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.