INVESTMENT ACUMEN: AXA INVESTMENT MANAGERS’ RESEARCH REVIEW

MODELS/MARKETS

Some clues about dispersion trades Why the implied correlation of dispersion has to be higher than the correlation strike

Saad Slaoui - Senior fund manager - AXA Investment Managers Jerôme Vierling - Hedge fund manager - AXA Investment Managers

hedge funds as the imbalance between If diversification is still the principle or the main technique bank demand and the lack of correlation- used to decrease the risk of an equity portfolio and improve sellers pushes the correlation price - also its risk return profile, many investors see induced correlation called correlation strike - to an attractive exposure as an extreme risk. In fact, one of the empirical level. characteristics of the correlation is its strong increase in stressed markets, and this is why correlation exposure could It is an empirical fact that the implied be considered as a bad risk by portfolio managers. correlation resulting from a dispersion trade is not equal to the strike of a correlation swap in the market. The main Usually banks manufacture equity subject of this article is to explore this structured products with embedded fact theoretically. A correlation exposure correlation exposure, mainly short should exhibit a constant gamma profile correlation exposure. High correlation - gamma being the convexity/concavity can “cheapen” dramatically structured of a price according to its products’ optional prices. This is why underlying share movements, or of a banks were used to holding and hedging portfolio/index according to its component correlation exposure using dispersion movements. After having presented the trades. They recently decided to create classical variance and gamma replication, an instrument to transfer this risk to other we will explore two different ways of counterparties: the correlation swap. being exposed to the realised correlation: the dispersion trade and the correlation To offset part of this exposure - namely swap. We will consider different dispersion “buying back” some correlation - they have weighting schemes: vega flat, gamma flat the between either pure correlation and theta flat – vega being the sensitivity swaps or dispersion trades. In the recent of a derivative price to its underlying years, the correlation risk started to be share volatilities, theta being its yield over traded more actively in the market. Selling the risk-free rate. We will see that these correlation became more popular for various dispersion trades contain not only 43 an exposure to the realised correlation, The replication for this option is based on correlation and a strike (the market price). but also to the realised of implied both the Itô formula and Carr-Madan [Ca A good proxy for the correlation strike volatility itself. We will show that this Ma] findings. A long gamma swap position – also known as implied correlation - is the exposure is concave according to the consists of: square of the ratio between the implied and could explain the n A long position on a continuum of calls index volatility and the average of the difference between the implied correlation and puts weighted by the inverse of equity component implied volatilities (as of the dispersion trade and the strike of the the strike implied volatility is a good estimate of correlation swap. n A rolling underlying share/index future the forward-realised volatility, we could position then also infer that implied correlation is

Realised volatility instruments n A zero-coupon bond holding a good estimate of the forward-realised correlation). The correlation swap pay-off is Variance swap versus gamma swap the following:

A variance swap has the following payoff: Both variance and gamma swaps Notional * ( Realised correlation Notional * ( Realised variance – variance provide exposure to volatility. However, – correlation swap strike ) swap strike ) one of the main differences - from a management point of view - is that The average historical spread between Demeterfi, Derman, Kamal and Zhou variance swaps offer a constant cash the average-realised correlation and the [De De Ka Zh] gave a closed formula for gamma exposure (constant convexity), correlation swap strike of a Dow Jones variance swap valuations and showed that whereas gamma swaps provide a correlation swap usually fluctuates the variance swap is fully replicable by an constant share/index gamma exposure between 4% and 6%. The Eurostoxx 50 infinite number of European call and put (convexity proportional to the underlying correlation swap strike is trading in line options. A long variance swap position share/index price). Furthermore, as with the average-realised correlation. consists in being long a continuum of calls gamma swaps are weighted by the and puts weighted by the inverse of the performance of the underlying share/ Dispersion trade strike square. index, they take into account any jumps in the former; hence traders do not need The dispersion trade consists of a short The variance swap is attractive both for to cap it as is the case for variance position on the index volatility and a long trading and risk management purposes swaps in order to avoid massive losses one on its component volatilities. as it provides a pure variance exposure. on downside jumps. A dispersion trade also gives an Nevertheless, the variance swap is not the exposure to the implied correlation of only instrument to build a dispersion trade, Correlation instruments the market, since as aforementioned, a and sometimes not the most efficient. good proxy for it is the square of the ratio Correlation swap between the implied index volatility and Gamma Swap the average of the equity components Correlation measures the degree to implied volatilities. The spread between A gamma swap strongly looks like a which share prices move together. We the implied volatility of the index and the variance swap, but the daily square of consider here the average correlation average volatility of the components is the returns are weighted by the price level in its between all the equity pairs of an index. result of some correlation between the pay-off, which is the following: A correlation swap is an OTC derivative stocks (the simple principle of diversification). contract that allows an investor to bet on Therefore any decrease of this spread Notional * this average correlation. It is an instrument could be interpreted as an implied Gamma swap strike ) very similar to the variance swap, which correlation increase and any increase as Pt being the index/share price at time t pays at maturity the notional multiplied an implied correlation decrease. by the difference between the realised A redundancy exists between the average 44

of the component implied volatilities, the implied correlation and the index implied volatility. One could refer to only two of trade using gamma swap or variance those parameters as the third one could be swap, we could create an exposure which calculated as a function of the two others. is independent of the stock level. trades in the following lines and covered by a short correlation swap- is positively Weighting schemes between the index We firstly omitted this fact for simplification yielding over the risk-free rate. This trade volatility and the component volatilities purposes, but even for the vega flat is insensitive to the realised correlation. dispersion trade, the level of implied This is true only because we considered an All the following schemes are delta-hedged volatility (index & components) has an -free market omitting some risk positions constructed with variance impact. An important implied volatility factors. A yielding position always hides swaps. In order to build the dispersion shift (not only a local one) will change the some risk. Our aim is now to identify this trade, the ratio between index options and characteristics of the trade in terms of hidden risk and to give some clues so as to component options could be set so as to volatility/convexity exposure and yield. value them. avoid any: In fact, the trade should be dynamically

n Vega exposure: position of equal adjusted. The cost of these adjustments P&L of a dispersion trade sensitivity to implied volatility local could reduce the expected return of the movements according to the index trade. This is why many traders could The level of the implied volatility impacts and components’ sum. If the index decide to manage the portfolio with a both the ratio between the index-implied and component-implied volatilities are theoretical surface of volatility close to volatility exposures versus the component- constant, the daily P&L of this trade the market one. But if the change of the implied volatility exposure and the marked- depends exclusively on the realised implied volatility level is significant, they will to-market P&L of the trade. Spectacular correlation. need to adjust. implied correlation increases were

n Gamma exposure: position of equal observed during the spring of 2005 and convexity according to the index The liquidity issue should be pointed out 2006, linked to implied volatility increases. and components’ sum. This position since some of the components do not The P&L of the variance dispersion trade is is insensitive to any index and/or have enough liquid options in order to approximately worth: component movements; the realised build the pure dispersion trade. In such volatilities will then not impact the P&L. cases, the replication of the volatility This gamma flat scheme holds positive basket is incomplete. This liquidity issue vega exposure and has a negative theta. will force traders not to pick all the index being the weight of the ith component th n Theta: position yielding at the risk-free components, but the most liquid and being the implied volatility of the i rate. The theta flat scheme holds relevant ones (proxied by business sectors component negative vega and gamma exposure. for example). An incomplete basket being the average realised correlation composition could also be motivated by We will focus on the vega flat dispersion trades market anticipations. Bossu [Bossu] proves that:

trade. The same study could be done n The correlation swap can be dynamically for the gamma flat and the theta flat Dispersion trade versus quasi-replicated by a dispersion trade.

dispersion trades. correlation swap n The P&L of a variance dispersion trade can be approximated by a function of the Any dispersion trade could be built using The trade consisting in a long vega flat component-implied volatilities and the calls and puts, variance swaps, and dispersion trade constructed with variance realised correlation. gamma swaps. If we built a dispersion swaps - also called variance dispersion With these approximations, one could say 45 that the main risk factor of the variance correlation and the strike of the correlation This risk could explain the difference dispersion trade is the realised correlation. swap. This volatility exposure is a second between the strike of the pure correlation But we cannot avoid an exposure to the order one. It consists in a concave trade (correlation swap) and the implied random movement of the implied volatility exposure to the change of the implied correlation resulting from the variance (volatility of implied volatility) of the index volatility (called volga exposure). It means dispersion trade. components. that if the implied volatility moves up, the exposure to volatility will decrease and In an extended arbitrage-free world, this vice versa. Then the P&L of a portfolio volatility exposure should be rewarded locally covered in vega will be negatively by the difference between the implied impacted by implied volatility moves.

Conclusion References The P&L of a dispersion trade will depend on the realised correlation but also on the [Avellaneda] Avellaneda, M., (2002), realised volatility of implied volatility. In the vega flat scheme, the difference between Empirical aspects of dispersion trading the implied correlation of the dispersion trade and the correlation swap strike could in US equity markets, Petit Déjeuner de la finance, be explained by the volga of the dispersion trade and hence by the volatility of the Nov 2002 implied volatility. One should be aware of the additional risk of a concave position [Blanc] Blanc, N., (2004), Index variance arbitrage: Arbitraging component correlation, BNP Paribas against volatility changes. A different weighting scheme will add a first order Technical Studies exposure to implied volatility movements. [Bossu] Bossu, S., (2006), A new approach This kind of exposure to the realised volatility of implied volatility could be translated for modelling and pricing correlation swaps in equity derivatives, Global Derivatives into a curvature exposure. An equivalent exposure could be taken using classic Trading & Risk Management vanilla option positions and strategies like butterflies, ratios, or using more exotic [Branger] Branger, N., Schlag, C., (2003), products such as equity default swaps. Why is the index smile so steep?, Taking this kind of position implies anticipating a realised volatility of implied volatility. EFMA 2003 Helsinki Meetings

The curvature of the implied volatility surface reflects the market anticipation of the [Brenner] Brenner, M., OU, E.Y., realised volatility of implied volatility - the “volga market price”. The over-return of the Zhang, J.E (2006), Hedging volatility risk, Journal of Banking & Finance 30, 811-821 dispersion trade cannot be compared to the one of a free-lunch trade of a liquidity [Ca Ma] Carr, P., Madan, D., (1998), premium reward trade. The dispersion trade embeds a short curvature position. From Towards a theory of volatility trading, volatility: the relative trade correlation swap hedged by dispersion, using a volatility of volatility new estimation techniques for pricing derivatives, 417-427 model, one could extract a level of implied volatility of volatility. [De De Ka Zh] Demeterfi, K., Derman, E., Kamal, M., Zhou, J. (1999), A Guide to volatility and variance swaps, Acknowledgement Journal of Derivativess, 6(4),9-32 This document is a vulgarisation of an internal study [Ja Sa]. We decided to skip numerous formulas as we think the results could be intuitively understood. The math is just a confirmation of the dispersion trading experience. [Dup] Dupire, B., (2004), Understanding Explicit formulas are useful in the case of valuing the level of the volatility of implied volatility. In this case, one needs and implementing volatility and correlation to specify the volatility dynamics. This specification would allow anyone to estimate the risk that matches specific , Global Derivatives Conference market anticipations. This model would then allow one to compare different instruments with the same kind of exposure and to decide which strategy is the most efficient, taking into account the volatility of implied volatility, [Ja sa] Jacquier, A., Slaoui, S. (2006) liquidity risks and bid-offer spreads. variance dispersion and correlation swaps This study was done in the case of variance dispersion trades since it is the most popular way to introduce a dispersion trade into a portfolio. Of course, the results still hold for other constant gamma instruments and option portfolios with a constant gamma. The study was done in the Black & Scholes framework. Later, we relaxed the hypothesis of a constant implied volatility but we did not specify the dynamics of the volatility process.

This document is for professional advisors only and must not be relied upon by private investors. Issued by AXA Investment Managers GS Ltd registered in England No 3886111. The registered office address is 7 Newgate Street, London EC1A 7NX. AXA Investment Managers GS Ltd is authorised and regulated by the Financial Services Authority.