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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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

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0.5

1

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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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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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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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Gamma P/L for an Index Option

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Gamma P/L for Dispersion Trade

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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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“Idiosyncratic”Gamma

Dispersion Gamma

Time-Decay

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

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σ

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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

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

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5

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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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VegaVega Vega P/L

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Market/Volatility Risk70

%

80%

90%

100%

110%

120%

130%

70

75

80

85

90

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100

105

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115

120

125

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vol % multiplier

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100% 11

5% 130%

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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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0.1

0.2

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-0.3

-0.2

-0.1 0

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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

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