Variance swap volatility dispersion
Izzy Nelken Super Computer Consulting, Inc., 3943 Bordeaux Drive, Northbrook, IL 60062, USA Tel: ϩ1 847 562 0600, e-mail: [email protected], websites: www.supercc.com and www.optionsprofessor.com Received: 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 Rutgers University and now lectures in the Department of Mathematics at the University of Chicago. Super Computer Consulting clients include several regulatory bodies, major broker-dealers, large and medium-sized banks as well 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 on the components, traditionally using at the money (ATM) straddles. A problem with this approach, however, is that as the orginal ATM options move out-of-the-money, they lose their vega exposure. To correct this problem, a new style of trading has emerged the ‘variance swap’ approach, which eliminates the constant rebalancing requirement associated with ATM options. Moreover, under variance swap methodology, the volatility difference appears more tame.
Abstract trading this strategy using a ‘variance swap’ Several trading institutions are actively engaged approach. This has the advantage that both legs in ‘volatility dispersion’ strategies. These involve of the trade have relatively constant vega selling volatility on the index and buying exposure, regardless of stock market volatility on the components. This trade was movements. traditionally done using at the money (ATM) straddles. An important practical problem with this approach is that market prices move and INTRODUCTION cause the original ATM options to become out of This paper reviews the volatility dispersion the money (OTM) and lose their vega exposure. trade and compares the two approaches. It Even if volatility moved as expected by the gives a heuristic motivation to the variance trader, the profit potential of the trade would be swap style of the trade. It also provides Derivatives Use, Trading & Regulation, greatly diminished as the options lost their vega. some empirical evidence that seems to Vol. 11 No. 4, 2006, pp. 334–344 To correct this problem, a new style of trading indicate that the variance swap approach is ᭧ Palgrave Macmillan Ltd has emerged in which some practitioners are more malleable to trading. 1747–4426/06 $30.00
334 Derivatives Use, Trading & Regulation Volume Eleven Number Four 2006 STYLES OF PROPRIETARY TRADING historical back tests but are not It is useful to distinguish between two guaranteed to work in the future. ‘extreme’ types of trading. Somewhere between these two extreme (1) Simple relationships: these seek to types of trading lies the field of statistical exploit slight price differences between trading that is based on a formula. The identical products on different formula is not exact and depends on several exchanges or products whose prices are unobservable variables. This type of trading tied to one another by an exact, can be characterised by medium risk and pre-defined formula. These trades medium returns. There is anecdotal typically have very low risk and low evidence from the markets that this type of expected profits. They rely on trading is becoming more and more extremely efficient execution. Owing to commonplace. Examples of this type of the proliferation of fast electronic trading include: trading tools in the last several years, the opportunities for these trades have — Convertible bond ‘arbitrage’ Here, the been rapidly diminishing. Examples are: trader will purchase a convertible bond (a) The existence of several different and short shares against it. Obviously, option exchanges in the USA, the price of the bond depends on the which, at times, allows one to find price of the underlying share. This is zero cost ‘butterflies’ or other not an exact formula however, since it combinations of options that have a also depends on many unobservable small chance of a positive payoff; parameters: implied volatility, credit (b) Trading an index (eg the NDX) spreads, recovery values etc. against its component stocks. — Volatility dispersion. (2) Historical relationships — these types of trades rely on historical price correlations that have been observed in WHAT IS VOLATILITY DISPERSION? the past but have no inherent reason. Assume that one sells index options. There is no formula that dictates the Obviously, one can hedge the exposure relationship between the products. Such to the index price by trading the trading is quite risky as ‘past underlying index (‘dynamic delta performance is no guarantee of future hedging’). But one is still exposed to the results’. Examples of this are: risk that the index implied volatility will (a) Long short hedge funds that drop and the price of the sold options purchase one share and sell another will also decline. One method of hedging based on historical, observed price the exposure to implied volatility is to relationships. purchase options on the shares that (b) All kinds of technical trading compose the same index. systems. These have worked well in Obviously, there is a strong relationship
Nelken 335 between the implied volatility of the index THE TWO STYLES and the implied volatility of the Assume that a trader identified that the components. This is not an exact formula implied volatility of a particular index is high however, as it depends on an implied relative to its components. Traditionally, one correlation matrix which can only be would trade this by selling at the money approximated. (ATM) index options (for example an ATM Most dispersion strategies sell index straddle) and purchasing ATM component volatility and purchase component volatility, options. It is common to trade in ATM but not the reverse. That is, very rarely do options since: people sell individual stock options and purchase the index options. This is for two (1) The ATM options have the largest vega main reasons: exposure. (2) If one trades in out of the money (1) It is well known that index options (OTM) options, they typically establish have high implied volatility numbers a position with a delta exposure. The by comparision with their historical trader will be obliged to delta hedge volatility. Schneeweis and Spurgin1 that exposure from the initial trade. have observed that the volatility (3) If one were to sell OTM index options, implied by index option prices is too one would have to decide which are high relative to the realised volatility the corresponding OTM component of the index. Bollen and Whaley2 options. There are many potential argue that there is excess buying choices, pressure on S&P500 Index put (a) options which are OTM by a options (as fund managers seek to similar percentage; purchase insurance against a decline in (b) options which are OTM by a the stock market). similar number of standard (2) One can be surprised by an ‘event’ in a deviations; single stock. A sudden takeover or (c) options which have similar deltas; bankruptcy can cause an individual (d) and more. stock price to react very strongly and (4) The ATM options are typically the very quickly. Selling single stock most liquid. options could be quite dangerous. Assume that the trader might decide to Market makers in index options who sell sell an ATM index straddle and purchase index options when the premiums are high ATM component straddles. Of course, for may choose to engage in volatility each stock in the index, the trader will dispersion strategies to reduce their have to determine how many component volatility risk. Many hedge fund managers straddles he/she will need to sell. If the also employ these strategies on an index is cap weighted and has many opportunistic basis. components, the trader might choose not
336 Nelken to trade options on the smaller cap stocks range of stock prices. This can be created at all. using the ‘variance swap’ technique. The trader hopes to profitwhenimplied Nowadays, some practitioners are trading volatilities come back into line. Perhaps the volatility dispersion using variance swap index implied volatility declines or the volatility. implied volatility of the components In what follows, the volatility dispersion increases. In any case, the trade will be a trade is described in more detail and the winner. Another source of potential profits two styles of trade are explained. The paper is that each of the component stocks might concludes with some experimental evidence experience a price ‘jump’. A jump may that seems to suggest that the variance swap occur owing to a takeover, bankruptcy or volatility is more amenable to trading. other material change in the company. In the case of a jump, one of the options in the component straddle suddenly becomes THE PORTFOLIO EQUATION deep in the money and rises in value. The well-known Markowitz Butwhatifstockpriceschangeslowly? Mean–Variance Portfolio Equation expresses In this case, the straddle that was originally the volatility of a portfolio as a function of purchased at the money, with close to zero the volatility of its component stocks.3 delta, now has a delta exposure. If the n n n Var ϭ 2w2 ϩ w w traders did not hedge the delta exposure, p i i i,j i j i j i=1 i=1 j=1 random changes in stock price would ϭ ͙Var overwhelm the profit potential (which is p p due to volatility) of the trade. Therefore, Here, a portfolio of n assets with a variance traders will typically delta hedge the stock of Va r p and a volatility of p is considered. exposure of each component and the index For asset i, i is the volatility of the ith
exposure of the index straddle. They do so asset and wi is its weight in the portfolio. because they trade attempts to profitfrom The quantity i,j is the pairwise correlation volatility changes. between assets i and j. Nevertheless, a serious problem remains. For historical volatility and correlation Once stock prices have moved, even if numbers, this equation is exact. Many implied volatility were to return to normal traders combine historical correlation levels, the trade will still not be profitable, estimates with implied volatility numbers. as the options have become OTM and The resulting theoretical volatility for the have no vega exposure. Some traders resort index is then compared with the actual to exiting OTM straddle positions and implied volatility. entering new ATM straddles — re they keep rebalancing (or ‘chasing the stock A hypothetical example: The two price’) to maintain a vega exposure. stock index What traders require is an instrument Assume that one had an index composed of that maintains a vega exposure for a wide two stocks. One could observe the implied
Nelken 337 volatility of options on the index and on the component stocks to move sharply the two component stocks. Plugging these (either up or down). numbers into the portfolio equation and solving for the correlation would give the The trade is exposed to a practical ‘implied correlation number’ between the danger. If the stocks and index move two stocks. If the implied correlation from the previous strikes, even if the number is quite high, it may be a good correlation returns to normal levels, the time for a trade. Certainly, if the implied option positions will not be exposed to correlation number is close to 1, there is vega in any significant way. The change little risk that the correlation will climb in implied volatility will not have a even further. substantial impact on the price of the Even for a two stock index though, the options. This may mean that the trader trade requires some assumptions. will continuously have to adjust the Suppose one sells 100 ATM straddles on straddle positions to remain close to the index. One must determine how many ATM. It is precisely because of this ATM straddles to purchase on each of the difficulty that some market participants components. use the variance swap volatility for these The choice of quantities and strikes is trades. typically accomplished by the technique of Another danger is that of correlation equating ‘Greeks’. ‘explosion’.Thistradeisabetthat implied correlation will come back to its — Deltas are hedged separately in each normal levels. However, if correlation instrument. continues to increase, the trade may, in — The trader than calculates how many fact, lose money. gamma or vega units he/she has sold An additional difficulty occurs in some and then purchases an equal amount of capital weighted (‘cap weighted’) indexes. units in the components. As the market moves and stock prices
change,sodothewis, the relative The potential profit of this trade comes weights of the companies within the from several sources: index.
— If the implied correlation comes back to Anindexwithmanystocks normal level, the trade will make Real-life indexes have more than two money. This will happen either because stocks. In that case, there is an entire the implied volatility of the index correlation matrix. This matrix cannot be option has decreased, or because the uniquely determined from the implied implied volatility of the components has volatility numbers. increased, or both. It is possible to use a historical — A ‘windfall’ gain. This happens when a correlation matrix in the portfolio equation. sudden idiosyncratic event causes one of Of course, one has to choose carefully the
338 Nelken historical period for which correlation positive semi-definiteness). It is possible should be computed, the averaging method to use techniques such as described in used etc. See Nelken4 for a discussion on Higham5 to find the ‘nearest’ correlation volatility and correlation measurement. In matrix to the resultant matrix. any case, it is well known that the resulting — The trader’sgoalistoestimatethe correlation matrix must be symmetrical and ‘implied correlation’ matrix, that is, to positive semi-definite. find a symmetric, positive semi-definite Using the historical correlation matrix in matrix with unit diagonal that is as close conjunction with the implied volatility as possible to the historical correlation numbers will cause the portfolio equation matrix and makes the ‘stock volatility’ to be inexact. In practice, however, it is match the ‘index volatility’. still usable as a signal of when to enter or — If the implied correlation matrix has exit a trade. large elements by comparison with the As it is impossible to determine the historical correlation matrix, it may be a implied correlation matrix precisely, many signal to enter the trade and bet on traders attempt to estimate it. declining correlations.
— Begin with a historical correlation matrix. WHICH IMPLIED VOLATILITY? — One now has two volatility numbers: If the trader uses ATM straddles to perform (a) the implied volatility of the index the trade, the market may ‘run away’ from (‘index volatility’); the strikes. In that case, even if volatilities (b) the volatility of the index as move in the expected directions, the trade computed using the implied will not make money, as the options that volatility of the components and the have become OTM have very little historical correlation matrix (’stock volatility exposure. volatility’). Some market participants are therefore — These volatility numbers will not match. using the variance swap volatility to create The market has its own view on future their volatility dispersion strategies. correlations (the so-called ‘implied correlation’ matrix), and it adjusts the historical correlation matrix in some VARIANCE SWAP VOLATILITY: A undeterminable fashion. For example, SHORT HISTORY when market participants are worried In 1996, Neuberger6 described the log about a crash, the historical correlation contract. This is an option whose payout is matrix is adjusted upwards. tied to the logarithm of the stock price. — Traders will typically adjust the historical Thus a call option would have a pay correlation matrix until these two out of volatility numbers match. The resultant matrix may lose its features (symmetric, max[ln(S) Ϫ X,0]
Nelken 339 where S is the stock price on expiration, X with a strike of k. It turns out that such a is the strike and ln is the natural logarithm portfolio has a constant ‘variance vega’.The function. The log contract has a unique dollar amount gained by such a portfolio feature in that it has stable ‘Greeks’.For for a unit change in the variance (volatility example, a contour plot of the vega of such squared) is insensitive to the stock price a contract versus the stock price results in a across a wide range of stock prices. The straight line. By contrast, a similar contour paper also described how this portfolio is plot for vanilla European options results in almost equivalent to a log contract. a non-linear curve. The main conclusion In 2003, the Chicago Board Options was that ‘the log contract provides a much Exchange (CBOE)9 announced a major easier and more reliable way of betting on change in the way that the volatility index volatility’. (VIX) was computed. This was followed by In 1998, in the aftermath of the Long the initiation of trading on the VIX in Term Capital Management (LTCM) 2004. The new VIX is fashioned after the debacle, implied volatility figures rose to variance swap portfolio previously unprecedented levels. As Gatheral7 explains introduced by Demeterfi et al.8 The generalised formula used by the ‘Variance swaps took off as a product in CBOE is the aftermath of the LTCM meltdown in 2 ⌬K 1 F 2 2 ϭ eRTQ(K ) Ϫ Ϫ 1 late 1998 when implied stock index 2 i ΄ ΅ T Ki T K0 volatility levels rose to unprecedented i levels. Hedge funds took advantage of this where T is the time to expiration; F is the by paying variance in swaps (selling the forward level of the stock (or index) as
realised volatility at high implied levels). derived from the options; Ki is the strike The key to their willingness to pay on a price of the ith OTM option, a call if K>F ⌬ variance swap rather than sell options was and a put if K realised volatility — no labour — typically $5); K0 is the first strike price below intensive delta hedging or other path the forward price F; R is the risk-free dependency is involved. Dealers were interest rate; and Q(Ki) is the quote for the happy to buy vega at these high levels specific option (it is taken as the mid price). because they were structurally short vega One of the main advantages of the way (in the aggregate) through sales of thenewVIXiscalculatedisthatitusesan guaranteed equity-linked investments to entire collection of options to come at the retail investors and were getting badly index rather than just a few options as hurt by high implied volatility levels.’ previously. Also, contrary to the normal implied volatility which relies on one’sown In 1999, Demeterfi et al.8 constructed a choice of model (eg Black-Scholes for portfolio of European options. Their European options), the VIX volatility does portfolio has ⌬k/k2 units of each option not rely on a particular choice of model. 340 Nelken Figure 1: Value of Dow Jones Index (DJI), index volatility and stock volatility March 2003–March 2004 11000 35 10000 30 25 9000 20 8000 15 10 7000 Annualised volatility Dow Jones Index level 5 6000 0 3/3/2003 5/3/2003 7/3/2003 9/3/2003 1/3/2004 3/3/2004 11/3/2003 DJI Index Vol Stock Vol Figure 2: Value of Dow Jones Index and difference between index volatility and stock volatility, March 2003–March 2004 11000 6 5 10000 4 9000 3 2 8000 1 0 7000 Dow Jones Index level –1 Differences (volatility points) 6000 –2 7/3/2003 3/3/2003 5/3/2003 9/3/2003 11/3/2003 1/3/2004 3/3/2004 DJI Index-Stock Vol SHOULD ONE TRADE VARIANCE DJI as given by the index options. To get SWAP VOLATILITY DISPERSION? to a 30-day running volatility, it is typical This paper looks at the Dow Jones Index. to use interpolation between the front The DJI is composed of 30 well-known month option and the second to front companies whose stocks are liquid. In month option. Also plotted is the volatility addition, it is a price-weighted index. which was obtained using the portfolio Figure 1 plots the index level on the formula, the implied volatility of the left-hand scale. The right-hand scale plots component options and a long-term the ATM 30-day implied volatility of the historical correlation matrix. In 2003, the Nelken 341 Figure 3: Value of Dow Jones Index (DJI), index volatility and stock volatility (from stock options using the variance swap methodology), March 2003–March 2004 11000 35 30 10000 25 9000 20 8000 15 10 7000 Dow Jones Index level 5 Differences (volatility points) 6000 0 3/3/2003 5/3/2003 9/3/2003 1/3/2004 7/3/2003 11/3/2003 3/3/2004 DJI Index Vol Stock Vol Figure 4: Value of Dow Jones Index (DJI), and difference between index volatility and stock volatility (computed from stock options using variance swap methodology) March 2003–March 2004 11000 7 6 10000 5 4 9000 3 2 8000 1 0 Dow Jones Index level 7000 –1 –2 Differences (volatility points) 6000 –3 3/3/2003 5/3/2003 7/3/2003 9/3/2003 11/3/2003 1/3/2004 DJI Index-Stock Vol index rose sharply and implied volatility volatility’) is different from the volatility numbers fell. computed from the portfolio formula Figure 2 plots the ATM volatility (‘stock volatility’). That formula used difference. This is the index implied historical correlation numbers. As volatility minus the index volatility as mentioned before, when market participants computed from the component (stock) are worried about a crash, the historical options. correlation matrix is adjusted upwards. As The index implied volatility (‘index the stock market trends upwards, fears of a 342 Nelken Figure 5: Volatility diferences, March 2003–March 2004 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00 Differences (volatility points) –1.00 –2.00 –3.00 5/3/2003 7/3/2003 9/3/2003 1/3/2004 3/3/2004 3/3/2003 11/3/2003 ATM VIX crash are ‘forgotten’, and the upward the ATM methodology. The volatility of adjustment to the correlation matrix the VIX chart is approximately 200 per decreases. This reduces the difference cent, while that of the ATM chart is about between the index volatility and the stock 338 per cent. volatility. This is clearly visible in Figure 2. Figures 3 and 4 repeat the same computations, but now using the implied SUMMARY volatility numbers that are computed using The theoretical results show that it may be the variance swap methodology. simpler to trade the volatility dispersion It is instructive to place both volatility strategy using the variance swap difference charts next to each other. Figure methodology. This is for two main reasons: 5 plots the difference that was computed using the ATM implied volatilities and the (1) It eliminates the constant rebalancing difference as computed by the volatility (‘chasing the spot’) requirement when dispersion methodology (the VIX using ATM options. approach). (2) The volatility difference, which is the It appears that the volatility difference quantity being traded, appears more using the VIX methodology is much less tame under the variance swap (VIX) volatile than the volatility difference using methodology. Nelken 343 It appears that some market participants have ‘Mean-Variance Analysis in Portfolio Choice and Capital Markets’, John Wiley, Chichester, UK. already adopted the variance swap (4) Nelken, I. (Ed.) (1997) ‘Volatility in the Capital methodology for volatility dispersion trading. Markets’, Glenlake, Chicago, IL. (5) Higham, N. (2002) ‘Computing the nearest correlation matrix — Aproblemfromfinance’, Acknowledgments IMA Journal of Numerical Analysis,Vol.22, pp. 329–343. The author wishes to thank Krag Gregory and (6) Neuberger, A. (1995) ‘The Log Contract and Venkatesh Balasubramanian at Goldman Sachs. Other Power Contracts’, in Nelken, I. (Ed.), ‘Handbook of Exotic Options’, Irwin Professional Publishing (now McGraw-Hill), Burr Ridge, IL. References (7) Gatheral, J. (2003) ‘Replication of Quadratic (1) Schneeweis, T. and Spurgin, R. (2001) ‘The Variation-based Payoffs’, Merrill Lynch, available benefits of index option-based strategies for at: http://www.math.nyu.edu/ institutional investors’, Journal of Alternative fellows_fin_math/gatheral/lecture7_2003.pdf. Investments, Spring, pp. 44–52. (8) Demeterfi,K.,Derman,E.,Kamal,M.andZou, (2) Bollen, N. P. and Whaley, R. (2002) ‘Does price J. (1999) ‘More Than You Ever Wanted to pressure affect the shape of implied volatility Know About Volatility Swaps’, Goldman Sachs functions?’, Working Paper, Duke University, Quantitative Strategies Research Notes,March. Durham, NC, USA. (9) For more details, see http://www.cboe.com/ (3) For an excellent guide to this see Markowitz, H. micro/vix/index.asp and also M., Todd, P. G. and Sharpe, W. F. (2000) http://www.cboe.com/micro/vix/vixwhite.pdf 344 Nelken