Volatility Derivatives

Menachem Brenner

Professor of Finance

Stern School of Business New York University

For presentation at the IRMC 2015

“The new risk management paradigm"

Luxembourg, June 15-16, 2015

OUTLINE

• BACKGROUND

• HISTORY OF VIX

• VIX METHODOLOGY

• VIX GRAPHS

• VIX RESEARCH

• VIX OBSERVATIONS

• VOLATILITY DERIVATIVES

• THE IMPLIED CORRELATION MARKET

2

BACKGROUND

BRIEF HISTORY OF DERIVATIVES MARKETS• PRE 1970 – COMMODOITIES (AGRIC, METALS)• POST 1970 – FINANCIALS (FX, DEBT, EQUITIES, CREDIT)

SIZE OF MARKETS (NOTIONAL)-GLOBAL• EXCHANGE TRADED – 60-70 $tr (NOTIONAL)• EQUITIES(ET) – 20-30 $tr (Notional) (e.g. Korea 100% of spot)• OVER THE COUNTER – 700 $tr (NOTIONAL)

OTC DERIVATIVE MARKETS:• INTEREST RATE DERIVATIVES – 550 $tr• FOREIGN EXCHNGE DERIVATIVES – 50 $tr• CREDIT DEFAULT SWAPS – 30 $tr• OTHERS (COMMODITIES, EQUITIES, CREDIT) – 70 $tr

• Spot Glob: Equity markets (40-50 tr), Bond markets (80-90 tr)

BACKGBACKBBABBBBCKGROUNDROUND

3

HISTORY OF VIX1973 OPTIONS TRADE ON CBOE, B-S-M MODEL PUBLISHED

1974 IMPLIED VOLATILITY IS DERIVED (Latane & Rendlman)

1983 INDEX OPTIONS START TRADING, PRICED BY B-S-MOPTIONS ARE QUOTED/TRADED BY IV (OEX)

1986 A VOLATILITY INDEX AND DERIVATIVES ON IT AREPROPOSED BY BRENNER AND GALAI, PUBLISHEDBY THE FINANCIAL ANALYSTS JOURNAL IN 1989

1987 OCT. 19 MARKET CRASHBefore the CRASH options on Equities prices with a “smile” due to stochastic volatility, jumps, “fat” tails

After the CRASH the “skew” was ‘born’ (world wide)Due to the observed negative correlation of S and σOther models suggested (e.g. CEV)

DEMAND FOR VOL. DERIVATIVES (variance swaps discussed by market players)

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HISTORY OF VIX (CONT’)1992 Missed Opportunity at the AMEX. VOL Index proposed

to AMEX board and at COLLOQUIUM IN NYC

1993 CBOE Launches VIX

THE JOURNAL OF DERIVATIVES PUBLISHES TWO PAPERS, ONE BY BRENNER-GALAI (methodology on next page) and ONEBY WHAELY. Same idea, METHODOLOGY somewhat different

2003 A GOLDMAN-SACHS TEAM LED BY DERMAN CONVINCES THE CBOE TO CHANGE METHODOLOGY (essentially model free (not B-S-M), use our basicapproach for a 30 day forecast based on 2 maturities

1990s There is an OTC market in Variance Swaps, dominatedby Goldman-Sachs

2004 FUTURES ON VIX START TRADING-LOW VOLUME

2005 FUTURES ON REALIZED VARIANCE-LOW VOLUME

2006 ETFs ON VIX ARE INTRODUCED (BASED ON FUTURES) 5

HISTORY OF VIX (CONT’)

2005 VSTOXX IS INTRODUCED

2006 OPTIONS ON VIX START TRADING-HUGE VOLUME

2007 CBOE COMPUTES VVIX

2012 NIKKEI VI IS INTRODUCED

2012-14 LARGE VOLUMES IN OPTIONS, FUTURES, ETFS(on April 15 2013; 450,000 futures) (see next chart)

THERE ARE VIXs ON EQUITY (INTERNATIONAL) INDICES, COMMODITIES, ETC.

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7

VIX METHODOLOGY (in brief)

VIX is a 30 days IV index derived from S&P500 index options using a weighted average of two near term maturities (today applied to many assets; e.g. oil, gold)

VIX (New Methodology): Using all OTM option prices weighted by the square of the strike price (model ‘free’)

VXO (Old Methodology): CBOE: a weighted average IV (B-S) of 8 ATM puts and calls of two near term options. Brenner-Galai version: Same as CBOE but using PRICES

VIX is a strike price and time weighted averages where the weights change with the strike price and time to expiration

A study by Carr & Wu (2006) shows a correlation of 98%between VIX and VXO

8

)1)(()(),30( 12111 wKCwKCKTC AA

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VIX(OLD)-METHODOLOGY (Brenner-Galai)

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VIX is obtained from converting C* using B-S

VIX-NEW METHODOLOGY

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VIX and VXN (Feb 2001-May 2015)

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(through 5/29/2015)

0

10

20

30

40

50

60

70

80

90

VXN VIX

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GARCH VOLATILIGARCH VS. VIX (Feb 1990-Feb 2015)

13

0

10

20

30

40

50

60

2/18/2010 8/18/2010 2/18/2011 8/18/2011 2/18/2012 8/18/2012 2/18/2013 8/18/2013 2/18/2014 8/18/2014 2/18/2015

VIX Index VSTOXX Index VHSI Index

VIX, VSTOXX and VHSI (Feb 2010- May 2015)

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(through 5/29/2015)

GARCH VOLATILITY(SP500 DAX HSI) (Feb 1990-Feb 2015)

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OVX and GVX (June 2008-Feb 2015)

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(through 2/18/2015)

0

20

40

60

80

100

120

OVX GVX

IMPLIED VOLATILITY OF $/Euro (EVZ) (Nov 2007-Feb 2015)

17

0

5

10

15

20

25

30

35

11/1/2007 11/1/2008 11/1/2009 11/1/2010 11/1/2011 11/1/2012 11/1/2013 11/1/2014

(through 2/18/2015)

VIX RESEARCH

IN GENERAL, THERE ARE THREE STRANDS OF

RESEARCH RELATED TO VIX:

1. USING THE INDEX AS A MEASURE/PROXY FOR MARKET VOLATILITY IN FINANCIAL MARKETS/ REAL ECONOMY STUDIES

2. STATISTICAL AND ECONOMIC ANALYSIS OF THEINDEX AND IT’S RELATION TO THE REALIZEDVARIANCE AND OTHER PARAMETERS(E.G. TRYINGTO EXPLAIN THE VARIANCE PREMIUM (VP)

3. ANALYZING THE VIX DERIVATIVES MARKETS; FUTURES , OPTIONS, ETFs

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OBSERVATIONS (Vol of Vol)

Observation I: Volatility is stochastic(see next graph)

Observation II: Volatility risk can cause disastrousloses (e.g. Barrings, LTCM)

Objective: Hedging Volatility Risk (Vol of Vol)

Dynamic Strategy with plain vanilla options (cost unknown until end, high TC)

Static Strategy; derivatives on a volatility Index (costKnown, low TC)

19

20

VIX and VVIX

0

20

40

60

80

100

120

140

160

6/1/06 6/1/07 6/1/08 6/1/09 6/1/10 6/1/11 6/1/12

VIX VVIX

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GARCH VOLATILITY OF VIX ( JAN 1990-MAR 2015)

VIX DERIVATIVES (observations)

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OPTIONS ON VIX ARE AMONG THE THREE MOST ACTIVE INDEX

OPTIONS (CALL VOLUME >> PUT VOLUME)

FUTURES & OPTION ON VIX; BIG VOLUMES BIG OPEN INTEREST.

BID-ASK SPREAD WIDE BY ANY MEASURE (10-20% FOR ATM)

PUT-CALL RATIO 0.6-0.8 (FOR SPX 1.2-1.4)

Futures on realized variance; no volume

ETFs (VXX, others) trade on NASDAQ; large volume

Futures & Options on VSTOXX (EUREX); moderate volume

Futures on Nikkie VI (OSAKA); launched (2012) low volume

VIX DERIVATIVES (Pricing)

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MAIN ISSUE: PRICING BY NO ARBITRAGE, underlying is

not traded

FUTURES: NO COC MODEL SINCE VIX IS NOT TRADED

OPTIONS: NO ARBITRAGE BASED MODEL (E.G. B-S-M)

How are the Options Priced? Of Spot? Of Futures?

Does Put-Call-Parity hold? What is the underlying?

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THE IMPLIED CORRELATION MARKET

THE IMPLIED CORRELATION MARKET IS NOT VERY NEW

BUT IT IS AN OTC MARKET WITH APPARENTLY FEW

PARTICIPANTS

IT IS POTENTIALLY A (STATISTICAL) ARBITRAGE MARKET

TWO EXAMPLES ARE PRESENTED NEXT

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AVERAGE CORRELATION (ALL STOCKS)

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AVERAGE CORRELATION (IMPLIED)

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23

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STEPS

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FX IMPLIED CORRELATION

TRIANGULAR RELATIONSHIP ($/€) = (¥/€)/(¥/$)

Ln($/€) = Ln[(¥/€)/(¥/$)] = Ln(¥/€) - Ln(¥/$) Ln($/€) =1 Ln(¥/€) =2 Ln(¥/$) =3

27

FX CORRELATIONS

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

USD/JPY(EUR)

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