1Empirical Aspects of Dispersion Trading in U.S. Equity Markets

Marco AvellanedaCourant Institute of Mathematical Sciences, New York University

& Gargoyle Strategic Investments

Petit Dejeuner de la FinanceParis, Nov 27, 2002

What is Dispersion Trading?

Sell index option, buy options on index components (sell correlation)

Buy index option, sell options on index components (buy correlation)

Motivation: to profit from price differences in volatility marketsusing index options and options on individual stocks

Opportunities: Market segmentation, temporary shifts in correlations between assets, idiosyncratic news on individual stocks

2Index Arbitrage versus Dispersion Trading

Stock 1

Index

Stock N

Stock 3

Stock 2

*

*

*

*

Index Arbitrage:Reconstructan index product (ETF)using thecomponent stocks

Dispersion Trading:Reconstruct an index optionusing options on the component stocks

Main U.S. indices and sectors

Major Indices: SPX, DJX, NDXSPY, DIA, QQQ (Exchange-Traded Funds)

Sector Indices: Semiconductors: SMH, SOX

Biotech: BBH, BTKPharmaceuticals: PPH, DRG

Financials: BKX, XBD, XLF, RKHOil & Gas: XNG, XOI, OSX

High Tech, WWW, Boxes: MSH, HHH, XBD, XCIRetail: RTH

3COMS CMGI LGTO PSFTADPT CNET LVLT PMCSADCT CMCSK LLTC QLGCADLAC CPWR ERICY QCOMADBE CMVT LCOS QTRNALTR CEFT MXIM RNWKAMZN CNXT MCLD RFMDAPCC COST MEDI SANMAMGN DELL MFNX SDLIAPOL DLTR MCHP SEBLAAPL EBAY MSFT SIALAMAT DISH MOLX SSCCAMCC ERTS NTAP SPLSATHM FISV NETA SBUXATML GMST NXTL SUNWBBBY GENZ NXLK SNPSBGEN GBLX NWAC TLABBMET MLHR NOVL USAIBMCS ITWO NTLI VRSNBVSN IMNX ORCL VRTSCHIR INTC PCAR VTSSCIEN INTU PHSY VSTRCTAS JDSU SPOT WCOMCSCO JNPR PMTC XLNXCTXS KLAC PAYX YHOO

COMS CMGI LGTO PSFTADPT CNET LVLT PMCSADCT CMCSK LLTC QLGCADLAC CPWR ERICY QCOMADBE CMVT LCOS QTRNALTR CEFT MXIM RNWKAMZN CNXT MCLD RFMDAPCC COST MEDI SANMAMGN DELL MFNX SDLIAPOL DLTR MCHP SEBLAAPL EBAY MSFT SIALAMAT DISH MOLX SSCCAMCC ERTS NTAP SPLSATHM FISV NETA SBUXATML GMST NXTL SUNWBBBY GENZ NXLK SNPSBGEN GBLX NWAC TLABBMET MLHR NOVL USAIBMCS ITWO NTLI VRSNBVSN IMNX ORCL VRTSCHIR INTC PCAR VTSSCIEN INTU PHSY VSTRCTAS JDSU SPOT WCOMCSCO JNPR PMTC XLNXCTXS KLAC PAYX YHOO

QQQ trades as a stock

QQQ options: largest daily traded volume in U.S.

NASDAQ-100Index (NDX)

and ETF (QQQ)

Capitalization-weighted

QQQ ~ 1/40 * NDX

Sector Exchange Traded FundsXNG

APAAPCBRBRREEXENEEOGEPGKMINBLNFGOEIPPPSTRWMB

SOX

ALTRAMATAMDINTCKLACLLTCLSCCLSIMOTMUNSMNVLSRMBSTERTXNXLNX

XOI

AHCBPCHVCOC.BXOMKMGOXYPREPRDSUNTXTOTUCLMRO

~ 20 - 40 stocksin samesector

Weightings by:

capitalization equal-dollar equal-stock

4Index Option Arbitrage (Dispersion Trading)

Takes advantage of differences in implied volatilities of index options and implied volatilities of individual stockoptions

Main source of arbitrage: correlations between asset pricesvary with time due to corporate events, earnings, and ``macro shocks

Full or partial index reconstruction

The trade in pictures

Index

Stock 1 Stock 2

Sell index call

Buy calls on different stocks.Also, buy index/sell stocks

5Profit-loss scenarios for a dispersion trade in a single day

-2

-1.5

-1

-0.5

0

0.5

1

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15stock #

sta

nda

rd m

ov

e

-3-2.5

-2-1.5

-1-0.5

00.5

11.5

22.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15stock #

sta

nda

rd m

ov

e

Scenario 1 Scenario 2

Stock P/L: - 2.30Index P/L: - 0.01Total P/L: - 2.41

Stock P/L: +9.41Index P/L: - 0.22Total P/L: +9.18

( ) ( )

( ) ( ) ,,,,

0,max0,max

1

1

1

TKSCwTKIC

KSwKI

KwK

iii

M

jiI

ii

M

ji

i

M

ji

=

=

=

=

First approximation to hedging:``Intrinsic Value Hedge

'``divisor'by scaled shares, ofnumber 1

===

ii

M

ii wSwI

IVH:premium from indexis less than premium from components Super-replication

Makes sense for deep--in-the-money options

IVH: use indexweights for optionhedge

6Intrinsic-Value Hedging is `exact only if stocks are perfectly correlated

( ) ( )

( )( ) ( )( ) TKTSwKTI

eFK

eFwKX

NN

eFwTSwTI

M

iiii

TX

ii

TX

i

M

ii

iij

TN

i

M

iii

M

ii

ii

ii

iii

=

=

=

=

==

=

=

==

0,max0,max

:Set

:in for Solve

normal edstandardiz 1

1

21

21

1

21

11

2

2

2

Similar to Jamshidian (1989)for pricing bond options in 1-factormodel

IVH : Hedge with ``equal-delta options

( )

constant tas Del constant moneyness-log

constant N 21ln1

21ln1

2

2

21 2

=

=

=

+

==

d

dTKF

TX

TFK

TXeFK

ii

i

i

ii

i

i

TTX

iiii

7What happens after you enter a trade:Risk/return in hedged option trading

!

" # $" # %" & $" & %" ' $" ' %(") $ $" ) $ %") ) $") ) %" ) * $") * %") + $

Unhedged call option Hedged option

Profit-loss for a hedged single option position (Black Scholes)

( )

==

==

+

CNVtS

Sn

dNVnLP

Vega normalized , (dollars),decay - time

1/ 2

n ~ standardized move

Gamma P/L for an Index Option

( )

( ) ( )

1 Index P/L

1 Gamma P/LIndex

22

12

22

1

2

1

1

2

ijjiji I

jijiIi

M

i I

iiI

ijjij

M

ijiI

M

jjj

iiii

M

i I

iiI

II

nnpp

np

pp

Sw

Swpnpn

n

+=

=

==

=

=

=

=

=

Assume 0=d

8Gamma P/L for Dispersion Trade

( )

( ) ( )ijjiji I

jijiIi

M

iI

I

iii

ii

nnpp

np

n

+

+

=

22

12

22

2th

1 P/LTrade Dispersion

1 stock P/L i

diagonal term:realized single-stock movements vs.implied volatilities

off-diagonal term:realized cross-market movements vs. implied correlation

Introducing the Dispersion Statistic

( )

( ) ( )

+=

++=

++=

+=

=

=

==

=

===

==

=

=

=

22

2

12

22

2

1

222

1

222

1

2

1

2

1

2

22

1

22

1

222

2

1

2

11 P/L

,

Dnnp

nnpnpn

nn

nn

nnpD

IIY

SSXYXpD

I

Ii

N

ii

I

iiiI

II

N

iiii

I

IN

iiii

I

IN

iii

I

N

iiII

N

iii

IIi

N

ii

II

N

iiii

i

iii

N

ii

9Summary of Gamma P/L for Dispersion Trade

+=

=

22

2

12

22

Gamma P/L DnnpI

Ii

N

ii

I

iiiI

Idiosyncratic Gamma

Dispersion Gamma

Time-Decay

Example: ``Pure long dispersion (zero idiosyncratic Gamma):

011 2

2

2

2

2

2

>

==

I

iii

II

iii

II

iiIi

ppp

70 75 80 85 90 9510

010

511

0

115

120

125

130

70

80

90

100

110

120

130

05

101520

25

30

70 75 80 85 90 95 100 105 110 115 120 125 13070

80

90

10 0110

120130

0

5

10

15

20

25

Payoff function for a tradewith short index/long options (IVH), 2 stocks

Value function (B&S) for the IVH position as a function ofstock prices (2 stocks)

In general: short index IVHis short-Gamma along the diagonal, long-Gamma for``transversal moves

10

5.80

10.31

20.49

70 75 80 85 90 95 100 105 110 115 120 125 13070

75

80

85

90

95

100

105

110

115

120

125

130

-6.80 +7.88

-2.29+10.84

Gamma Risk: Negative exposure for parallel shifts, positiveexposure to transverse shifts

5.%40%30

12

2

1

=

=

=

-0.

15-0.

08

-0.

01

0.06

0.13

1.21

0.3

0.07

0.01

2 0

-1.E+06-1.E+06-8.E+05-6.E+05-4.E+05-2.E+050.E+002.E+054.E+056.E+058.E+051.E+06

inde

x

normalized dispersion

Gamma-Risk for Baskets

D= Dispersion, or cross-sectional move, D/(Y*Y)= Normalized Dispersion

( )

( )2

1

2

2

1

1//

=

=

=

=

=

=

N

iii

N

iii

i

ii

YXpYD

YXpD

IIY

SSX

From realistic portfolio

11

Vega Risk

Sensitivity to volatility: move all single-stock implied volatilitiesby the same percentage amount

( ) ( )

( ) ( )

==

+=

+

=

+=

=

=

=

VNV

NVNV

NVNV

I

M

jj

I

II

j

jM

jj

IIj

M

jj

vega normalized

VegaVega Vega P/L

1

1

1

Market/Volatility Risk

70%

80%

90%

100%

110%

120%

130%

7075

80859095100105110115120

125130

vol % multiplier

mar

ket l

ev

el

70% 85

%

100% 11

5% 130%

707580859095

100

105

110

115

120

125

130

0123456789

1011121314151617181920

Vol % multiplerMarket level

Short Gamma on a perfectly correlated move Monotone-increasing dependence on volatility (IVH)

12

``Rega: Sensitivity to correlation

( ) ( )[ ]( ) ( )

( ) ( ) ( ) ( )( ) ( )( ) ( )II

II

I

III

I

II

I

I

j

M

jjIj

M

jjIIII

jijji

iijjij

M

ijiI

ijij

NVNV

pp

pppp

ji

==

=

===

+

+

==

=

2

2021

2

2)0(2)1(

2

2)0(2)1(

2

1

2)0(

1

)1(2)0(2)1(2

1

2

21

ega R 21

P/LnCorrelatio

21

, ,

Market/Correlation Sensitivity

-0.

3

-0.

2

-0.

1 0

0.1

0.2

0.3

70

90

110130

00.30.60.91.21.51.82.12.42.7

33.33.63.94.24.54.85.1

corr change

market level

-0.

3

-0.

2

-0.

1 0

0.1

0.2

0.370

7580859095100105110115120125130

corr change

market level

Short Gamma on a perfectly correlated move Monotone-decreasing dependence on correlation

13

Valuation Method I: Weighted Monte Carlo

Simulate scenarios (paths) for the group of stocks that comprisethe index or indices under consideration

Simulate the cash-flows of options on all the stocks and theindex options

Select weights or probabilities on the scenarios in such a waythat all options/forward prices are correctly reproduced by averaging over the paths

Use ``weighted Monte Carlo to derive fair-value of target options and compare with market values

Entering a trade

time

dtBdWdX +=

Avellaneda, Buff, Friedman, Kruk, Grandchamp: IJTAF, 1999

14

time

1p

2p

3p

dtBdWdX +=

Avellaneda, Buff, Friedman, Kruk, Grandchamp: IJTAF, 1999

Computation of weights: Max-Entropy Method

Market pricesof single-stockoptions

Risk-neutralpricing probabilities

cash-flow matrix

15

Example of Pricing with WMCIndex Market Vols vs. Model Vols : January 03 expiration

0.00

10.00

20.00

30.00

40.00

50.00

60.00

360 380 400 420 440 460Index Strike Price

impl

iedv

ol BidVol

AskVolModelVolRHO=1

Another Valuation Example with WMC (From Aug 2002, front month)

Implied vol Expiration Sep02

05

10152025303540

440 445 450 455 460 465 470 475 480 485 490 495 500 505

Index Strike

Vol Bid

AskModel

16

Another Valuation Example with WMC (From Aug 2002, second month)

Implied vol Expiration Oct02

05

10152025303540

430

440

450

460

470

480

490

500

510

520

Index Strike

Vol Bid

AskModel

Another Valuation Example with WMC (From Aug 2002, third month)

Implied vol Expiration Nov02

05

101520253035

430 440 450 460 470 480 490 500 510 520 530

Index Strike

Vol Bid

AskModel

17

Another Valuation Example with WMC (From Aug 2002, 4th month)

Implied vol Expiration Dec02

05

101520253035

420 440 460 470 480 490 500 510 520 530 540

Index Strike

Vol Bid

AskModel

Valuation Method II: (WKB) Steepest-Descent Approximation

Improvement on Standard Volatility Formula for Index Options

ijjiji

jij

N

jjI ppp

=

+= 2

1

22

Assume that the correlation is given

Use markets on single-stock volatilities taking into accountvolatility skew

How can we integrate volatility skew information into (*)?

(*)

(Avellaneda, Boyer-Olson, Busca, Friz: RISK 2002, C.R.A.S. Paris 2003)

18

Approximate this conditional expectation using the mostlikely stock configuration given that

Steepest-Descent Approximation

( ) ( )dttIdWtII

dIII ,, +=

( ) ( )( ) ( )( ) ( )

==

= =

N

jk

N

jjjkjjkkkjjI ItSwppttSttStI

1 1

2,,E,

( )**1 ,..., NSS

Define a risk-neutral 1-factor modelfor the index process

Local index vol= conditional expectation of local variance (rigorous)

( ) ItSwi

ii =

( ) ( ) ( )tStSSSpptI jjiijijNij

iI ,,,****

1

2 =

Steepest descent vs. Market vs. WMC (Aug 20, 2002, front month)

Expiration: Sep 02

15

20

25

30

35

40

440

445

450

455

460

465

470

475

480

485

490

495

500

505

strike

impl

ied

vo

l BidVolAskVolWMC volSteepest Desc

19

Steepest descent vs. Market vs. WMC (Aug 20, 2002, 2nd month)

Expiration: Nov 02

15

20

25

30

35

40

430

440

450

460

470

480

490

500

510

520

strike

impl

ied

vo

l BidVolAskVolW MC volSteepest Desc

Gargoyle Dispersion Fund

Joint venture between Gargoyle Strategic Partners andMarco Avellaneda (manager)

Started Trading: May 2001

Uses proprietary system to detect trades and executeselectronically and through network of brokers in 5 U.S. exchanges

1 FT junior trader, 3 PT senior traders, 1 FT risk manager

20

May-0

1

Jun-

01Ju

l-01

Aug-0

1

Sep-0

1Oc

t-01

Nov-

01

Dec-

01

Jan-

02

Feb-0

2

Mar-0

2

Apr-

02

May-0

2

Jun-

02Ju

l-02

Aug-0

2

Sep-0

2Oc

t-02

$0.50$0.55$0.60$0.65$0.70$0.75$0.80$0.85$0.90$0.95$1.00$1.05$1.10$1.15$1.20$1.25$1.30$1.35$1.40$1.45$1.50$1.55$1.60$1.65

GargoyleDispersionFund

$1

ROI May01-Oct02

Trading History: Monthly Returns

-1.38%10.10%

-7.56%1.82%

3.58%9.18%

13.97%3.78%

0.49%6.09%

-1.02%3.27%

-2.04%5.20%

-8.49%-16.17%

-3.17%12.54%

0.67%-2.43%

-0.98%-6.26%

-8.07%1.90%

7.67%0.88%

-1.46%-1.93%

3.76%-6.06%

-0.74%-7.12%

-7.79%0.66%

-10.87%8.80%

-20% -15% -10% -5% 0% 5% 10% 15% 20%

M a y- 0 1

J u n - 0 1

J u l- 0 1

A u g - 0 1

S e p - 0 1

O c t - 0 1

N o v- 0 1

D e c - 0 1

Ja n - 0 2

F e b - 0 2

M a r - 0 2

A p r - 0 2

M a y - 0 2

Ju n - 0 2

J u l- 0 2

Au g - 0 2

S e p - 0 2

O c t - 0 2

S&P 500

GargoyleDispersion Fund

21

Dispersion Fund PerformanceTrading Period: 15 months

Cumulative ROI* since inception: 28.33%

Annualized Rate of Return: 22.65%

Annualized Standard Deviation: 26.59%

Worst monthly loss: August 02, -16%

Correlation with S&P 500: 35%

Correlation with VIX Index: - 33%

* After paying brokerage fees and commissions, etc

0%

10%

20%

30%

40%

50%

60%

Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov

Average CorrWeighted Corr

Dow IndustrialAverage (DJX)

Volatility

Correlation

22

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Dec Ja n Fe b Mar Apr May Jun Jul Aug Se p Oct Nov

Average Corr

Weighted Corr

Volatility

Correlation

Amex Biotech-nology Index (BTK)

DJX expiration 9/ 21/ 2002 strike 86

0

0.2

0.4

0.6

0.8

1

1.2

7/11/2

002

7/13/2

002

7/15/2

002

7/17/2

002

7/19/2

002

7/21/2

002

7/23/2

002

7/25/2

002

7/27/2

002

7/29/2

002

7/31/2

002

8/2/20

02

8/4/20

02

8/6/20

02

8/8/20

02

8/10/2

002

8/12/2

002

8/14/2

002

8/16/2

002

8/18/2

002

8/20/2

002

8/22/2

002

8/24/2

002

8/26/2

002

8/28/2

002

8/30/2

002

Corr

ela

tion

0

10

20

30

40

50

60

70

80

90

Delta

ImpliedCorrBidRhoAskRhoDelta

DJX Correlation Blowout, July 2002

DJX Sep 86 Call

23

Conclusions Dispersion trading: a form of ``statistical correlation arbitrage

Sell correlation by selling index options and buying optionson the components

Buy correlation by buying index options and selling optionson the components

``Convergence trading style.

Price discovery using model and market data on vol skews

Sophisticated trading strategy. Potentially very profitable, with moderate (but not low) risk profile.