European Equity Derivatives Research J.P. Morgan Securities Ltd. London, 17 November, 2006 Variance Swaps • Variance swaps offer straightforward and direct exposure to the volatility of an underlying . 28 O • Returns from variance swaps can act as a diversifying asset • Variance swaps can be used for hedging volatility exposures : N or generating alpha Overview In this note we discuss the variance swap market, mechanics, pricing and uses. TRATEGIES Variance swaps offer straightforward and direct exposure to the volatility of an underlying asset. They are liquid across major equity indices and large cap S stocks, and increasingly across emerging market indices and other asset classes. Major uses include taking a volatility view, diversifying returns, hedging and relative value trading. Variance swaps can also be used to trade forward volatility and correlation. Variance swaps can be replicated by a delta-hedged portfolio of vanilla options, NVESTMENT I so that pricing reflects volatilities across the entire skew surface. In practice this means that variance swaps trade at a small premium to ATM implied volatilities. Variance swap cash-flows Buyer pays variance swap strike Buyer of Seller of Peter Allen variance variance (44-20) 7325-4114 swap swap [email protected] Seller pays Stephen Einchcomb realised variance (44-20) 7325-9064 at expiry [email protected] Nicolas GrangerAC (44-20) 7325-7033 Source: JPMorgan [email protected] The certifying analyst is indicated by an AC. See page 102 for analyst certification and important legal and regulatory disclosures. www.morganmarkets.com Peter Allen European Equity Derivatives Strategy (44-20) 7325-4114 17 November 2006 Stephen Einchcomb (44-20) 7325-9064 Nicolas Granger (44-20) 7325-7033 Table of Contents Part 1: Variance Swap Mechanics........................................................................................................................ 8 1.1: Realised volatility.......................................................................................................................................... 8 1.2: The variance swap contract ....................................................................................................................... 11 1.3: Vega notional and variance notional.......................................................................................................... 12 1.4: Variance swap convexity............................................................................................................................ 13 1.5: Variance swap example – accruing realised volatility................................................................................ 14 1.6: Variance swap mark-to-market .................................................................................................................. 15 1.7: Forward variance........................................................................................................................................ 17 1.8: Variance swap contract specifications ....................................................................................................... 19 1.9: Example Variance Swap Term Sheet ........................................................................................................ 22 Part 2: The Variance Swap Market..................................................................................................................... 24 2.1: Market development................................................................................................................................... 24 2.2: Historical prices.......................................................................................................................................... 26 2.3: Variance swaps and option volatilities ....................................................................................................... 28 2.4: Pricing rules of thumb ................................................................................................................................ 30 2.5: What drives variance swap levels?............................................................................................................ 33 2.6: The volatility risk premium.......................................................................................................................... 34 2.7: Variance swaps as predictor of future volatility.......................................................................................... 36 2.8: Is variance swap convexity fairly priced?................................................................................................... 38 2.9: Variance term structure.............................................................................................................................. 41 Part 3: Uses of Variance Swaps......................................................................................................................... 44 3.1: Exploiting a volatility view........................................................................................................................... 45 3.2: Specific hedging purposes ......................................................................................................................... 46 3.3: Rolling short variance................................................................................................................................. 47 3.4: Diversification............................................................................................................................................. 50 3.5: Index variance spreads .............................................................................................................................. 51 3.6: Relative value single-stock volatility........................................................................................................... 54 3.7: Variance dispersion and correlation trading............................................................................................... 56 3.8: Forward variance and volatility spikes ....................................................................................................... 58 3.9: Trading the variance swap term structure.................................................................................................. 64 3.10: Skew and convexity trades ...................................................................................................................... 66 3.11: Cross asset class trades: trading equity volatility against credit.............................................................. 68 Part 4: Replication and Hedging ........................................................................................................................ 72 4.1: Delta hedging and dollar gamma ............................................................................................................... 73 4.2: Theta - the cost of gamma ......................................................................................................................... 76 4.3: Options path-dependency: can volatility be captured by delta-hedging? .................................................. 77 4.4: From options to variance swaps ................................................................................................................ 79 4.5: Variance swap replication in one page ...................................................................................................... 81 4.6: Sensitivity to skew and convexity............................................................................................................... 82 4.7: Variance swap Greeks ............................................................................................................................... 84 4.8: Setting up a replicating portfolio................................................................................................................. 87 4.9: Replication and hedging in practice ........................................................................................................... 89 4.10: Effects of variance swap hedging ............................................................................................................ 90 4.11: Why not volatility swaps? ......................................................................................................................... 92 Part 5: Future developments.............................................................................................................................. 94 References ........................................................................................................................................................... 98 2 Peter Allen European Equity Derivatives Strategy (44-20) 7325-4114 17 November 2006 Stephen Einchcomb (44-20) 7325-9064 Nicolas Granger (44-20) 7325-7033 Introduction Introduction Variance swaps are instruments which offer investors straightforward and direct exposure to the volatility of an underlying asset such as a stock or index. They are swap contracts where the parties agree to exchange a pre-agreed variance level for the actual amount of variance realised over a period. Variance swaps offer investors a means of achieving direct exposure to realised variance without the path-dependency issues associated with delta-hedged options. Buying a variance swap is like being long volatility at the strike level; if