Lecture 11:Quantitative Option StrategiesVolatility Statistical Arbitrage

Marco Avellaneda

G63.2936.001

Spring Semester 2009

The theory…

Weighted MC Approach

• Use a simple model for the dynamics of the single stock relative to its

ETF• Model the residual volatility as a fraction of the total implied (ATM) vol

of the stock

• Calibrate this to all options on ETF and to the forward for the stock underconsideration, using Weighted Monte Carlo

regression of sense in the 1 2

,,

ii

Statmii

etftatmIi

it

it

R

dZdWS

dS

−=

+=

γ

σγσβ

Model Value vs. Market Value

( )( )

( )( )

( ) ( )( )

( ) ( ) ( )],,,,, [ ,,,

:Compare

,,,,,,,,

),,(,,,,,,

IVOLfor Solve

0,max),,(

offerbidmodel

model

TKSCTKSCTKSC

TKdrKTSCallAmericanBSTKSC

TKSCTKdrKTSBSCall

KSEeTKSC

imp

eurimp

TWMCrT

eur

σ

σ

=

=

−= −

A Brief History of Volatility 1990 – 2008:the CBOE VIX Index

GW1

Tequila

AsianFlu

LTCMRuble

.com

9/11

2002Crash

GW2

EM

Subprime

Volatility decreases from 2003 to 2006-2007

S&P 500 Volatility 8/2006-4/2008

Volatility Continuesto drop

Equity mkt.Ignores financials;Volatility doubles inat the end

Liquidity crunchRegime change in volatility

XLF (Financial S&P ETF) vs.VIX

Volatility drops are synchronous to Fed moves

Bear Stearns

Implementation of L/S Strategyin 2007

• Few ``buy’’ signals in early 2007

• Buying volatility gave poor results 2004-2006

• Theta-neutral portfolios gave poor results in 2004-2006

• Suggested selling volatility on strong signals in small amounts per name

• Hedging with ETFs (sector neutral)

• Hedging with SPY (``globally neutral’’)

• Synthetic insurance company: selling protection (Gamma) on many single names

• Hedging is the key to maintain market neutrality and get desired results(as we will see)

• In back-testing we use variance swaps instead of options for simplicity

Market-Neutrality matching different Greeks

stock

etfstocketf

etf

stockstocketf

stocketf

VV

VV

VV

σσ

σσ

⋅=

⋅=

=Vega Neutral

Theta Neutral

Gamma Neutral

net long veganet long gamma

collect time decaynet long vega

collect time decaynet long gamma

Portfolio profile Vega Ratio Exposure

Vega-N

eutral Per S

ector 2007C

apital ~ $1000 (all sectors)

Bias: long etfgam

ma, collect decay (“etflight”)

0 10 20 30 40 50 60

2007/01/03

2007/01/18

2007/02/01

2007/02/15

2007/03/02

2007/03/16

2007/03/30

2007/04/16

2007/04/30

2007/05/14

2007/05/29

2007/06/12

2007/06/26

2007/07/11

2007/07/25

2007/08/08

2007/08/22

2007/09/06

2007/09/20

2007/10/04

2007/10/18

2007/11/01

2007/11/15

2007/11/30

2007/12/14

+1 %, d

rawd

ow

n 4%

Vega-N

eutral 2006-2007C

apital ~ $1000 (all sectors)

Bias: long etfgam

ma, collect decay (“etflight”)

-50 0 50

100

150

200

2006/02/03

2006/03/08

2006/04/07

2006/05/10

2006/06/12

2006/07/13

2006/08/14

2006/09/14

2006/10/16

2006/11/15

2006/12/18

2007/01/23

2007/02/23

2007/03/27

2007/04/27

2007/05/30

2007/06/29

2007/08/01

2007/08/31

2007/10/03

2007/11/02

2007/12/05

+ 13%, d

rawd

ow

n 8%

Vega-N

eutral With S

PY

hedge

VE

GA

-NE

UT

RA

L HE

DG

E w

ith SP

Y C

AP

ITA

L = $1000

-6 -4 -2 0 2 4 6 8 10 12

2007/01/03

2007/01/16

2007/01/26

2007/02/07

2007/02/20

2007/03/02

2007/03/14

2007/03/26

2007/04/05

2007/04/18

2007/04/30

2007/05/10

2007/05/22

2007/06/04

2007/06/14

2007/06/26

2007/07/09

2007/07/19

2007/07/31

2007/08/10

2007/08/22

2007/09/04

2007/09/14

2007/09/26

2007/10/08

2007/10/18

2007/10/30

2007/11/09

2007/11/21

2007/12/04

2007/12/14

2007/12/27

Global vega hedging does not w

ork: too much decay, im

precise

Gamma-Neutral/Sector 2007Capital=1000

Bias: collect decay, long vega

0102030405060708090

100

2007

/01/

0320

07/0

1/19

2007

/02/

0520

07/0

2/21

2007

/03/

0820

07/0

3/23

2007

/04/

1020

07/0

4/25200

7/05

/1020

07/0

5/25

2007

/06/

1220

07/0

6/27

2007

/07/

1320

07/0

7/30

2007

/08/

1420

07/0

8/29

2007

/09/

1420

07/1

0/01

2007

/10/

1620

07/1

0/31

2007

/11/

15200

7/12

/0320

07/1

2/18

+ 4%, drawdown 5%

Gam

ma-N

eutral 2006-2007C

apital=1000

Bias: collect decay, long vega

-40-20 0 20 40 60 80100120140160

2006/02/03

2006/03/07

2006/04/05

2006/05/05

2006/06/06

2006/07/06

2006/08/04

2006/09/05

2006/10/04

2006/11/02

2006/12/04

2007/01/05

2007/02/06

2007/03/08

2007/04/09

2007/05/08

2007/06/07

2007/07/09

2007/08/07

2007/09/06

2007/10/05

2007/11/05

2007/12/05

+8%, d

rawd

ow

n 7%

Theta-N

eutral/Sector

2007 Capital=

1000

Bias: long gam

ma, long vega

Not enough collected decay

and liquidity crunch

-50

-40

-30

-20

-10 0 10 20 30

2007/01/03

2007/01/16

2007/01/26

2007/02/07

2007/02/20

2007/03/02

2007/03/14

2007/03/26

2007/04/05

2007/04/18

2007/04/30

2007/05/10

2007/05/22

2007/06/04

2007/06/14

2007/06/26

2007/07/09

2007/07/19

2007/07/31

2007/08/10

2007/08/22

2007/09/04

2007/09/14

2007/09/26

2007/10/08

2007/10/18

2007/10/30

2007/11/09

2007/11/21

2007/12/04

2007/12/14

-4%, d

rawd

ow

n 6%