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Empirical 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
2
Index 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
3
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
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 Funds
XNG
APAAPCBRBRREEXENEEOGEPGKMINBLNFGOEIPPPSTRWMB
SOX
ALTRAMATAMDINTCKLACLLTCLSCCLSIMOTMUNSMNVLSRMBSTERTXNXLNX
XOI
AHCBPCHVCOC.BXOMKMGOXYPREPRDSUNTXTOTUCLMRO
~ 20 - 40 stocksin samesector
Weightings by:
� capitalization� equal-dollar� equal-stock
4
Index 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
5
Profit-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 15
stock #
sta
nd
ard
mo
ve
-3
-2.5
-2
-1.5
-1
-0.50
0.51
1.5
2
2.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
stock #
sta
nd
ard
mo
ve
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
6
Intrinsic-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
σσ
σσ
σσ
ρ
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
2
1ln
1
2
1ln
1
2
2
2
1 2
≈≈
=
=−���
����
�=−
+���
����
�=∴=
−
d
dTK
F
TX
TF
K
TXeFK
ii
i
i
ii
i
i
TTX
ii
ii
σσ
σσ
σσ
7
What happens after you enter a trade:Risk/return in hedged option trading
���
���
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��� �
��� �
��� �
��� �
� �� � � � � � � � � �� � ��� � ���� ����� � ���� � ��� � � �� � � ����� � � ���
���
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" # $�" # %�" & $�" & %�" ' $�" ' %("�) $ $�" ) $ %�"�) ) $�"�) ) %�" ) * $�"�) * %�"�) + $
Unhedged call option Hedged option
Profit-loss for a hedged single option position (Black –Scholes)
( )
σσ
σθ
σσθ
∂∂==
∆∆==
⋅+−⋅≈
CNV
tS
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
Swpn
pn
n
ρσ
σσθ
σσθ
ρσσσ
σσ
θ
−+−=
=
==
−=
��
�
��
≠=
=
=
=
Assume 0=σd
8
Gamma P/L for Dispersion Trade
( )
( ) ( )ijjiji I
jijiIi
M
iI
I
iii
ii
nnpp
np
n
ρσ
σσθθ
σσθ
θ
−+−���
����
�+≈
−⋅≈
��≠=
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
( )
( ) ( )
Θ−−���
����
�+=
Θ−+−+=
+≡ΘΘ−+=
−+−=
−=
∆=∆=−=
�
���
��
�
�
�
=
===
==
=
=
=
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
I
IY
S
SXYXpD
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
σθθ
σσθ
θσσθσ
σθθ
θθθθ
θθ
σσ
9
Summary of Gamma P/L for Dispersion Trade
Θ−−���
����
�+=�
=
22
2
12
22
Gamma P/L Dnnp
I
Ii
N
ii
I
iiiI
σθθ
σσθ
“ Idiosyncratic” Gamma
Dispersion Gamma
Time-Decay
Example: ``Pure long dispersion” (zero idiosyncratic Gamma):
011 2
2
2
2
2
2
>
�����
�
�
�����
�
�
−��
���
�
≥���
�
�
���
�
�
−=Θ−=��
I
iii
II
iii
II
iiIi
ppp
σ
σθ
σ
σθ
σσθθ
70 75 80
85 90 95
100
105
110
115
120
125
130
70
80
90
100
110
120
130
0
5
10
15
20
25
30
70 75 80 85 90 95 100 105 110 115 120 125 13070
80
90
10 0
110
120
130
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, positive‘exposure’ to transverse shifts
5.
%40
%30
12
2
1
===
ρσσ
-0.1
5
-0.0
8
-0.0
1
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
I
IY
S
SX
From realistic portfolio
11
Vega Risk
Sensitivity to volatility: move all single-stock implied volatilitiesby the same percentage amount
( ) ( )
( ) ( )
σσ
σσ
σσ
σσ
σσ
∂∂==
∆
��
�+=
∆+∆
=
∆+∆=
�
�
�
=
=
=
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%
70
75
80
85
90
95
100
105
110
115
120
125
130
vol % multiplier
mar
ket
leve
l
70% 85
%
100% 11
5% 130%
707580859095100
105
110
115
120
125
130
0123456789
1011121314151617181920
Vol % multipler
Market 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
2
1ega R
2
1 P/LnCorrelatio
2
1
, ,
σσσρ
σσσ
ρσ
σσσσ
σσσσρσσσ
ρσσρσσσ
ρρρ
Market/Correlation Sensitivity
-0.3
-0.2
-0.1 0
0.1
0.2
0.3
70
90
110
130
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
0.3
70
75
80
85
90
95
100
105
110
115
120
125
130
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 ⋅+⋅Σ=
Avellaneda, Buff, Friedman, Kruk, Grandchamp: IJTAF, 1999
14
time
1p
2p
3p
dtBdWdX ⋅+⋅Σ=
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 WMC
Index 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
imp
lied
vol BidVol
AskVolModelVolRHO=1
Another Valuation Example with WMC (From Aug 2002, front month)
Implied vol Expiration Sep02
0
510
15
20
25
30
35
40
440 445 450 455 460 465 470 475 480 485 490 495 500 505
Index Strike
Vo
l Bid
Ask
Model
16
Another Valuation Example with WMC (From Aug 2002, second month)
Implied vol Expiration Oct02
05
10
152025303540
43044
045
046
047
048
049
050
051
052
0
Index Strike
Vo
l Bid
Ask
Model
Another Valuation Example with WMC (From Aug 2002, third month)
Implied vol Expiration Nov02
0
5
10
15
20
25
30
35
430 440 450 460 470 480 490 500 510 520 530
Index Strike
Vo
l Bid
Ask
Model
17
Another Valuation Example with WMC (From Aug 2002, 4th month)
Implied vol Expiration Dec02
0
5
10
15
20
25
30
35
420 440 460 470 480 490 500 510 520 530 540
Index Strike
Vo
l Bid
Ask
Model
Valuation Method II: (WKB) Steepest-Descent Approximation
� Improvement on Standard Volatility Formula for Index Options
ijjiji
jij
N
jjI ppp ρσσσσ ��
≠=
+= 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 ,, µσ +=
( ) ( )( ) ( )( ) ( )
�
��
�== � �
= =
N
jk
N
jjjkjjkkkjjI ItSwppttSttStI
1 1
2 ,,E, ρσσσ
( )**1 ,..., NSS
� Define a risk-neutral 1-factor modelfor the index process
� Local index vol= conditional expectation of local variance (rigorous)
( ) ItSwi
ii =�
( ) ( ) ( )tStSSSpptI jjiijij
N
ijiI ,,, ****
1
2 σσσ �=
≅
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
vol
BidVol
AskVol
WMC vol
Steepest Desc
19
Steepest descent vs. Market vs. WMC (Aug 20, 2002, 2nd month)
Expiration: Nov 02
15
20
25
30
35
40
43044
045
046
0470
480
490
500
51052
0
strike
impl
ied
vol BidVol
AskVol
W MC vol
Steepest 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-
01
Jun-0
1
Jul-01
Aug-01
Sep-01
Oct-01
Nov-01
Dec-01
Jan-02
Feb-02
Mar-02
Apr-02
May-0
2
Jun-02
Jul-02
Aug-02
Sep-02
Oct-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 Performance
Trading 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 Corr
Weighted 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
/200
2
7/13
/200
2
7/15
/200
2
7/17
/200
2
7/19
/200
2
7/21
/200
2
7/23
/200
2
7/25
/200
2
7/27
/200
2
7/29
/200
2
7/31
/200
2
8/2/
2002
8/4/
2002
8/6/
2002
8/8/
2002
8/10
/200
2
8/12
/200
2
8/14
/200
2
8/16
/200
2
8/18
/200
2
8/20
/200
2
8/22
/200
2
8/24
/200
2
8/26
/200
2
8/28
/200
2
8/30
/200
2
Co
rrel
atio
n
0
10
20
30
40
50
60
70
80
90
Del
ta
ImpliedCorr
BidRho
AskRho
Delta
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.
