Volatility Futures at EurexAxel Vischer Eurex Business Development Equity & Index Derivatives November 2006 Agenda n Trading Volatility: Available Instruments and Concepts Volatility Futures: Trading Volatility made easy Basics on Futures Pricing Applications n n n 2 prices observed in the past Standard deviation of a stock’s or index’s returns over the last N days Return: natural logarithm of close-to-close price observations n Implied Volatility – – – Implied by observed option prices Iterative numerical procedure needed to extract implied volatility out of option prices Forward looking: market’s opinion about the expected volatility of the stock or index n Most of the time implied volatility is larger than realized volatility – The difference is the risk premium payable to the holder of the short volatility position.g.Realized vs. 3 . Implied Volatility n Realized Volatility – – – – Also referred to as historical volatility Based on historical market data. e. Introduction of Volatility indices VDAX-New. VSTOXX.Volax-Future at DTB (today Eurex) .First Volatility & Variance Swaps in the OTC market . VSTOXX & VSMI . VSMI at Eurex 4 2004 2005 .VIX Future at the CBOE listed since March 2004 .Volatility-Futures on VDAX-New.Different types of measuring & trading Vola (Brief History) 1993 1994 1996 1997 1998 Introduction of VIX index at CBOE Introduction of VDAX at Deutsche Börse (Realized) Vola Futures by OMX Introduction of VDAX real-time + Sub-indices .Variance Future at the CBOE . euro.) per a unit of volatility. 5 . = the volatility specified by the swap.Trading Volatility OTC (1/2) n Volatility Swaps Contract on forward realized volatility Payoff = Notional * ( realized volatility – volatility strike ) = N * (σv-Dvol) Variance Swaps Contract on forward realized variance Payoff = Notional * ( realized volatility2 – variance strike ) = N * (σ2v-Dvar) = is the actual volatility of an index over the life of the contract. etc. n where: σv Dvol N The fair value of a Volatility / Variance swap is the volatility/variance strike that makes the swap have zero value at inception. = the notional amount of the swap (in dollar. estim.Gain for swap owner from 20% volatility increase > Loss from 20% decrease . 6 .Market Volume pretty small Variance Swap . in Europe > 80 bn EUR p.Dynamic hedge .a.Gain for swap owner from 20% volatility increase ~ Loss from 20% decrease .Non-linear Payoff .Linear Payoff .Static hedge .Trading Volatility OTC (2/2) Volatility Swap . Market vol. Hedging Static Hedge: – – – – Hedge is implemented in an initial transaction No more adjustments necessary afterwards Hedged position is insensitive to market changes Preferred procedure for all hedgers Dynamic Hedge: – – – – – Hedge is implemented in an initial transaction Frequent adjustments necessary afterwards Hedged position is sensitive to market changes Frequency of readjustments can be regular or irregular. but generally driven by market changes The more readjustments the hedger has to perform during the life-time of the hedged position. the more expensive and inconvenient the dynamic hedging procedure is 7 . Agenda n Trading Volatility: Available Instruments and Concepts Volatility Futures: Trading Volatility made easy Basics on Futures Pricing Applications n n n 8 . Index Design: Link between Index and Future Index concept and design need to anticipate demands from the users of volatility futures: Issues facing derivative user Ability to Hedge Index Must represent Volatility Buy-Side Investor Derivative Market Participant Market Maker / Issuers of structured products 9 . Index Design n n Implied ATM Vola n n n n Methodology established: VDAX®. Square Root of Implied Variance n n n n Stable and straightforward formula for evaluation using options directly not implied volatilities More representative (using a strip of options) Can only be hedged dynamically Is closely related to implied volatility but not identical. 10 . VXO (old VIX) Calculation method complicated (implied volatilities necessary) Based upon a limited number of near ATM option prices Derivative would need to be hedged dynamically In terms of hedging worst case scenario OTC Swap Market negligible Implied Variance n n n n n Stable and straightforward formula for evaluation using option prices directly not implied volatilities More representative (using a strip of options) Static hedge possible Swap market well established Is NOT and does NOT look like volatility. – The information about the full skew is contained in the strips of options. Volatility Indices: Methodology n Evaluation of Sub-index per option expiry based on the square-root of implied variance – – 100 times the square root of σ2 The first 8 expirations are covered by sub-indices n All sub-indices are calculated real time – Update frequency of 1 minute n Construction of rolling index at 30 days to expiration through linear interpolation of the two nearest sub-indices Index formula represents the fair strike for a variance swap to a given expiry. ∆K 2 1 F σ = ∑ 2j × R × M ( K j ) − − 1 T j Kj T K 0 2 2 n 11 . 01.04.10.2003 03.07.04.10.2000 03.2005 03.10.2005 03.04.04.01.07.2000 03.2003 03.10.2000 03.07.2005 03.2002 03.2006 Volatility Indices: Comparison 5 VDAX-NEW VSMI VSTOXX 12 .04.2004 03.10.2000 03.2001 03.2002 03.01.01.01.07.2001 03.07.2001 03.2002 03.2004 03.2004 03.2003 03.01.01.2001 03.07.2003 03.2002 03.04.10.2004 03.Volatility Levels % 15 25 35 45 55 65 75 03.2005 03. FVDX).m. no index interpolation necessary) Contract Months: The three nearest calendar months and the next quarterly month of the February. Eurex will establish the official settlement price. daily settlement price will be the last traded price within the last 15 minutes of Continuous Trading. Exception for FVSM. until 17:30 p. VDAX-NEW (for FVDX) and VSMI (for FVSM) Contract Value: EUR1000 per index point (FVSX.Volatility Futures: Contract Specs n n Underlying: Volatility Index VSTOXX (for FVSX). FVDX: 12:30-13:00 CET). equivalent to a value of EUR 50 and CHF 50 respectively – Typical minimum tick size in variance swap market is 0. CHF 1000 per index point (FVSM) – The vega of the contract Minimum Price Movement: 0. average over last 60 minutes: 9:00-10:00 CET. Trading hours: 09:00 a. August.05 of a point. If no price can be determined in the closing auction or if the price so determined does not reasonably reflect current market conditions. and November cycle Daily Settlement: The closing price determined within the closing auction. Final Settlement: Cash settled Final Settlement Price: Average over the index ticks of the last 30 minutes before expiration (FVSX: 11:30-12:00 CET. May. If the last traded price is older than 15 minutes or does not reasonably reflect current market conditions.10 points Last Trading Day: The Wednesday prior to the second last Friday of the expiring month (exactly 30 days before the next index option expiry. CET n n n n n n n 13 .m. for fulfilling the monthly obligations Merrill Lynch Optiver Currently all maturities are quoted with spreads of around 0. i.7 points 85 percent of the total trading period on a monthly average All 4 available expirations have to be covered 100%-fee rebate until 31st of March.Volatility Futures: Designated Market Making n n Minimum contract Size: Maximum Spread: Required Coverage: Expiry Range: Incentive: 10 contracts 10% of bid price.5 volatility point n n n n Current Market Makers: n Typical Spreads: 14 . spread ≤ 1.e. 2007. Future bid = 17%. Agenda n Trading Volatility: Available Instruments and Concepts Volatility Futures: Trading Volatility made easy Basics on Futures Pricing Applications n n n 15 . 17th) 18 16 14 12 10 Oct 05 Nov 05 Dec 05 Mar 06 Jun 06 Sep 06 Dec 06 Jun 07 16 .Implied Volatility Term Structure 22 VDAX-NEW VSTOXX VSMI 20 (as of Oct. this approach gives us an upper bound for the fair value One can show that a corresponding lower bound can be calculated by the forward vola swap rate Lower bound Fair Value Upper bound E RVT1 . due to the so-called convexity bias.T2 ≤ E E (RVT1 . VDAX-New subindices) 17 . Forward variance swap curve approx. „old“ VDAX subind.g. by vola (e. by ATM vola (e.g.) calculated via imp. Forward vola swap curve approx.T2 ) ≤ E(RVT1 .Arbitrage Bounds for Futures price n Variance is additive: n n Mathematically.T2 ) calculated via imp. Nov 18.Mar 18 16. 17th 2005) 18.Feb 20.Oct . 16.d (~ 16.41 30.Oct 21. 17. 16.Fair Future Values VSTOXX Subindices 3 d.Jan 17.Nov 16.d Upper bounds 16.88 17.d 16.66 120 d. Oct.88 31 d. 16.17 interpolated Subindices 1 d. (example: VSTOXX.05 29 d. 16. 19.d 17.Dec 21.Oct 19.63 150 d.50 59 d.67 64 d.59 30.Dec 15. 16.99 30.9) 30. 1 05 0 18 . 20 .20 .20 .1 05 0 19 .0 05 9 25 .20 .0 05 9 29 .20 . 1 05 0 09 . 20 .0 05 9 22 .1 05 0. 20 .0 005 9 21 .20 .20 .20 . 20 .20 . 20 05 19 .20 .1 05 0.0 05 9.20 . 1 05 0 06 .1 05 0 16 . 1 05 0 06 .20 . 0 05 9 24 . 20 .0 05 9 28 .20 . 20 . 20 .20 .19 Fair Future Values 12 14 16 18 20 How good are the bounds? Vola-Futures FVDX (OCT 05) Upper Bound Settlement Price Lower Bound . 20 .20 .20 .0 9 20 .1 05 0 11 .20 . 0 05 9 27 .0 05 9 29 .1 05 0 05 . 1 05 0 21 . 20 . 20 . 20 . 1 05 0 03 .1 05 0 07 .0 05 9 01 . 1 05 0 10 .0 05 9 30 .20 . 1 05 0 02 .20 .1 05 0 17 . 20 .20 . 20 . 2 . 20 .20 .20 .0 9. 0 05 9 30 . 20 . 20 05 19 14 15 16 17 18 19 20 20 Fair Values for the different expiries Daily Settlement Prices for FVDX FV2 FV4 FV1 .20 .1 05 0 08 . 1 05 0 13 .1 05 0 02 .1 05 0 09 . 0 05 9 28 .1 05 0 04 . 0 05 9 21 .20 .20 . 20 . 0 05 9 23 . 20 . 20 .0 05 9 01 . 1 05 0 15 .20 . 20 .20 .20 .20 .20 . 1 05 0 14 .0 05 9 26 .1 05 0 05 .20 .1 05 0 19 .1 05 0 14 .1 05 0 21 . 20 .0 05 9 25 .1 05 0 11 .1 05 0 04 .20 .20 .1 05 0 20 .1 05 0 07 . 20 .0 05 9 23 .1 05 0 08 .20 . 1 05 0 15 .20 . 20 . 20 . 0 05 9 24 .1 05 0 13 .1 05 0 20 .1 05 0 16 . 0 05 9 27 .20 . 22 20 . 12 20 .1 05 0 10 . 20 .20 .1 05 0 17 .1 05 0 03 . 1 05 0.20 . 1 05 0 18 .0 05 9 26 . 1 05 0 12 . Price Comparison FVSX FVDX FVSM Currency Euro Euro CHF Order Book 0.3 EUR…replication cost: 3 EUR 20 .g.20 Block Trades 0.1000 EUR (vega) n ATM-options on the EURO STOXX 50 index – – – – Vega ~ M S √T / 100 ~ 95 EUR M…multiplier (10 EUR/index point) T…annualized time to maturity ( e. around 3300 index points) n About at least 10 ATM EURO STOXX 50 Index options are needed to provide the same vega as the VSTOXX future – – VSTOXX contract fee: 0.10 1.50 0.5 EUR Euro Stoxx 50 index options contract fees: • A.75 1.80 n Compare Vegas for a 1% Volatility change between volatility futures and their underlying index options – Measure about how much cheaper the new volatility futures are to build up pure vega positions n VSTOXX future – A 1% volatility increase/decrease changes the contract value by +/. 1month: √T = √(1/12) ) S…underlying index level (e.g.75 1.account (customer): 0. Risk Management n n n Only vega exposure (sensitivity to volatility). no delta exposure (sensitivity to underlying cash index) Vega exposure is constant over time – Vega of a variance swap declines linearly over time and is zero at settlement of the contract Value at risk can be based on the sensitivity to changes in volatility VSTOXX Index Weekly Returns 2000-2005 22 17 Frequency 12 7 VAR 2 -25 -23 -21 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 -3 Return in % 21 . 6 / 1000 Euro = 385 Futures VST O XX Re tu rn in % 50 40 30 20 10 0 -10 -20 -30 -40 -50 -15 based on weekly returns 2000-2005 Regression: Y = -0.6% rise in the VSTOXX E.g.g. Euro Euro STOXX 50 index exposure – The VSTOXX future delivers a 1000 Euro Vega exposure Hedge Ratio VSTOXX vs. Euro n n n Goal: Hedge a 100 Mio.Hedge Ratio n Negative correlation between volatility and the underlying cash index – E. a 1% fall in the Euro STOXX 50 level costs 1 Mio. a 1% fall in the EURO STOXX 50 level triggers roughly a 2. EuroStoxx – 1 Mio.6 * X -10 -5 0 5 10 15 Euro STOXX Return in % 22 .5 -2. Euro / 2. Agenda n Trading Volatility: Available Instruments and Concepts Volatility Futures: Trading Volatility made easy Basics on Futures Pricing Applications n n n 23 . Hedging n Hedge n Hedge n Hedge . . Spread Trading n Trading n Trading Market Spreads Calendar Spreads .volatility and credit spreads are linked.Uses for Volatility Futures (I) n Hedge equity market exposure: crash risk: correlation exposure: credit spread exposure: .volatility rises significantly and rapidly in a crash scenario. .e. Volatility spread between US (VIX) and Europe (VSTOXX) .g.Trading on changes in the volatility term strucuture 24 .equity returns and volatility changes are negatively correlated.g.e. stock-picking becomes harder in a high correlation environment. trade implied versus realized volatility .g.Uses for Volatility Futures (II) Speculating n Invest in volatility as an asset class in itself: asset allocation: . outright views on volatility: .e. n Tactical n Take .g.e.trade Volatility futures against a delta-neutral portfolio of options 25 . speculators can take advantage of this characteristic. volatility can be a key input and risk in the asset allocation process. Arbitrage n Volatility arbitrage: .volatility can improve the risk and return characteristics of a balanced portfolio. volatility tends to be mean-reverting. Further Information n At the institutional investor section of www. CISDM – n For further questions please send an e-mail to – Axel. – Thesis: Volatility and its Measurements: The Design of a Volatility Index and the Evaluation of its Historical Time Series at the Deutsche Börse AG Various papers on derivatives in alternative investment from Edhec Business School and Tom Schneeweiss.com further information is available.eurexchange..com 26 .g.com – [email protected]@eurexchange. e.