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
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ijiI
ijij
NVNV
pp
pppp
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==
=
===
+
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=
2
2021
2
2)0(2)1(
2
2)0(2)1(
2
1
2)0(
1
)1(2)0(2)1(2
1
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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.