R&D /61 Volatility arbitrage indices – a primer

Keith Loggie, director global research & design at Standard & Poor’s Index Services, looks at the ­instruments and their context

n broad terms, volatility arbitrage This strategy is often implemented can be used to describe trading through a delta-neutral portfolio consist- strategies based on the difference in ing of an option and its underlying asset. Ivolatility between related assets – for The return on such a portfolio will be instance, the implied volatility of two based not on the future returns of the options based on the same underly- underlying asset, but rather on the ing asset. However, the term is most volatility of its future price movements. commonly used to describe strategies Buying an option and selling the that take advantage of the difference underlying asset results in a long between the forecasted future volatility volatility position, while selling an option of an asset and the implied volatility of and buying the underlying asset results options based on that asset. For more in a short volatility position. A long detail on volatility see ‘box’ on page 67. volatility position will be profitable to the

Payoff Implied Realised of Volatility Volatility – Volatility = Arbitrage

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extent that the realised volatility on the Implied Variance A long variance swap position will underlying is ultimately higher than the profit if the realised variance of the " The variance swap market has grown implied volatility on the option at the time Variance Variance underlying asset is greater than the ­exponentially over the past decade and is of the trade. implied variance at the time the swap is While delta-neutral options-based swap buyer swap seller struck. A variance swap provides pure among the most liquid equity derivatives trades provide a means for investing exposure to volatility, as, unlike options contracts in the over-the-counter markets." based on a view of future volatility, they Realised Variance prices, its value is based solely on do present some drawbacks. Since the changes to volatility. delta of an option changes as the price The payoff of a variance swap is equal of the underlying asset changes over feasible for traders as they cannot the other leg, the strike, is set at the to the difference between realised time, a portfolio consisting of an option constantly alter their hedge. Thus the ­inception of the swap and is based variance and implied variance, multiplied and its underlying asset that is initially position will generally not be solely on the squared amount of the implied by the number of contract units. The delta-neutral will soon no longer be so. dependent on volatility and is therefore ­volatility of the underlying asset at the number of contract units is set such that At this point, the performance of the not an ideal means of trading volatility. time the swap is struck. if the realised volatility is one volatility portfolio is no longer based solely on The strike price of the swap is point above the strike, the payoff to the volatility of the underlying asset but also Variance swaps determined by the implied volatility of the receiver of realised volatility will be equal on the performance of the underlying An alternative to options-based ­volatility options currently traded in the market to the notional value of the contract. asset. This can be prevented by trading is to use variance swaps. In based on the underlying asset. Thus a The variance swap market has grown ­continuous delta hedging, or ­rebalancing a variance swap, one leg is valued variance swap position is equivalent to a exponentially over the past decade and of the portfolio to ensure that it is delta based on the realised variance ­(volatility portfolio of options on the underlying is among the most liquid equity deriva- such as variance swaps will tend to be against the actual realised volatility for neutral. However, this not only creates squared) of the underlying asset, as asset and can be hedged in such tives contracts in the over-the-counter priced at a higher implied volatility than the following one-month period. The transaction costs, but also it is not measured by logarithmic returns, while a manner. markets. While the most liquid underly- is actually expected to be realised. start and end dates of each monthly ing index is the S&P 500, there are also Based on the above discussion, period are determined by the expiration markets in EuroStoxx 50, FTSE 100 and writers of options and investors who take date of VIX options, generally the third Exhibit 1: Implied and Realised Volatility of S&P 500 Nikkei 225 contracts as well. a short volatility position in a variance Friday of each month. One can see that An interesting characteristic of swap or similar product should profit in general the implied volatility is higher volatility is that for most traded assets, over time. Historical data bears this out. than the realised volatility. implied volatility tends to be higher than As noted above, the most popular The trend for implied volatility to be actual realised volatility. This is due to underlying index for variance swaps is higher than realised volatility is quite % the fact that options are often used as a the S&P 500 and the implied volatility of consistent. Exhibit 2 shows that implied 35 hedge or insurance. An investor will the S&P 500 is measured by VIX, the volatility exceeds realised volatility in 86 purchase a put option to hedge against CBOE Volatility Index. VIX measures the per cent of the months. Furthermore, large downward price movements or a implied volatility of the S&P 500 based the average and median differences 30 call option to hedge against large on the entire strip of S&P 500 options where implied exceeds realised is more upward price movements. contracts and uses the near-term and than the months where realised The options market is in many ways next-term options to calculate the exceeds implied. 25 similar to the home insurance market. implied volatility of the S&P 500 over the Investors using options buy protection next 30 calendar days. Indexation of volatility arbitrage 20 from changes in prices, much as a Exhibit 1 tracks the implied volatility of Over the past year, transparent homeowner purchases insurance the S&P 500 as measured by VIX ­indexation of volatility arbitrage has been against fire, flood damage, etc. In both 15 cases, the provider of the insurance is paid a premium to compensate for the risk that is being taken. 10 Just as insurance companies price Exhibit 2: One-Month Implied versus Realised Volatility their policies so that they expect to Implied Exceeds Realised Exceeds 5 make a profit based on estimates of their Realised Implied Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb Feb eventual payouts, options writers will Number (%) of Observations 190 (86.36%) 30 (13.64%) 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 only be willing to write options if they can Average Monthly Difference 5.34% 3.63% Implied Volatility Realised Volatility expect a sufficient profit to compensate for the risks they are assuming. Thus, Median Monthly Difference 5.00% 3.13% Source: Standard & Poor’s, CBOE. options and options-based structures Source: Standard & Poor’s. Data from February 1990 to June 2008.

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VIX between 12.00pm and 1.00pm on " The choice of vega is an important the roll date, less 1 per cent to account Index methodology ­determinant, and probably the principal for implementation slippage. The performance of the index is calculated As noted in the main text, the S&P 500 Volatility Arbitrage where: qualitative element in design of volatility based on the difference in the implied Index is rebalanced on the third Friday of each month. The Index = The closing price of the S&P 500 on day i. ­arbitrage indices." variance less the realised variance of the methodology approximates a one-month variance swap i S&P 500 from the prior rebalancing date ­position that receives implied variance on the S&P 500, as n = The number of trading days from, and including, through the current date. measured by the VIX, the Chicago Board Option Exchange the prior rebalancing date, t-1, but excluding the This difference is multiplied by the Volatility Index, and pays realised variance on the S&P 500. current rebalancing date, t. variance notional, equal to vega The total return version of the index incorporates interest

(a volatility exposure modifier) divided accrual on the notional value of the index. Interest accrues VarianceNotionalt = * Vega (4) introduced to serve as benchmarks for until the third Friday of the following by twice the strike price (which converts based on the one-month US dollar Libor rate. volatility arbitrage strategies and to serve month, at which time the position is volatility points into dollar amounts). where: as underlyings for index-linked products. rolled over. The index is calculated on a For more details see the ‘box’ on Index calculations Vega = A limit to the exposure on the spread. The S&P 500 Volatility Arbitrage Index price return (unfunded) and total return page 65. On any rebalancing date, t, the index is calculated as follows: 30 per cent is used for this index. measures the performance of a variance (funded) basis. The total return index The choice of vega is an important IndexValue = IndexValue (1 + VolArbStrategy ) (1) swap strategy that consists of receiving includes interest accrual on the notional determinant, and probably the principal t t-1 * t Total Return Index the implied variance of the S&P 500 value of the index based on the one- qualitative element in design of volatility where: A total return version of the index is calculated, which and paying the realised variance of the month US dollar Libor rate. arbitrage indices. As Exhibit 3 shows, the IndexValue = The index value as of the last includes interest accrual on the notional value of the index S&P 500. The implied variance used for the higher the vega, the higher the expected t-1 rebalancing date (t-1). based on the one-month US dollar Libor rate, as follows: The index assumes a one-month index, equivalent to the strike price of a return and risk from the index. For the variance swap is entered into on the corresponding variance swap, is S&P 500 Volatility Arbitrage index, the VolArbStrategy = A percentage, as determined t TRIV = TRIV (1+LIBOR (d /360) + VolArbStrategy ) third Friday of a given month and is held measured based on the average level of vega is set at a level of 30 per cent. by the following formula: t t-1 * t-1 * t-1,t t where: VolArbStrategy = VarianceNotional (IVS )2 – (RVS )2 (2) t t-1 * [ t-1 t-1,t ] TRIV = Total return index value as of the current Exhibit 3: Impact of Vega on Volatility Arbitrage Index Returns t where: rebalancing date, t.

35% t-1 = The last rebalancing date TRIVt-1 = Total return index value as of the last Return 34% rebalancing date, t-1. IVSt = Implied Volatility Strike. On any

Risk ­rebalancing date, t, it is represented by dt-1,t = Count of calendar days from the prior 30% the equally-weighted average of the levels rebalancing date, t-1, to the current rebalancing of VIX, the CBOE Volatility Index as date, t, but excluding the prior rebalancing date. published by the CBOE, minus 1 per 25% LIBOR = 1 month US dollar Libor rate as of the last cent. The average is calculated using the t-1 rebalancing date, t-1. index levels as published by the CBOE every five minutes from, and including, 20% Base Date 12:00pm to, and including, 01:00pm (ET), The index base date is February 16 1990 at a base value 19% divided by 100. For index history prior to of 100. 15% January 18 2007, IVSt is represented by 15% the average of the high value and low Index committee 13% value of VIX as published by the CBOE, The S&P Arbitrage Index Committee maintains the S&P 10% divided by 100, minus 1 per cent. 500 Volatility Arbitrage Index. The Index Committee meets regularly. At each meeting, the Index Committee reviews any RVS = Realised Volatility Strike of the S&P 500 7% 7% t-1,t significant market events. In addition, the Committee may 5% calculated using the following formula: revise index policy for timing of rebalancings or other matters. 4% Standard & Poor’s considers information about changes to 2% 0% its indices and related matters to be potentially market 10% 30% 50% 100% moving and material. Therefore, all Index Committee discus- RVS = (3) Vega t-1,t sions are confidential.

Source: Standard & Poor’s. Data from February 1990 to June 2008.

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Exhibit 4: Return and Risk Characteristics of S&P 500 Volatility Arbitrage Index Implied and realised volatility explained S&P 500 Volatility Arbitrage S&P 500 Implied volatility different implied volatility than those options prices. An example of this is Annualised Return Standard Deviation Annualised Return Standard Deviation Implied volatilities are derived from that expire in three months. This is the VIX calculation, a measure of 3 Year 8.26% 4.44% 4.69% 13.02% ­options prices. The implied volatil- referred to as the term structure of implied volatility of the S&P 500. 5 Year 10.31% 3.66% 7.75% 11.94% ity of an option is the level of volatility volatility. Also, options expiring at the 10 Year 11.49% 4.94% 3.49% 17.36% consistent with the current market same time but at different strike prices Realised volatility Since Inception 12.72% 4.38% 9.21% 14.75% price of the option. Of the factors that tend to be priced based on different The realised volatility of an asset is the Correlation with S&P 0.47 1.00 determine the value of an option, strike implied volatilities. This is due volatility of the actual historical ­returns 500 8 0 price, stock price, time to expiration, to what is known as the volatility skew. of an asset. Thus, unlike implied volatility, expected cash flows and the Options pricing models such as ­volatility which is a forward-looking Correlation with Lehman – 0.01 risk-free interest rate, volatility is the Black-Scholes make the simplifying prediction of volatility, realised volatility Aggregate 0.029 3 only factor that is unknown. assumption that returns of the is backward looking. For example, one Per cent of Month Positive 85.91% 61.36% The value of an option is dependent ­underlying asset are normally could calculate the realised volatility of Largest Drawdown 6.35% 41.71% on the future volatility of the underlying ­distributed. However, in reality this is the S&P 500 of June 2008 by ­taking Source: Standard & Poor’s. Data from February 1990 to June 2008. asset. As future volatility is unknown, not generally the case. Stock returns, the standard deviation of the daily the price of an option in the market will for example, tend to be long-tailed with returns of the index within that month. be determined by investors’ expecta- more observations far from the mean For variance swaps, realised ­volatility Exhibit 5: Performance of S&P Volatility Arbitrage since Inception tion of what the volatility of the asset than would be expected based on a is calculated based on logarithmic will be. Thus, for a given options normal distribution. Since the chances returns rather than arithmetic returns. 1,200 pricing model such as Black-Scholes, of an extreme observation are more The difference between implied the implied volatility of an option can likely than would be implied by a volatility and realised volatility is that be inferred based on the current normal distribution, the implied volatility implied volatility is a forecast of the 1,000 market price of the option. of far in the money and far out of future realised volatility of an asset. For a given underlying asset there the money options will tend to be While implied volatility is measured 800 will often be numerous options traded higher than that of at the based on the prices of options based on it, based on different strike prices money options. on the underlying asset, realised

600 and expiration dates. These options Because of the skew in the term volatility is calculated based on actual will generally not trade with the same structure of volatility there is not a historical returns of the underlying implied volatility. For example, single direct measure of implied asset. Implied and realised volatility will 400 ­investors may expect a quite different volatility of an underlying asset but differ for a given asset over a given volatility for an asset over the rather many measures based on the time period to the extent that investors’

200 next three months than over the various different options available on it. estimates of volatility as measured by next year. A benchmark of implied volatility for an options prices prior to the period in Thus options that expire in one underlying asset is generally calculated question differ from the actual volatility 0 year’s time will be priced with a by taking a weighted blend of multiple of the asset over that time. Feb 90 Feb 92 Feb 94 Feb 96 Feb 98 Feb 00 Feb 02 Feb 04 Feb 06 Feb 08 arbitrage indices, such as the S&P 500 Disclaimer This article is written by Standard & Poor’s, 55 ­Water Street, New York, NY 10041. Copyright © 2008. Standard & Poor’s (S&P) is a S&P Volatility Arbitrage Index S&P 500 Volatility Arbitrage Index, have shown division of The McGraw-Hill Companies, Inc. All rights reserved. Standard & Poor’s does not undertake to advise of changes in the information in this document. “S&P” and “S&P 500” are registered trademarks of The McGraw-Hill Companies, Inc. Other indices consistent positive performance and low are trademarks of respective index providers. These materials have been prepared solely for ­informational purposes based upon ­information generally available to the public Source: Standard & Poor’s. Data from February 1990 to June 2008. volatility. As with any arbitrage strat- from sources ­believed to be reliable. Standard & Poor’s makes no representation with respect to the accuracy or ­completeness of these materials, whose content may change without notice. Standard & Poor’s disclaims any and all liability relating to these egy that gains in popularity, increase in ­materials, and makes no express or implied representations or warranties concerning the statements made in, or omissions from, these materials. No portion of this publication may be reproduced in any format or by any means including electronically arbitrageurs may tighten the spreads or ­mechanically, by photocopying, recording or by any information storage or retrieval system, or by any other form or manner ­whatsoever, without the prior written ­consent of Standard & Poor’s. Alternative formulations of volatility strategy has outperformed the US equity negative return in only 31 months. The between implied and realised volatilities Standard & Poor’s does not guarantee the ­accuracy and/or completeness of any Standard & Poor’s Index, any data included therein, or any data from which it is based, and Standard & Poor’s shall have no liability for any errors, omissions, or interruptions arbitrage indices could use a different market as measured by the S&P 500 by strategy has yet negative return for any in the future. therein. Standard & Poor’s makes no warranty, express or implied, as to results to be obtained from the use of the S&P Indices. Standard & Poor’s makes no express or implied warranties, and expressly disclaims all warranties of merchantability or fitness for a variance swap tenor (for example more than 300 basis points per year. 12-month period and has a much lower However, as long as there is an particular purpose or use with respect to the S&P Indices or any data included therein. Without limiting any of the foregoing, in no one-month instead of three-months), a More importantly, the strategy has insurance premium in the options event shall Standard & Poor’s have any liability for any special, punitive, indirect, or consequential damages (including lost profits), maximum drawdown. even if notified of the possibility of such damages. different source of the strike (traders’ resulted in consistent positive returns. It market, there will be room for volatility Standard & Poor’s does not sponsor, endorse, sell, or promote any investment fund or other vehicle that is offered by third parties and that seeks to provide an investment return based on the returns of S&P Indices. A decision to invest in any such investment variance swap quotes instead of VIX), has a very low annualised standard Conclusion arbitrage. The key issue that will drive fund or other vehicle should not be made in reliance on any of the statements set forth in this document. Prospective investors are advised to make an investment in any such fund or vehicle only after carefully considering the risks associated with investing in such and a different vega. deviation of 4.38 per cent versus 14.75 Volatility arbitrage is morphing from returns in the future will be the size of the funds, as detailed in an offering memorandum or similar document that is prepared by or on behalf of the issuer of the investment fund or vehicle. The S&P 500 Volatility Arbitrage Index per cent for the S&P 500. Out of 220 a niche institutional strategy to mass protection-seeker market versus the size The S&P 500 Volatility Arbitrage Index was officially launched on January 17, 2008. The returns of the index from February 20, 1990 to January 17, 2008 are back-tested by applying the published index methodology. has history back to February 1990. The monthly observations, it has had a market, index-linked products. Volatility of the arbitrageur market.

the markit magazine – Autumn 08 Autumn 08 – the markit magazine