Volatility Futures at Eurex

Axel Vischer

Eurex Business Development Equity & Index Derivatives

November 2006

2

Agenda

n Trading Volatility: Available Instruments and Concepts

n Volatility Futures: Trading Volatility made easy

n Basics on Futures Pricing

n Applications

3

Realized vs. Implied Volatility

n Realized Volatility– Also referred to as historical volatility– Based on historical market data, e.g. 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.

4

Different types of measuring & trading Vola(Brief History)

1993 Introduction of VIX index at CBOE

1994 Introduction of VDAX at Deutsche Börse

1996 (Realized) Vola Futures by OMX

1997 Introduction of VDAX real-time + Sub-indices

1998 - Volax-Future at DTB (today Eurex)- First Volatility & Variance Swaps in the OTC market

2004 - VIX Future at the CBOE listed since March 2004- Variance Future at the CBOE

2005 - Introduction of Volatility indices VDAX-New, VSTOXX & VSMI- Volatility-Futures on VDAX-New, VSTOXX, VSMI at Eurex

5

Trading Volatility OTC (1/2)

n Volatility Swaps Contract on forward realized volatilityPayoff = Notional * ( realized volatility – volatility strike )

= N * (σv-Dvol)

n Variance Swaps Contract on forward realized variancePayoff = Notional * ( realized volatility2 – variance strike )

= N * (σ2v-Dvar)

where: σv = is the actual volatility of an index over the life of the contract,Dvol = the volatility specified by the swap,N = the notional amount of the swap (in dollar, euro, etc.) per a unit of volatility.

The fair value of a Volatility / Variance swap is the volatility/variance strike that makes the swap have zero value at inception.

6

Trading Volatility OTC (2/2)

Volatility Swap Variance Swap

- Linear Payoff - Non-linear Payoff

- Gain for swap owner from - Gain for swap owner from

20% volatility increase ~ 20% volatility increase >

Loss from 20% decrease Loss from 20% decrease

- Dynamic hedge - Static hedge

- Market Volume pretty small - estim. Market vol. in Europe > 80 bn EUR p.a.

7

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

8

Agenda

n Trading Volatility: Available Instruments and Concepts

n Volatility Futures: Trading Volatility made easy

n Basics on Futures Pricing

n Applications

9

Index Design: Link between Index and Future

Index concept and design need to anticipate demands from the users of volatility futures:

Issues facing Derivative Marketderivative user Participant

Ability to Hedge Market Maker / Issuers of structured products

Must represent Buy-SideVolatility Investor

Index

10

Index Designn Methodology established: VDAX®, VXO (old VIX)n Calculation method complicated (implied volatilities necessary)n Based upon a limited number of near ATM option pricesn Derivative would need to be hedged dynamicallyn In terms of hedging worst case scenarion OTC Swap Market negligible

n Stable and straightforward formula for evaluation using option prices directly not implied volatilitiesn More representative (using a strip of options)n Static hedge possiblen Swap market well establishedn Is NOT and does NOT look like volatility.

n Stable and straightforward formula for evaluation using options directly not implied volatilitiesn More representative (using a strip of options)n Can only be hedged dynamicallyn Is closely related to implied volatility but not identical.

– The information about the full skew is contained in the strips of options.

ImpliedATM Vola

ImpliedVariance

Square Root of ImpliedVariance

11

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

n Index formula represents the fair strike for a variance swap to a given expiry.

2

02

2 11)(2

−−××

∆= ∑ K

FT

KMRKK

T jj j

jσ

12

Volatility Indices: Comparison

5

15

25

35

45

55

65

75

03.0

1.20

00

03.0

4.20

00

03.0

7.20

00

03.1

0.20

00

03.0

1.20

01

03.0

4.20

01

03.0

7.20

01

03.1

0.20

01

03.0

1.20

02

03.0

4.20

02

03.0

7.20

02

03.1

0.20

02

03.0

1.20

03

03.0

4.20

03

03.0

7.20

03

03.1

0.20

03

03.0

1.20

04

03.0

4.20

04

03.0

7.20

04

03.1

0.20

04

03.0

1.20

05

03.0

4.20

05

03.0

7.20

05

03.1

0.20

05

03.0

1.20

06

Vola

tility

Lev

els

%

VDAX-NEWVSMIVSTOXX

13

Volatility Futures: Contract Specs

n Underlying: Volatility Index VSTOXX (for FVSX), VDAX-NEW (for FVDX) and VSMI (for FVSM)

n Contract Value: EUR1000 per index point (FVSX, FVDX), CHF 1000 per index point (FVSM)– The vega of the contract

n Minimum Price Movement: 0.05 of a point, equivalent to a value of EUR 50 and CHF 50 respectively– Typical minimum tick size in variance swap market is 0.10 points

n Last Trading Day: The Wednesday prior to the second last Friday of the expiring month (exactly 30 days before the next index option expiry, no index interpolation necessary)

n Contract Months: The three nearest calendar months and the next quarterly month of the February, May, August, and November cycle

n Daily Settlement: The closing price determined within the closing auction; If no price can be determined in the closing auction or if the price so determined does not reasonably reflect current market conditions, daily settlement price will be the last traded price within the last 15 minutes of Continuous Trading. If the last traded price is older than 15 minutes or does not reasonably reflect current market conditions, Eurex will establish the official settlement price.

n Final Settlement: Cash settled

n Final Settlement Price: Average over the index ticks of the last 30 minutes before expiration (FVSX: 11:30-12:00 CET, FVDX: 12:30-13:00 CET). Exception for FVSM, average over last 60 minutes: 9:00-10:00 CET.

n Trading hours: 09:00 a.m. until 17:30 p.m. CET

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Volatility Futures: Designated Market Making

n Minimum contract Size: 10 contracts

n Maximum Spread: 10% of bid price, i.e. Future bid = 17%, spread ≤ 1.7 points

n Required Coverage: 85 percent of the total trading period on a monthly average

n Expiry Range: All 4 available expirations have to be covered

n Incentive: 100%-fee rebate until 31st of March, 2007, for fulfilling the monthly obligations

n Current Market Makers: Merrill LynchOptiver

n Typical Spreads: Currently all maturities are quoted with spreads of around 0.5 volatility point

15

Agenda

n Trading Volatility: Available Instruments and Concepts

n Volatility Futures: Trading Volatility made easy

n Basics on Futures Pricing

n Applications

16

Implied Volatility Term Structure (as of Oct. 17th)

10

12

14

16

18

20

22

Oct 05 Nov 05 Dec 05 Mar 06 Jun 06 Sep 06 Dec 06 Jun 07

VDAX-NEWVSTOXXVSMI

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Arbitrage Bounds for Futures price

n Variance is additive:

n Mathematically, dueto the so-calledconvexity bias, thisapproach gives us an upper bound for thefair value

n One can show that a corresponding lowerbound can becalculated by theforward vola swaprate

Lower bound

calculated via imp. Forward vola swapcurve

approx. by ATM vola (e.g. „old“ VDAX subind.)

)E(RV)RV(EERVE212121 T,TT,TT,T ≤≤

Fair Value Upper bound

calculated via imp. Forward variance swapcurve

approx. by vola(e.g. VDAX-Newsubindices)

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Fair Future Values (example: VSTOXX, Oct. 17th 2005)

VSTO

XX

Subi

ndic

es 18.88

16.50

16.63

17.17

17.O

ct

21.O

ct

18.N

ov

16.D

ec

17.M

ar

19.O

ct

16.N

ov

21.D

ec

15.F

eb

20.J

an

inte

rpol

ated

Subi

ndic

esU

pper

bou

nds

19.05

16.67

16.66

16.99

3 d.

31 d.

59 d.

150 d.

1 d.

29 d.

64 d.

120 d.

30.d

30.d

30.d

30.d

16.41

16.59

17.88

(~ 16.9)

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Fair Future Values

12

14

16

18

20

19.09

.2005

20.09

.2005

21.09

.2005

22.09

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23.09

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24.09

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25.09

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26.09

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27.09

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28.09

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29.09

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30.09

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01.10

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02.10

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03.10

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04.10

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05.10

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06.10

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07.10

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08.10

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09.10

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10.10

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11.10

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12.10

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13.10

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14.10

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15.10

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16.10

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17.10

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18.10

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19.10

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20.10

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21.10

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Upper BoundSettlement PriceLower Bound

Vola-Futures FVDX (OCT 05)

14

15

16

17

18

19

20

19.09

.2005

20.09

.2005

21.09

.2005

22.09

.2005

23.09

.2005

24.09

.2005

25.09

.2005

26.09

.2005

27.09

.2005

28.09

.2005

29.09

.2005

30.09

.2005

01.10

.2005

02.10

.2005

03.10

.2005

04.10

.2005

05.10

.2005

06.10

.2005

07.10

.2005

08.10

.2005

09.10

.2005

10.10

.2005

11.10

.2005

12.10

.2005

13.10

.2005

14.10

.2005

15.10

.2005

16.10

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17.10

.2005

18.10

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19.10

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20.10

.2005

21.10

.2005

FV1FV2FV4

Daily Settlement Prices for FVDX

How good are the bounds? Fair Values for the different expiries

20

Price Comparison

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 +/- 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.g. 1month: √T = √(1/12) )– S…underlying index level (e.g. 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: 0.5 EUR– Euro Stoxx 50 index options contract fees:

• A- account (customer): 0.3 EUR…replication cost: 3 EUR

BlockTrades

OrderBook

Currency

1.801.20CHFFVSM

1.100.75EuroFVDX

0.750.50EuroFVSX

21

Risk Management

n Only vega exposure (sensitivity to volatility), no delta exposure (sensitivity to underlying cash index)

n Vega exposure is constant over time– Vega of a variance swap declines linearly over time and is zero at settlement of the contract

n Value at risk can be based on the sensitivity to changes in volatilityVSTOXX Index Weekly Returns 2000-2005

-3

2

7

12

17

22

-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

Return in %

Freq

uenc

y

VAR

22

Hedge Ratio

n Negative correlation between volatility and the underlying cash index– E.g. a 1% fall in the EURO STOXX 50 level triggers roughly a 2.6% rise in the

VSTOXX

n Goal: Hedge a 100 Mio. Euro Euro STOXX 50 index exposure– E.g. a 1% fall in the Euro STOXX 50 level costs 1 Mio. Euro

n The VSTOXX future delivers a 1000 Euro Vega exposuren Hedge Ratio

– 1 Mio. Euro / 2.6 / 1000 Euro = 385 FuturesVSTOXX vs. EuroStoxx

based on weekly returns 2000-2005

-50-40-30-20-10

01020304050

-15 -10 -5 0 5 10 15

Euro STOXX Return in %

VSTO

XX R

etur

n in

%

Regression:Y = -0.5 -2.6 * X

23

Agenda

n Trading Volatility: Available Instruments and Concepts

n Volatility Futures: Trading Volatility made easy

n Basics on Futures Pricing

n Applications

24

Uses for Volatility Futures (I)

nHedge equity market exposure:- equity returns and volatility changes are negatively correlated.

nHedge crash risk:- volatility rises significantly and rapidly in a crash scenario.

nHedge correlation exposure:- e.g. stock-picking becomes harder in a high correlation environment.

nHedge credit spread exposure:- volatility and credit spreads are linked.

n Trading Market Spreads- e.g. Volatility spread between US (VIX) and Europe (VSTOXX)

n Trading Calendar Spreads- Trading on changes in the volatility term strucuture

Hed

ging

Spr

ead

Trad

ing

25

Uses for Volatility Futures (II)

n Invest in volatility as an asset class in itself:- volatility can improve the risk and return characteristics of a balanced portfolio.

n Tactical asset allocation:- e.g. volatility can be a key input and risk in the asset allocation process.

n Take outright views on volatility:- e.g. volatility tends to be mean-reverting, speculators can take advantage of this characteristic.

nVolatility arbitrage:- trade implied versus realized volatility - trade Volatility futures against a delta-neutral portfolio of options

Spe

cula

ting

Arb

itrag

e

26

Further Information

n At the institutional investor section of www.eurexchange.comfurther information is available, e.g.,

– 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, CISDM

n For further questions please send an e-mail to

– Axel.Vischer@eurexchange.com– Investors@eurexchange.com