Transcript
Lecture 10:Dispersion Trading
Marco Avellaneda
G63.2936.001
Spring Semester 2009
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
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
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
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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Intuition…
Fair value relation forvolatilities assuming a given correlation matrix
The trade in pictures
Index
Stock 1 Stock 2
Sell index call
Buy calls on different stocks.
Delta-hedge using index and stocks
Profit-loss scenarios for a dispersion trade in a single day
-2
-1.5
-1
-0.5
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0.5
1
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
stock #
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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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First approximation to the dispersion package: ``Intrinsic Value Hedge’’
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IVH:premium from indexis less than premium from components “Super-replication”
Makes sense for deep--in-the-money options
IVH: use indexweights for optionhedge
Intrinsic-Value Hedging is `exact’ only if stocks are perfectly correlated
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normal edstandardiz 1
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Similar to Jamshidian (1989)for pricing bond options in 1-factormodel
IVH : Hedge with ``equal-delta’’options
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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)
( )
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∂∂==
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Vega normalized , (dollars),decay - time
1/ 2
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Gamma P/L for an Index Option
( )
( ) ( )
1 Index P/L
1 Gamma P/LIndex
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Assume 0=σd
Gamma P/L for Dispersion Trade
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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
Dispersion Statistic
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Summary of Gamma P/L for Dispersion Trade
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22
Gamma P/L Dnnp
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“Idiosyncratic”Gamma
Dispersion Gamma
Time-Decay
Example: ``Pure long dispersion” (zero idiosyncratic Gamma):
011 2
2
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σθ
σ
σθ
σσθθ
70 75 80 85 90 95
100
105
110
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120
125
130
70
80
90
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110
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130
0
5
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30
70 75 80 85 90 95 100 105 110 115 120 125 13070
8 0
90
10 0
110
120
130
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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
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.
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12
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inde
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Gamma-Risk for Baskets
D= Dispersion, or cross-sectional move, D/(Y*Y)= Normalized Dispersion
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From realistic portfolio
Vega Risk
Sensitivity to volatility: perturb all single-stock implied volatilitiesby the same percent amount
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vega normalized
VegaVega Vega P/L
1
1
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Market/Volatility Risk70
%
80%
90%
100%
110%
120%
130%
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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)
``Rega’’: Sensitivity to correlation
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Market/Correlation Sensitivity-0
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-0.2
-0.1 0
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0.2
0.3
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90
110
130
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market level
-0.3
-0.2
-0.1 0
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0.2
0.3
70
75
80
85
90
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115
120
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130
corr change
market level
� Short Gamma on a perfectly correlated move� Monotone-decreasing dependence on correlation
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.
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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