lecture 10: dispersion trading
TRANSCRIPT
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Lecture 10:Dispersion Trading
Marco Avellaneda
G63.2936.001
Spring Semester 2009
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What is dispersion trading?
• Dispersion trading refers to trades in which one
-- sells index options and buys options on the index components, or
-- buys index options and sells options on the index components
• All trades are delta-neutral (hedged with stock)
• The package is maintained delta-neutral over the horizon of the trade
Dispersion trading:
-- selling index volatility and buying volatility of the index components-- buying index volatility and selling volatility on the index components
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Why Dispersion Trading?
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
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Index Arbitrage versus Dispersion Trading
Stock 1
Index
Stock N
Stock 3
Stock 2
*
*
*
*
Index Arbitrage:Reconstructan index or ETFusing thecomponent stocks
Dispersion Trading:Reconstruct an indexoptionusing options on the component stocks
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Main U.S. indices and sectors
• Major Indices: SPX, DJX, NDXSPY, DIA, QQQQ (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
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ijjij
ijiI
ij j
j
i
iji
n
i i
iiI
iii
n
i i
ii
n
i i
iiin
iii
i
n
iii
pp
S
dS
S
dSCovpp
S
dSpVar
I
dIVar
I
Swp
S
dSp
S
dS
I
SwdSw
II
dI
wSwI
ρσσσ
σ
∑
∑
∑
∑
∑∑
∑
=
=
=
=
==
==
==
=
=
==
=
,
1
indexin shares ofnumber
2
1
2
1
11
1
Intuition…
Fair value relation forvolatilities assuming a given correlation matrix
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The trade in pictures
Index
Stock 1 Stock 2
Sell index call
Buy calls on different stocks.
Delta-hedge using index and stocks
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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 #
stan
dar
d m
ove
-3
-2.5-2
-1.5-1
-0.50
0.51
1.52
2.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
stock #
stan
dar
d m
ove
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
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( ) ( )
( ) ( ) ,,,,
0,max0,max
1
1
1
TKSCwTKIC
KSwKI
KwK
iii
M
jiI
ii
M
ji
i
M
ji
∑
∑
∑
=
=
=
≤
−≤−
⇒=
First approximation to the dispersion package: ``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
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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
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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
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What happens after you enter an option trade ?
€ 0
€ 5
€ 10
€ 15
€ 20
€ 25
€ 30
€ 35
€ 70 € 75 € 80 € 85 € 90 € 95 € 100 € 105 € 110 € 115 € 120 € 125 € 130 -€ 1
€ 0
€ 1
€ 2
€ 3
€ 4
€ 5
€ 6
€ 7
€ 70 € 75 € 80 € 85 € 90 € 95 € 100 € 105 € 110 € 115 € 120 € 125 € 130
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
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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
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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
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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
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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
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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
8 0
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
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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
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-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
xnormalized 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
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Vega Risk
Sensitivity to volatility: perturb all single-stock implied volatilitiesby the same percent 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
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Market/Volatility Risk70
%
80%
90%
100%
110%
120%
130%
70
75
80
85
90
95
100
105
110
115
120
125
130
vol % multiplier
mar
ket l
evel
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)
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``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
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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
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A model for dispersion tradingsignals (taking into account volatility skews)
• Given an index (DJX, SPX, NDX) construct a proxy for the index withsmall residual.
)regression (multiple 1
εβ +=∑= k
km
kk S
dS
I
dI
• Alternatively, truncate at a given capitalization level and keep the original weights, modeling the remainder as a stock w/o options.
• Build a Weighted Monte Carlo simulation for the dynamics of the m stocksand value the index options with the model
• Compare the model values with the bid/offer values for the index optionstraded in the market.
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Morgan Stanley High-Technology 35 Index (MSH)
ADP JDSUAMAT JNPRAMZN LUAOL MOTBRCM MSFTCA MUCPQ NTCSCO ORCLDELL PALMEDS PMTCEMC PSFTERTS SLRFDC STMHWP SUNWIBM TLABINTC TXNINTU XLNX
YHOO
�35 Underlying Stocks
� Equal-dollar weighted index, adjustedannually
�Each stock has typically O(30) options over a 1yr horizon
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Test problem: 35 tech stocks
Number of constraints: 876
Number of paths: 10,000 to 30,000 paths
Optimization technique: Quasi-Newton method (explicit
gradient)
Price options on basket of 35 stocks underlying the MSH index
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ZQN AC-E AMZN 1/20/01 15 Call 0 4.125 4.375 13 3058 16.6875 12/20/00ZQN AT-E AMZN 1/20/01 16.75 Call 0 3.125 3.375 0 1312 16.6875 12/20/00ZQN AO-E AMZN 1/20/01 17.5 Call 0 2.875 3.25 20 10 16.6875 12/20/00ZQN AU-E AMZN 1/20/01 18.375 Call 0 2.625 2.875 10 338 16.6875 12/20/00ZQN AD-E AMZN 1/20/01 20 Call 0 1.9375 2.125 223 5568 16.6875 12/20/00ZQN BC-E AMZN 2/17/01 15 Call 0 5.125 5.625 30 1022 16.6875 12/20/00ZQN BO-E AMZN 2/17/01 17.5 Call 0 4 4.375 0 0 16.6875 12/20/00ZQN BD-E AMZN 2/17/01 20 Call 0 3.125 3.5 10 150 16.6875 12/20/00ZQN DC-E AMZN 4/21/01 15 Call 0 5.875 6.375 0 639 16.6875 12/20/00ZQN DO-E AMZN 4/21/01 17.5 Call 0 5 5.375 0 168 16.6875 12/20/00ZQN DD-E AMZN 4/21/01 20 Call 0 3.875 4.125 5 1877 16.6875 12/20/00ZQN DS-E AMZN 4/21/01 22.5 Call 0 3.125 3.375 20 341 16.6875 12/20/00ZQN GC-E AMZN 7/21/01 15 Call 0 6.875 7.375 0 134 16.6875 12/20/00ZQN GO-E AMZN 7/21/01 17.5 Call 0 5.625 6.125 0 63 16.6875 12/20/00ZQN GD-E AMZN 7/21/01 20 Call 0 4.875 5.25 5 125 16.6875 12/20/00ZQN GS-E AMZN 7/21/01 22.5 Call 0 4.125 4.5 0 180 16.6875 12/20/00ZQN GE-E AMZN 7/21/01 25 Call 0 3.5 3.875 65 79 16.6875 12/20/00AOE AZ-E AOL 1/20/01 32.5 Call 0 6.6 7 20 1972 37.25 12/20/00AOE AO-E AOL 1/20/01 33.75 Call 0 5.6 6 0 596 37.25 12/20/00AOE AG-E AOL 1/20/01 35 Call 0 4.7 5.1 153 5733 37.25 12/20/00AOE AU-E AOL 1/20/01 37.5 Call 0 3.4 3.7 131 3862 37.25 12/20/00AOE AH-E AOL 1/20/01 40 Call 0 2.5 2.7 1229 19951 37.25 12/20/00AOE AR-E AOL 1/20/01 41.25 Call 0 2 2.3 6 1271 37.25 12/20/00AOE AV-E AOL 1/20/01 42.5 Call 0 1.65 1.85 219 4423 37.25 12/20/00AOE AS-E AOL 1/20/01 43.75 Call 0 1.3 1.5 44 3692 37.25 12/20/00AOE AI-E AOL 1/20/01 45 Call 0 1.2 1.25 817 11232 37.25 12/20/00AOE BZ-E AOL 2/17/01 32.5 Call 0 7 7.4 0 0 37.25 12/20/00AOE BG-E AOL 2/17/01 35 Call 0 5.4 5.8 31 4 37.25 12/20/00AOE BU-E AOL 2/17/01 37.5 Call 0 4.1 4.5 0 0 37.25 12/20/00AOE BH-E AOL 2/17/01 40 Call 0 3.1 3.4 299 48 37.25 12/20/00AOE BV-E AOL 2/17/01 42.5 Call 0 2.15 2.45 191 266 37.25 12/20/00AOE BI-E AOL 2/17/01 45 Call 0 1.55 1.75 235 1385 37.25 12/20/00AOE DZ-E AOL 4/21/01 32.5 Call 0 8.4 8.8 16 10 37.25 12/20/00AOE DG-E AOL 4/21/01 35 Call 0 6.9 7.3 32 179 37.25 12/20/00AOE DU-E AOL 4/21/01 37.5 Call 0 5.5 5.9 36 200 37.25 12/20/00AOE DH-E AOL 4/21/01 40 Call 0 4.5 4.9 264 2164 37.25 12/20/00AOE DV-E AOL 4/21/01 42.5 Call 0 3.6 3.9 209 632 37.25 12/20/00AOE DI-E AOL 4/21/01 45 Call 0 2.9 3.1 415 3384 37.25 12/20/00AOE DW-E AOL 4/21/01 47.5 Call 0 2.15 2.45 37 1174 37.25 12/20/00AOO DJ-E AOL 4/21/01 50 Call 0 1.75 1.95 224 7856 37.25 12/20/00AOE GZ-E AOL 7/21/01 32.5 Call 0 9.4 9.8 0 0 37.25 12/20/00
OptionNameStockTickerExpDate Strike Type Intrinsic Bid Ask Volume OpenInterestStockPriceQuoteDate
Fragment of data forcalibration with 876 constraints
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Near-month options(Pricing Date: Dec 2000)
MSH Basket option: model vs. marketFront Month
6062646668707274767880
600 610 620 630 640 650 660 670 680 690 700strike
imp
lied
vo
l model
midmarket
bid
offer
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Second-month options
Basket option: model vs. market
50
52
54
56
58
60
62
64
66
68
70
600 620 640 660 680 700
strike
imp
lied
vo
l model
midmarket
bid
offer
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Third-month options
Basket option: model vs. market
45
50
55
60
65
70
600 640 680 720 760
strike
imp
lied
vo
l model
midmarket
bid
offer
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Six-month options
Basket option: model vs. market
35
40
45
50
55
60
600 640 680 720 760 840
strike
imp
lied
vo
l model
midmarket
bid
offer
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Skew Graph
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
500 510 520 530 540 550 555 560 565 570 580
Strike Price
Vo
latil
ity---- Bid Price---- Ask Price---- Model Fair Value
Broad Market Index Options (OEX)Pricing Date: Oct 9, 2001
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Hedging
• Covering the ``wings’’ in every name implies an excess Vega risk.Intrinsic Value Hedge implies long Volatility
• Use the WMC sensitivity method (regressions) to determine the bestsingle co-terminal option to use for each component.
• Implement a Theta-Neutral hedge using the most important names withthe corresponding Betas.
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Simulation for OEX Group:$10MM/ Targeting 1% daily stdev
Constant-VaR portfolio (1% stdev per day)
Capital is allocated evenly among signals
Transaction costs in options/ stock trading included
SIGNALSTRENGTH > threshold 1080 tradesOEX 2001 2002 2003 2001-2003turnover time 60 daysannualized return $4,239,794 $3,029,015 $1,339,717 $2,966,986percentage 42.40 30.29 13.40 29.67Sharpe Ratio 2.83 2.02 0.89 1.98
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Dispersion OEX (return on $100)
0
10
20
30
40
50
60
70
80
90
100
déc-
00
juil-01
févr
-02
août-
02
mar
s-03
sept-
03
avr-0
4
$-re
turn
signal
realized
Results of Back-testing
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signal >threshold trades 296QQQ 2001 2002 2003 2001-2003turnover time 76annualized return -$1,369,462 $1,078,541 $5,339,452 $1,533,241percentage -13.69 10.79 53.39 15.33Sharpe Ratio -0.91 0.72 3.56 1.02
Simulation for QQQ group $10MM with 1% target daily stdev
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QQQ, return on $100
-20
0
20
40
60
80
100
120
sept-01 déc-01 avr-02 juil-02 oct-02 janv-03 mai-03 août-03 nov-03 mars-04
$-re
turn signal
realized
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QQQ; number of signals
0
2
4
6
8
10
12
14
16
18
20
mai-01
juil-01
sept-
01no
v-01
janv-0
2fév
r-03
avr-0
2ju in-
02ao
ût-02
oct-0
2dé
c-02
févr-0
3av
r-03
ju in-03
août-
03oc
t-03
sig
nal
s p
er d
ay
QQQ
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QQQ + OEX 2001 2002 2003 2001-2003turnover time 65
annualized return $3 054 673 $2 878 561 $2 264 803 $2 672 645percentage 30.5 28.8 22.6 26.7Sharpe Ratio 1.9 1.8 1.4 1.7
Simulation for QQQ+OEX $10MM with 1% daily stdev
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OEX + QQQ, return on $100
0.00
20.00
40.00
60.00
80.00
100.00
120.00
févr-0
1
sept
-01
avr-0
2
oct-0
2
mai-
03
nov-0
3
juin-
04
$-re
turn
signal
realized
Includes T.C., in options and stock trading
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Dispersion Capacity Estimate
USD 10 MM ~ 100 OEX contracts per day
If we assume 1000 contracts to be a liquiditylimit, capacity is 100 MM just for OEX
Capacity is probably around 200 MMif we use sectors and Europe
Dispersion has higher Sharpe Ratio:It is an arb strategy based on waiting for profit opportunities