Lecture 10: Dispersion Trading
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1 Lecture 0: Dserso Tradg Marco Avellaeda G Srg Semester 009
2 What s dserso tradg? Dserso tradg refers to trades whch oe -- sells dex otos ad buys otos o the dex comoets, or -- buys dex otos ad sells otos o the dex comoets All trades are delta-eutral (hedged wth stock) The ackage s mataed delta-eutral over the horzo of the trade Dserso tradg: -- sellg dex volatlty ad buyg volatlty of the dex comoets -- buyg dex volatlty ad sellg volatlty o the dex comoets
3 Why Dserso Tradg? Motvato: to roft from rce dffereces volatlty markets usg dex otos ad otos o dvdual stocks Oortutes: Market segmetato, temorary shfts correlatos betwee assets, dosycratc ews o dvdual stocks
4 dex Arbtrage versus Dserso Tradg Stock N * * * * Stock 3 dex dex Arbtrage: Recostruct a dex or ETF usg the comoet stocks Stock Stock Dserso Tradg: Recostruct a dex oto usg otos o the comoet stocks
5 Ma U.S. dces ad sectors Maor dces: SPX, DJX, NDX SPY, DA, QQQQ (Exchage-Traded Fuds) Sector dces: Semcoductors: SMH, SOX Botech: BBH, BTK Pharmaceutcals: PPH, DRG Facals: BKX, XBD, XLF, RKH Ol & Gas: XNG, XO, OSX Hgh Tech, WWW, Boxes: MSH, HHH, XBD, XC Retal: RTH
6 S ds S ds Cov S ds Var d Var w S S ds S ds w S w ds d w w S ρ, shares dex umber of tuto Far value relato for volatltes assumg a gve correlato matrx
7 The trade ctures Sell dex call dex Buy calls o dfferet stocks. Stock Stock Delta-hedge usg dex ad stocks
8 Proft-loss scearos for a dserso trade a sgle day Scearo Scearo stadard move stadard move stock # stock # Stock P/L: -.30 dex P/L: Total P/L: -.4 Stock P/L: +9.4 dex P/L: - 0. Total P/L: +9.8
9 Frst aroxmato to the dserso ackage: ``trsc Value Hedge M w S w umber of shares, scaled by ``dvsor' ' K M w K VH: use dex weghts for oto hedge max C ( K,0) w max( S K,0) M (, K, T ) w C ( S, K, T ) M VH: remum from dex s less tha remum from comoets Suer-relcato Makes sese for dee- --the-moey otos
10 trsc-value Hedgg s `exact oly f stocks are erfectly correlated ( T ) w S ( T ) M ρ N M w Fe N T N stadardzed ormal Solve for X : K Set : max K M Fe w Fe ( ( T ) K,0) w max( S ( T ) K,0) T M X T X T Smlar to Jamshda (989) for rcg bod otos -factor model
11 VH : Hedge wth ``equal-delta otos ( ) costat Deltas costat log- moeyess costat N l l + d d T K F T X T F K T X Fe K T T X
12 What haes after you eter a oto trade? Uhedged call oto Hedged oto Proft-loss for a hedged sgle oto osto (Black Scholes) P / L ( ) tme - decay (dollars), + d NV S S t, NV ormalzed Vega C ~ stadardzed move
13 Gamma P/L for a dex Oto ( ) ( ) ( ) dex P/L dex Gamma P/L M M M M w S w S ρ ρ + Assume 0 d
14 Gamma P/L for Dserso Trade th stock P/L ( ) DsersoTrade P/L M + ( ) + ( ρ ) dagoal term: realzed sgle-stock movemets vs. mled volatltes off-dagoal term: realzed cross-market movemets vs. mled correlato
15 Dserso Statstc ( ) ( ) ( ) Θ + Θ Θ Θ + + P/L, D D Y S S X Y X D N N N N N N N N N
16 Summary of Gamma P/L for Dserso Trade Θ + Gamma P/L D N dosycratc Gamma Dserso Gamma Tme-Decay Examle: ``Pure log dserso (zero dosycratc Gamma): 0 > Θ
17 30 Payoff fucto for a trade wth short dex/log otos (VH), stocks Value fucto (B&S) for the VH osto as a fucto of stock rces ( stocks) geeral: short dex VH s short-gamma alog the dagoal, log-gamma for ``trasversal moves
18 Gamma Rsk: Negatve exosure for arallel shfts, ostve exosure to trasverse shfts % 40% ρ
19 Gamma-Rsk for Baskets.E+06 8.E+05 6.E+05 4.E+05.E+05 0.E+00 -.E+05-4.E+05-6.E+05-8.E+05 -.E+06 -.E+06 ormalzed dserso dex X D D S S N N / Y Y ( X Y ) ( ) X / Y D Dserso, or cross-sectoal move, D/(Y*Y) Normalzed Dserso From realstc ortfolo
20 Vega Rsk Sestvty to volatlty: erturb all sgle-stock mled volatltes by the same ercet amout Vega P/L M M M Vega + Vega ( NV ) + ( NV ) ( NV ) + ( NV ) NV ormalzed vega V
21 Market/Volatlty Rsk 70% 80% 90% 00% 0% vol % multler 0% 30% market level Market level % 70 85% 00% 5% 30% Vol % multler Short Gamma o a erfectly correlated move Mootoe-creasg deedece o volatlty (VH)
22 ``Rega : Sestvty to correlato ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) M M M NV NV (0) () (0) () (0) () (0) () Rega Correlato P/L,, ρ ρ ρ ρ ρ ρ ρ ρ
23 Market/Correlato Sestvty corr chage market level corr chage market level Short Gamma o a erfectly correlated move Mootoe-decreasg deedece o correlato
24 A model for dserso tradg sgals (takg to accout volatlty skews) Gve a dex (DJX, SPX, NDX) costruct a roxy for the dex wth small resdual. d ds β + ε m k k k Sk (multle regresso) Alteratvely, trucate at a gve catalzato level ad kee the orgal weghts, modelg the remader as a stock w/o otos. Buld a Weghted Mote Carlo smulato for the dyamcs of the m stocks ad value the dex otos wth the model Comare the model values wth the bd/offer values for the dex otos traded the market.
25 Morga Staley Hgh-Techology 35 dex (MSH) ADP AMAT AMZN AOL BRCM CA CPQ CSCO DELL EDS EMC ERTS FDC HWP BM NTC NTU JDSU JNPR LU MOT MSFT MU NT ORCL PALM PMTC PSFT SLR STM SUNW TLAB TXN XLNX YHOO 35 Uderlyg Stocks Equal-dollar weghted dex, adusted aually Each stock has tycally O(30) otos over a yr horzo
26 Test roblem: 35 tech stocks Prce otos o basket of 35 stocks uderlyg the MSH dex Number of costrats: 876 Number of aths: 0,000 to 30,000 aths Otmzato techque: Quas-Newto method (exlct gradet)
27 OtoNameStockT ckerexdate Strke Tye trsc Bd Ask Volume Oeterest StockPrceQuoteDate ZQN AC-E AMZN /0/0 5 Call /0/00 ZQN AT-E AMZN /0/ Call /0/00 ZQN AO-E AMZN /0/0 7.5 Call /0/00 ZQN AU-E AMZN /0/ Call /0/00 ZQN AD-E AMZN /0/0 0 Call /0/00 ZQN BC-E AMZN /7/0 5 Call /0/00 ZQN BO-E AMZN /7/0 7.5 Call /0/00 ZQN BD-E AMZN /7/0 0 Call /0/00 ZQN DC-E AMZN 4//0 5 Call /0/00 ZQN DO-E AMZN 4//0 7.5 Call /0/00 ZQN DD-E AMZN 4//0 0 Call /0/00 ZQN DS-E AMZN 4//0.5 Call /0/00 ZQN GC-E AMZN 7//0 5 Call /0/00 ZQN GO-E AMZN 7//0 7.5 Call /0/00 ZQN GD-E AMZN 7//0 0 Call /0/00 ZQN GS-E AMZN 7//0.5 Call /0/00 ZQN GE-E AMZN 7//0 5 Call /0/00 AOE AZ-E AOL /0/0 3.5 Call /0/00 AOE AO-E AOL /0/ Call /0/00 AOE AG-E AOL /0/0 35 Call /0/00 AOE AU-E AOL /0/ Call /0/00 AOE AH-E AOL /0/0 40 Call /0/00 AOE AR-E AOL /0/0 4.5 Call /0/00 AOE AV-E AOL /0/0 4.5 Call /0/00 AOE AS-E AOL /0/ Call /0/00 AOE A-E AOL /0/0 45 Call /0/00 AOE BZ-E AOL /7/0 3.5 Call /0/00 AOE BG-E AOL /7/0 35 Call /0/00 AOE BU-E AOL /7/ Call /0/00 Fragmet of data for AOE BH-E AOL /7/0 40 Call /0/00 AOE BV-E AOL /7/0 4.5 Call 0.5 calbrato.45 wth costrats 37.5 /0/00 AOE B-E AOL /7/0 45 Call /0/00 AOE DZ-E AOL 4//0 3.5 Call /0/00 AOE DG-E AOL 4//0 35 Call /0/00 AOE DU-E AOL 4// Call /0/00 AOE DH-E AOL 4//0 40 Call /0/00 AOE DV-E AOL 4//0 4.5 Call /0/00 AOE D-E AOL 4//0 45 Call /0/00 AOE DW-E AOL 4// Call /0/00 AOO DJ-E AOL 4//0 50 Call /0/00 AOE GZ-E AOL 7//0 3.5 Call /0/00
28 Near-moth otos (Prcg Date: Dec 000) MSH Basket oto: model vs. market Frot Moth mled vol strke model mdmarket bd offer
29 Secod-moth otos Basket oto: model vs. market mled vol strke model mdmarket bd offer
30 Thrd-moth otos Basket oto: model vs. market mled vol model mdmarket bd offer strke
31 Sx-moth otos Basket oto: model vs. market mled vol model mdmarket bd offer strke
32 Broad Market dex Otos (OEX) Prcg Date: Oct 9, Bd Prce ---- Ask Prce ---- Model Far Value Skew Grah Volatlty Strke Prce
33 Hedgg Coverg the ``wgs every ame mles a excess Vega rsk. trsc Value Hedge mles log Volatlty Use the WMC sestvty method (regressos) to determe the best sgle co-termal oto to use for each comoet. mlemet a Theta-Neutral hedge usg the most mortat ames wth the corresodg Betas.
34 Smulato for OEX Grou: $0MM/ Targetg % daly stdev SGNALSTRENGTH > threshold 080 trades OEX turover tme 60 days aualzed retur $4,39,794 $3,09,05 $,339,77 $,966,986 ercetage Share Rato Costat-VaR ortfolo (% stdev er day) Catal s allocated evely amog sgals Trasacto costs otos/ stock tradg cluded
35 avr-04 set Dserso OEX (retur o $00) févr-0 août-0 mars-03 Results of Back-testg sgal realzed ul-0 déc-00 $-retur
36 Smulato for QQQ grou $0MM wth % target daly stdev sgal >threshold trades 96 QQQ turover tme 76 aualzed retur -$,369,46 $,078,54 $5,339,45 $,533,4 ercetage Share Rato
37 QQQ, retur o $ $ -re tu r sgal realzed 0 0 set-0 déc-0 avr-0 ul-0 oct-0 av-03 ma-03 août-03 ov-03 mars-04-0
38 QQQ; umber of sgals QQQ août-03 oct-03 u-03 avr-03 févr-03 a v-0 févr-03 avr-0 u-0 août-0 oct-0 déc-0 ov-0 set-0 ul-0 ma-0 sgals er day
39 Smulato for QQQ+OEX $0MM wth % daly stdev QQQ + OEX turover tme 65 aualzed retur $ $ $ $ ercetage Share Rato
40 OEX + QQQ, retur o $ $-retur sgal realzed 0.00 févr-0 set-0 avr-0 oct-0 ma-03 ov-03 u-04 cludes T.C., otos ad stock tradg
41 Dserso Caacty Estmate USD 0 MM ~ 00 OEX cotracts er day f we assume 000 cotracts to be a lqudty lmt, caacty s 00 MM ust for OEX Caacty s robably aroud 00 MM f we use sectors ad Euroe Dserso has hgher Share Rato: t s a arb strategy based o watg for roft oortutes
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