implied volatility of the indexand the implied volatility of thecomponents. This is not an exact formulahowever, as it depends on an impliedcorrelation matrix which can only beapproximated.

Most dispersion strategies sell indexvolatility and purchase component volatility,but not the reverse. That is, very rarely dopeople sell individual stock options andpurchase the index options. This is for twomain reasons:

(1) It is well known that index optionshave high implied volatility numbersby comparision with their historicalvolatility. Schneeweis and Spurgin1

have observed that the volatilityimplied by index option prices is toohigh relative to the realised volatilityof the index. Bollen and Whaley2

argue that there is excess buyingpressure on S&P500 Index putoptions (as fund managers seek topurchase insurance against a decline inthe stock market).

(2) One can be surprised by an ‘event’ in asingle stock. A sudden takeover orbankruptcy can cause an individualstock price to react very strongly andvery quickly. Selling single stockoptions could be quite dangerous.

Market makers in index options who sellindex options when the premiums are highmay choose to engage in volatilitydispersion strategies to reduce theirvolatility risk. Many hedge fund managersalso employ these strategies on anopportunistic basis.

THE TWO STYLES

Assume that a trader identified that theimplied volatility of a particular index is highrelative to its components. Traditionally, onewould trade this by selling at the money(ATM) index options (for example an ATMstraddle) and purchasing ATM componentoptions. It is common to trade in ATMoptions since:

(1) The ATM options have the largest vegaexposure.

(2) If one trades in out of the money(OTM) options, they typically establisha position with a delta exposure. Thetrader will be obliged to delta hedgethat exposure from the initial trade.

(3) If one were to sell OTM index options,one would have to decide which arethe corresponding OTM componentoptions. There are many potentialchoices,(a) options which are OTM by a

similar percentage;(b) options which are OTM by a

similar number of standarddeviations;

(c) options which have similar deltas;(d) and more.

(4) The ATM options are typically themost liquid.

Assume that the trader might decide tosell an ATM index straddle and purchaseATM component straddles. Of course, foreach stock in the index, the trader willhave to determine how many componentstraddles he/she will need to sell. If theindex is cap weighted and has manycomponents, the trader might choose not

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range of stock prices. This can be createdusing the ‘variance swap’ technique.Nowadays, some practitioners are tradingvolatility dispersion using variance swapvolatility.

In what follows, the volatility dispersiontrade is described in more detail and thetwo styles of trade are explained. The paperconcludes with some experimental evidencethat seems to suggest that the variance swapvolatility is more amenable to trading.

THE PORTFOLIO EQUATION

The well-known MarkowitzMean–Variance Portfolio Equation expressesthe volatility of a portfolio as a function ofthe volatility of its component stocks.3

Varp �n�

i=1

� 2iwi

2 �n�

i=1

n�j=1

�i,j�i�jwiwj

�p � �Varp

Here, a portfolio of n assets with a varianceof Varp and a volatility of �p is considered.For asset i, �i is the volatility of the ithasset and wi is its weight in the portfolio.The quantity �i,j is the pairwise correlationbetween assets i and j.

For historical volatility and correlationnumbers, this equation is exact. Manytraders combine historical correlationestimates with implied volatility numbers.The resulting theoretical volatility for theindex is then compared with the actualimplied volatility.

A hypothetical example: The two

stock index

Assume that one had an index composed oftwo stocks. One could observe the implied

to trade options on the smaller cap stocksat all.

The trader hopes to profit when impliedvolatilities come back into line. Perhaps theindex implied volatility declines or theimplied volatility of the componentsincreases. In any case, the trade will be awinner. Another source of potential profitsis that each of the component stocks mightexperience a price ‘jump’. A jump mayoccur owing to a takeover, bankruptcy orother material change in the company. Inthe case of a jump, one of the options inthe component straddle suddenly becomesdeep in the money and rises in value.

But what if stock prices change slowly?In this case, the straddle that was originallypurchased at the money, with close to zerodelta, now has a delta exposure. If thetraders did not hedge the delta exposure,random changes in stock price wouldoverwhelm the profit potential (which isdue to volatility) of the trade. Therefore,traders will typically delta hedge the stockexposure of each component and the indexexposure of the index straddle. They do sobecause they trade attempts to profit fromvolatility changes.

Nevertheless, a serious problem remains.Once stock prices have moved, even ifimplied volatility were to return to normallevels, the trade will still not be profitable,as the options have become OTM andhave no vega exposure. Some traders resortto exiting OTM straddle positions andentering new ATM straddles — re theykeep rebalancing (or ‘chasing the stockprice’) to maintain a vega exposure.

What traders require is an instrumentthat maintains a vega exposure for a wide

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volatility of options on the index and onthe two component stocks. Plugging thesenumbers into the portfolio equation andsolving for the correlation would give the‘implied correlation number’ between thetwo stocks. If the implied correlationnumber is quite high, it may be a goodtime for a trade. Certainly, if the impliedcorrelation number is close to 1, there islittle risk that the correlation will climbeven further.

Even for a two stock index though, thetrade requires some assumptions.

Suppose one sells 100 ATM straddles onthe index. One must determine how manyATM straddles to purchase on each of thecomponents.

The choice of quantities and strikes istypically accomplished by the technique ofequating ‘Greeks’.

— Deltas are hedged separately in eachinstrument.

— The trader than calculates how manygamma or vega units he/she has soldand then purchases an equal amount ofunits in the components.

The potential profit of this trade comesfrom several sources:

— If the implied correlation comes back tonormal level, the trade will makemoney. This will happen either becausethe implied volatility of the indexoption has decreased, or because theimplied volatility of the components hasincreased, or both.

— A ‘windfall’ gain. This happens when asudden idiosyncratic event causes one of

the component stocks to move sharply(either up or down).

The trade is exposed to a practicaldanger. If the stocks and index movefrom the previous strikes, even if thecorrelation returns to normal levels, theoption positions will not be exposed tovega in any significant way. The changein implied volatility will not have asubstantial impact on the price of theoptions. This may mean that the traderwill continuously have to adjust thestraddle positions to remain close toATM. It is precisely because of thisdifficulty that some market participantsuse the variance swap volatility for thesetrades.

Another danger is that of correlation‘explosion’. This trade is a bet thatimplied correlation will come back to itsnormal levels. However, if correlationcontinues to increase, the trade may, infact, lose money.

An additional difficulty occurs in somecapital weighted (‘cap weighted’) indexes.As the market moves and stock priceschange, so do the wis, the relativeweights of the companies within theindex.

An index with many stocks

Real-life indexes have more than twostocks. In that case, there is an entirecorrelation matrix. This matrix cannot beuniquely determined from the impliedvolatility numbers.

It is possible to use a historicalcorrelation matrix in the portfolio equation.Of course, one has to choose carefully the

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positive semi-definiteness). It is possibleto use techniques such as described inHigham5 to find the ‘nearest’ correlationmatrix to the resultant matrix.

— The trader’s goal is to estimate the‘implied correlation’ matrix, that is, tofind a symmetric, positive semi-definitematrix with unit diagonal that is as closeas possible to the historical correlationmatrix and makes the ‘stock volatility’match the ‘index volatility’.

— If the implied correlation matrix haslarge elements by comparison with thehistorical correlation matrix, it may be asignal to enter the trade and bet ondeclining correlations.

WHICH IMPLIED VOLATILITY?

If the trader uses ATM straddles to performthe trade, the market may ‘run away’ fromthe strikes. In that case, even if volatilitiesmove in the expected directions, the tradewill not make money, as the options thathave become OTM have very littlevolatility exposure.

Some market participants are thereforeusing the variance swap volatility to createtheir volatility dispersion strategies.

VARIANCE SWAP VOLATILITY: A

SHORT HISTORY

In 1996, Neuberger6 described the logcontract. This is an option whose payout istied to the logarithm of the stock price.Thus a call option would have a payout of

max[ln(S) � X,0]

historical period for which correlationshould be computed, the averaging methodused etc. See Nelken4 for a discussion onvolatility and correlation measurement. Inany case, it is well known that the resultingcorrelation matrix must be symmetrical andpositive semi-definite.

Using the historical correlation matrix inconjunction with the implied volatilitynumbers will cause the portfolio equationto be inexact. In practice, however, it isstill usable as a signal of when to enter orexit a trade.

As it is impossible to determine theimplied correlation matrix precisely, manytraders attempt to estimate it.

— Begin with a historical correlationmatrix.

— One now has two volatility numbers:(a) the implied volatility of the index

(‘index volatility’);(b) the volatility of the index as

computed using the impliedvolatility of the components and thehistorical correlation matrix (’stockvolatility’).

— These volatility numbers will not match.The market has its own view on futurecorrelations (the so-called ‘impliedcorrelation’ matrix), and it adjusts thehistorical correlation matrix in someundeterminable fashion. For example,when market participants are worriedabout a crash, the historical correlationmatrix is adjusted upwards.

— Traders will typically adjust the historicalcorrelation matrix until these twovolatility numbers match. The resultantmatrix may lose its features (symmetric,

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where S is the stock price on expiration, Xis the strike and ln is the natural logarithmfunction. The log contract has a uniquefeature in that it has stable ‘Greeks’. Forexample, a contour plot of the vega of sucha contract versus the stock price results in astraight line. By contrast, a similar contourplot for vanilla European options results ina non-linear curve. The main conclusionwas that ‘the log contract provides a mucheasier and more reliable way of betting onvolatility’.

In 1998, in the aftermath of the LongTerm Capital Management (LTCM)debacle, implied volatility figures rose tounprecedented levels. As Gatheral7 explains

‘Variance swaps took off as a product inthe aftermath of the LTCM meltdown inlate 1998 when implied stock indexvolatility levels rose to unprecedentedlevels. Hedge funds took advantage of thisby paying variance in swaps (selling therealised volatility at high implied levels).The key to their willingness to pay on avariance swap rather than sell options wasthat a variance swap is a pure play onrealised volatility — no labour —intensive delta hedging or other pathdependency is involved. Dealers werehappy to buy vega at these high levelsbecause they were structurally short vega(in the aggregate) through sales ofguaranteed equity-linked investments toretail investors and were getting badlyhurt by high implied volatility levels.’

In 1999, Demeterfi et al.8 constructed aportfolio of European options. Theirportfolio has �k/k2 units of each option

with a strike of k. It turns out that such aportfolio has a constant ‘variance vega’. Thedollar amount gained by such a portfoliofor a unit change in the variance (volatilitysquared) is insensitive to the stock priceacross a wide range of stock prices. Thepaper also described how this portfolio isalmost equivalent to a log contract.

In 2003, the Chicago Board OptionsExchange (CBOE)9 announced a majorchange in the way that the volatility index(VIX) was computed. This was followed bythe initiation of trading on the VIX in2004. The new VIX is fashioned after thevariance swap portfolio previouslyintroduced by Demeterfi et al.8

The generalised formula used by theCBOE is

� 2 �2T�

i

�KKi

2 eRTQ(Ki) �1T � F

K0� 1�

2

where T is the time to expiration; F is theforward level of the stock (or index) asderived from the options; Ki is the strikeprice of the ith OTM option, a call if K > Fand a put if K < F; �Ki is the intervalbetween strike prices (for single equities it istypically $5); K0 is the first strike price belowthe forward price F; R is the risk-freeinterest rate; and Q(Ki) is the quote for thespecific option (it is taken as the mid price).

One of the main advantages of the waythe new VIX is calculated is that it uses anentire collection of options to come at theindex rather than just a few options aspreviously. Also, contrary to the normalimplied volatility which relies on one’s ownchoice of model (eg Black-Scholes forEuropean options), the VIX volatility doesnot rely on a particular choice of model.

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DJI as given by the index options. To getto a 30-day running volatility, it is typicalto use interpolation between the frontmonth option and the second to frontmonth option. Also plotted is the volatilitywhich was obtained using the portfolioformula, the implied volatility of thecomponent options and a long-termhistorical correlation matrix. In 2003, the

SHOULD ONE TRADE VARIANCE

SWAP VOLATILITY DISPERSION?

This paper looks at the Dow Jones Index.The DJI is composed of 30 well-knowncompanies whose stocks are liquid. Inaddition, it is a price-weighted index.

Figure 1 plots the index level on theleft-hand scale. The right-hand scale plotsthe ATM 30-day implied volatility of the

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Figure 1: Value of Dow Jones Index (DJI), index volatility and stock volatility March2003–March 2004

Figure 2: Value of Dow Jones Index and difference between index volatility and stockvolatility, March 2003–March 2004

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index rose sharply and implied volatilitynumbers fell.

Figure 2 plots the ATM volatilitydifference. This is the index impliedvolatility minus the index volatility ascomputed from the component (stock)options.

The index implied volatility (‘index

volatility’) is different from the volatilitycomputed from the portfolio formula(‘stock volatility’). That formula usedhistorical correlation numbers. Asmentioned before, when market participantsare worried about a crash, the historicalcorrelation matrix is adjusted upwards. Asthe stock market trends upwards, fears of a

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Figure 3: Value of Dow Jones Index (DJI), index volatility and stock volatility (from stockoptions using the variance swap methodology), March 2003–March 2004

Figure 4: Value of Dow Jones Index (DJI), and difference between index volatility andstock volatility (computed from stock options using variance swap methodology) March2003–March 2004

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the ATM methodology. The volatility ofthe VIX chart is approximately 200 percent, while that of the ATM chart is about338 per cent.

SUMMARY

The theoretical results show that it may besimpler to trade the volatility dispersionstrategy using the variance swapmethodology. This is for two main reasons:

(1) It eliminates the constant rebalancing(‘chasing the spot’) requirement whenusing ATM options.

(2) The volatility difference, which is thequantity being traded, appears moretame under the variance swap (VIX)methodology.

crash are ‘forgotten’, and the upwardadjustment to the correlation matrixdecreases. This reduces the differencebetween the index volatility and the stockvolatility. This is clearly visible in Figure 2.

Figures 3 and 4 repeat the samecomputations, but now using the impliedvolatility numbers that are computed usingthe variance swap methodology.

It is instructive to place both volatilitydifference charts next to each other. Figure5 plots the difference that was computedusing the ATM implied volatilities and thedifference as computed by the volatilitydispersion methodology (the VIXapproach).

It appears that the volatility differenceusing the VIX methodology is much lessvolatile than the volatility difference using

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Figure 5: Volatility diferences, March 2003–March 2004

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It appears that some market participants havealready adopted the variance swapmethodology for volatility dispersion trading.

Acknowledgments

The author wishes to thank Krag Gregory andVenkatesh Balasubramanian at Goldman Sachs.

References

(1) Schneeweis, T. and Spurgin, R. (2001) ‘Thebenefits of index option-based strategies forinstitutional investors’, Journal of AlternativeInvestments, Spring, pp. 44–52.

(2) Bollen, N. P. and Whaley, R. (2002) ‘Does pricepressure affect the shape of implied volatilityfunctions?’, Working Paper, Duke University,Durham, NC, USA.

(3) For an excellent guide to this see Markowitz, H.M., Todd, P. G. and Sharpe, W. F. (2000)

‘Mean-Variance Analysis in Portfolio Choice andCapital Markets’, John Wiley, Chichester, UK.

(4) Nelken, I. (Ed.) (1997) ‘Volatility in the CapitalMarkets’, Glenlake, Chicago, IL.

(5) Higham, N. (2002) ‘Computing the nearestcorrelation matrix — A problem from finance’,IMA Journal of Numerical Analysis, Vol. 22,pp. 329–343.

(6) Neuberger, A. (1995) ‘The Log Contract andOther Power Contracts’, in Nelken, I. (Ed.),‘Handbook of Exotic Options’, Irwin ProfessionalPublishing (now McGraw-Hill), Burr Ridge, IL.

(7) Gatheral, J. (2003) ‘Replication of QuadraticVariation-based Payoffs’, Merrill Lynch, availableat: http://www.math.nyu.edu/fellows_fin_math/gatheral/lecture7_2003.pdf.

(8) Demeterfi, K., Derman, E., Kamal, M. and Zou,J. (1999) ‘More Than You Ever Wanted toKnow About Volatility Swaps’, Goldman SachsQuantitative Strategies Research Notes, March.

(9) For more details, see http://www.cboe.com/micro/vix/index.asp and alsohttp://www.cboe.com/micro/vix/vixwhite.pdf

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