Correlation Trading using Options Derivatives-based Portfolio Solutions Due to diversification (or less than 100% correlation), the volatility of indices tends to trade significantly less than its constituents. The flow from both institutions and structured products tends to put upward pressure on implied correlation, making index implied volatility expensive. Hedge funds and proprietary trading desks try to profit from this anomaly by either selling correlation swaps, or through dispersion trading (going short index implied volatility and long single stock implied volatility). Selling correlation became an unpopular strategy following losses during the credit crunch, but demand is now recovering. INDEX IMPLIED LESS THAN SINGLE STOCKS DUE TO DIVERSIFICATIONThe volatility of an index is capped by the weighted average volatility of its members. In order to show this we shall construct a simple index of two equal weighted members who have the same volatility. If the two members are 100% correlated with each other, then the volatility of the index is equal to the volatility of the members (as they have the same volatility and weight, this is the same as the weighted average volatility of the constituents). If we take a second example of two equal weighted index members with the same volatility, but with a negative 100% correlation (ie, as low as possible), then the index is a straight line with zero volatility.Volatility of index has floor at zero when there is very low correlation Index volatility is bounded by zero and weighted average single stock volatilityWhile the simple examples above have an index with only two members, results for a bigger index are identical. Therefore, the equation below is true. While we are currently examining historical volatility, the same analysis can be applied to implied volatility. In this way, we can get an implied correlation surface from the implied volatility surfaces of an index and its single-stock members. However, it is usually easiest to look at variance swap levels rather than implied volatility to remove any strike dependency. where σi = single stock volatility (of ith member of index) wi = single stock weight in index (of ith member of index) n = number of members of index CORRELATION OF INDEX CAN BE ESTIMATED FROM VARIANCEIf the correlation of all the different members of an index is assumed to be identical (a heroic assumption, but a necessary one if we want to have a single measure of correlation), the correlation implied by index and single-stock implied volatility can be estimated as the variance of the index divided by the weighted average single-stock variance. This measure is a point or two higher than the actual implied correlation but is still a reasonable approximation.
STRUCTURED PRODUCTS LIFT IMPLIED CORRELATIONUsing correlation to visually cheapen payouts through worst-of/best-of options is common practice for structured products. Similarly, the sale of structured products, such as Altiplano (which receives a coupon provided none of the assets in the basket has fallen), Everest (payoff on the worst performing) and Himalayas (performance of best share of index), leave their vendors short implied correlation. This buying pressure tends to lift implied correlation above fair value. We estimate that the correlation exposure of investment banks totals approx. USD 250M per percentage point of correlation. The above formulae can show that two correlation points is equivalent to 0.3 to 0.5 (single-stock) volatility points. Similarly, the fact that institutional investors tend to call overwrite on single stocks but buy protection on an index also leads to buying pressure on implied correlation. The different methods of trading correlation are shown below. **Dispersion trading**. Going short index implied volatility and going long single-stock implied volatility is known as a dispersion trade. As a dispersion trade is short Volga, or vol of vol, the implied correlation sold should be approx. 10pts higher value than for a correlation swap. A dispersion trade was historically put on using variance swaps, but the large losses from being short single stock variance led to the single stock market becoming extinct. Now dispersion is either put on using straddles, or volatility swaps. Straddles benefit from the tighter bid-offer spreads of ATM options (variance swaps need to trade a strip of options of every strike). Using straddles does imply greater maintenance of positions, but some firms offer delta hedging for 5-10bp. A disadvantage of using straddles is that returns are path dependent. For example, if half the stocks move up and half move down, then the long single stocks are away from their strike and the short index straddle is ATM.**Correlation swaps**. A correlation swap is simply a swap between the (normally equal weighted) average pairwise correlation of all members of an index and a fixed amount determined at inception. Market value-weighted correlation swaps are approx. 5 correlation points above equal weighted correlation, as larger companies are typically more correlated than smaller companies. While using correlation swaps to trade dispersion is very simple, the relative lack of liquidity of the product is a disadvantage. We note the levels of correlation sold are typically approx. 5pts above realized correlation.**Covariance swaps**. While correlation swaps are relatively intuitive and are very similar to trading correlation via dispersion, the risk is not identical to the covariance risk of structured product sellers (from selling options on a basket). Covariance swaps were invented to better hedge the risk on structure books, and they pay out the correlation multiplied by the volatility of the two assets.**Basket options**. Basket options (or options on a basket) are similar to an option on an index, except the membership and weighting of the members does not change over time. The most popular basket option is a basket of two equal weighted members, usually indices.**Worst-of / Best-of option**. The pricing of worst-of and best-of options has a correlation component.**Outperformance options**. Outperformance options pricing has as an input the correlation between the two assets.
Implied correlation of dispersion and level of correlation swap are not the same measure We note that the profit from theta-weighted variance dispersion is roughly the difference between implied and realized correlation multiplied by the average single-stock volatility. As correlation is correlated to volatility, this means the payout when correlation is high is increased (as volatility is high) and the payout when correlation is low is decreased (as volatility is low). A short correlation position from going long dispersion (short index variance, long single-stock variance) will suffer from this as profits are less than expected and losses are greater. Dispersion is therefore short vol of vol; hence, implied correlation tends to trade approx.10 correlation points more than correlation swaps (which is approx. 5 points above realized correlation). We note this does not necessarily mean a long dispersion trade should be profitable (as dispersion is short vol of vol, the fair price of implied correlation is above average realized correlation). Implied vs realized correlation increases for low levels of correlation For example, in normal market conditions the SX5E and S&P500 will have an implied correlation of 50-70 and a realized of 30-60. If realized correlation is 30, implied will tend to be at least 50 (as investors price in the fact correlation is unlikely to be that low for very long; hence, the trade has more downside than upside). The NKY tends to have correlation levels ten points below the SX5E and SPX. SIZE OF DISPERSION MARKET SHRANK AFTER THE CREDIT CRUNCHSelling correlation led to severe losses when the market collapsed in 2008, as implied correlation spiked to approx. 90%, which led many investors to cut back exposures or leave the market. Similar events occurred in the market during the May 2010 correction. The amount of crossed vega has been reduced from up to $125M at some firms to $5-25M now (crossed vega is the amount of offsetting single-stock and index vega, i.e. $10M crossed vega is $10M on single stock and $10M on index). Similarly, the size of trades has declined from a peak of $3M to $0.5M vega now. CBOE INDICES ALLOW IMPLIED CORRELATION TO BE PLOTTEDThere are now correlation indices for calculating the implied correlation of dispersion trades calculated by the CBOE. As there are 500 members of the S&P500, the CBOE calculation only takes the top 50 stocks (to ensure liquidity). There are three correlation indices tickers (ICJ, JCJ and KCJ), but only two correlation indices are calculated at any one time. On any date one correlation index has a maturity up to one year, and another has a maturity between one and two years. The calculation uses December expiry for S&P500 options, and the following January expiry for the top 50 members as this is the only listed expiry (US single stocks tend to be listed for the month after index triple witching expiries). The index is calculated until the previous November expiry, as the calculation tends to be very noisy for maturities only one month to December index expiry. On the November expiry, the one month maturity (to S&P500 expiry) index ceases calculation, and the previously dormant index starts calculation as a two-year (and one-month) maturity index. |

### Trading Correlation ρ

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