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Keith Loggie, Director Global Research & Design at Standard & Poor's Index

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Keith Loggie, Director Global Research & Design at Standard & Poor's Index 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 Autumn 08 – the markit magazine /62 R&D R&D /63 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. the markit magazine – Autumn 08 Autumn 08 – the markit magazine /64 R&D R&D /65 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. variance less the realised variance of the methodology approximates a one-month variance swap i arbitrage indices." 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.
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