Lecture 0: Dserso Tradg Marco Avellaeda G63.936.00 Srg Semester 009
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
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
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
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
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
The trade ctures Sell dex call dex Buy calls o dfferet stocks. Stock Stock Delta-hedge usg dex ad stocks
Proft-loss scearos for a dserso trade a sgle day Scearo Scearo.5.5.5 stadard move 0.5 0-0.5 - stadard move 0.5 0-0.5 - -.5 -.5 - -.5-3 4 5 6 7 8 9 0 3 4 5-3 3 4 5 6 7 8 9 0 3 4 5 stock # stock # Stock P/L: -.30 dex P/L: - 0.0 Total P/L: -.4 Stock P/L: +9.4 dex P/L: - 0. Total P/L: +9.8
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
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
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
What haes after you eter a oto trade? Uhedged call oto Hedged oto 35 7 30 6 5 5 0 4 5 3 0 5 0 70 75 80 85 90 95 00 05 0 5 0 5 30 0-70 75 80 85 90 95 00 05 0 5 0 5 30 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
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
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
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
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
30 Payoff fucto for a trade wth short dex/log otos (VH), stocks 5 0 5 30 0 0 5 0 70 75 80 85 90 95 00 05 0 5 0 5 30 0 00 90 80 70 5 0 5 Value fucto (B&S) for the VH osto as a fucto of stock rces ( stocks) 0 5 0 70 75 80 85 90 95 00 05 0 5 0 5 30 70 8 0 30 0 0 0 0 90 geeral: short dex VH s short-gamma alog the dagoal, log-gamma for ``trasversal moves
Gamma Rsk: Negatve exosure for arallel shfts, ostve exosure to trasverse shfts +0.84 0.3 30 5 -.9 0 5 0 05 00 95 30% 40% ρ.5 5.80 0.49 90 85 80 75-6.80 +7.88 70 70 75 80 85 90 95 00 05 0 5 0 5 30
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. 0.3 0.07 0.0 0 0.3 0.06-0.0-0.08-0.5 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
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
Market/Volatlty Rsk 70% 80% 90% 00% 0% vol % multler 0% 30% 30 5 0 5 0 05 00 95 90 85 80 75 70 market level 0 9 8 7 6 5 4 3 0 9 8 7 6 5 4 3 0 Market level 30 5 0 5 0 05 00 95 90 85 80 75 70% 70 85% 00% 5% 30% Vol % multler Short Gamma o a erfectly correlated move Mootoe-creasg deedece o volatlty (VH)
``Rega : Sestvty to correlato ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) M M M NV NV + + 0 (0) () (0) () (0) () (0) () Rega Correlato P/L,, ρ ρ ρ ρ ρ ρ ρ ρ
Market/Correlato Sestvty 30 5. 4.8 4.5 4. 3.9 3.6 3.3 3.7.4..8.5. 0.9 0.6 0.3 0-0.3-0. -0. corr chage 0 0. 0. 0.3 70 90 0 30 market level -0.3-0. -0. 0 0. corr chage 0. 5 0 5 0 05 00 market level 95 90 85 80 75 0.3 70 Short Gamma o a erfectly correlated move Mootoe-decreasg deedece o correlato
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.
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
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)
OtoNameStockT ckerexdate Strke Tye trsc Bd Ask Volume Oeterest StockPrceQuoteDate ZQN AC-E AMZN /0/0 5 Call 0 4.5 4.375 3 3058 6.6875 /0/00 ZQN AT-E AMZN /0/0 6.75 Call 0 3.5 3.375 0 3 6.6875 /0/00 ZQN AO-E AMZN /0/0 7.5 Call 0.875 3.5 0 0 6.6875 /0/00 ZQN AU-E AMZN /0/0 8.375 Call 0.65.875 0 338 6.6875 /0/00 ZQN AD-E AMZN /0/0 0 Call 0.9375.5 3 5568 6.6875 /0/00 ZQN BC-E AMZN /7/0 5 Call 0 5.5 5.65 30 0 6.6875 /0/00 ZQN BO-E AMZN /7/0 7.5 Call 0 4 4.375 0 0 6.6875 /0/00 ZQN BD-E AMZN /7/0 0 Call 0 3.5 3.5 0 50 6.6875 /0/00 ZQN DC-E AMZN 4//0 5 Call 0 5.875 6.375 0 639 6.6875 /0/00 ZQN DO-E AMZN 4//0 7.5 Call 0 5 5.375 0 68 6.6875 /0/00 ZQN DD-E AMZN 4//0 0 Call 0 3.875 4.5 5 877 6.6875 /0/00 ZQN DS-E AMZN 4//0.5 Call 0 3.5 3.375 0 34 6.6875 /0/00 ZQN GC-E AMZN 7//0 5 Call 0 6.875 7.375 0 34 6.6875 /0/00 ZQN GO-E AMZN 7//0 7.5 Call 0 5.65 6.5 0 63 6.6875 /0/00 ZQN GD-E AMZN 7//0 0 Call 0 4.875 5.5 5 5 6.6875 /0/00 ZQN GS-E AMZN 7//0.5 Call 0 4.5 4.5 0 80 6.6875 /0/00 ZQN GE-E AMZN 7//0 5 Call 0 3.5 3.875 65 79 6.6875 /0/00 AOE AZ-E AOL /0/0 3.5 Call 0 6.6 7 0 97 37.5 /0/00 AOE AO-E AOL /0/0 33.75 Call 0 5.6 6 0 596 37.5 /0/00 AOE AG-E AOL /0/0 35 Call 0 4.7 5. 53 5733 37.5 /0/00 AOE AU-E AOL /0/0 37.5 Call 0 3.4 3.7 3 386 37.5 /0/00 AOE AH-E AOL /0/0 40 Call 0.5.7 9 995 37.5 /0/00 AOE AR-E AOL /0/0 4.5 Call 0.3 6 7 37.5 /0/00 AOE AV-E AOL /0/0 4.5 Call 0.65.85 9 443 37.5 /0/00 AOE AS-E AOL /0/0 43.75 Call 0.3.5 44 369 37.5 /0/00 AOE A-E AOL /0/0 45 Call 0..5 87 3 37.5 /0/00 AOE BZ-E AOL /7/0 3.5 Call 0 7 7.4 0 0 37.5 /0/00 AOE BG-E AOL /7/0 35 Call 0 5.4 5.8 3 4 37.5 /0/00 AOE BU-E AOL /7/0 37.5 Call 0 4. 4.5 0 0 37.5 /0/00 Fragmet of data for AOE BH-E AOL /7/0 40 Call 0 3. 3.4 99 48 37.5 /0/00 AOE BV-E AOL /7/0 4.5 Call 0.5 calbrato.45 wth 9 876 66costrats 37.5 /0/00 AOE B-E AOL /7/0 45 Call 0.55.75 35 385 37.5 /0/00 AOE DZ-E AOL 4//0 3.5 Call 0 8.4 8.8 6 0 37.5 /0/00 AOE DG-E AOL 4//0 35 Call 0 6.9 7.3 3 79 37.5 /0/00 AOE DU-E AOL 4//0 37.5 Call 0 5.5 5.9 36 00 37.5 /0/00 AOE DH-E AOL 4//0 40 Call 0 4.5 4.9 64 64 37.5 /0/00 AOE DV-E AOL 4//0 4.5 Call 0 3.6 3.9 09 63 37.5 /0/00 AOE D-E AOL 4//0 45 Call 0.9 3. 45 3384 37.5 /0/00 AOE DW-E AOL 4//0 47.5 Call 0.5.45 37 74 37.5 /0/00 AOO DJ-E AOL 4//0 50 Call 0.75.95 4 7856 37.5 /0/00 AOE GZ-E AOL 7//0 3.5 Call 0 9.4 9.8 0 0 37.5 /0/00
Near-moth otos (Prcg Date: Dec 000) MSH Basket oto: model vs. market Frot Moth mled vol 80 78 76 74 7 70 68 66 64 6 60 600 60 60 630 640 650 660 670 680 690 700 strke model mdmarket bd offer
Secod-moth otos Basket oto: model vs. market mled vol 70 68 66 64 6 60 58 56 54 5 50 600 60 640 660 680 700 strke model mdmarket bd offer
Thrd-moth otos Basket oto: model vs. market 70 65 mled vol 60 55 model mdmarket bd offer 50 45 600 640 680 70 760 strke
Sx-moth otos Basket oto: model vs. market 60 55 mled vol 50 45 model mdmarket bd offer 40 35 600 640 680 70 760 840 strke
Broad Market dex Otos (OEX) Prcg Date: Oct 9, 00 ---- Bd Prce ---- Ask Prce ---- Model Far Value Skew Grah 40.00 35.00 30.00 Volatlty 5.00 0.00 5.00 0.00 5.00 0.00 500 50 50 530 540 550 555 560 565 570 580 Strke Prce
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.
Smulato for OEX Grou: $0MM/ Targetg % daly stdev SGNALSTRENGTH > threshold 080 trades OEX 00 00 003 00-003 turover tme 60 days aualzed retur $4,39,794 $3,09,05 $,339,77 $,966,986 ercetage 4.40 30.9 3.40 9.67 Share Rato.83.0 0.89.98 Costat-VaR ortfolo (% stdev er day) Catal s allocated evely amog sgals Trasacto costs otos/ stock tradg cluded
avr-04 set-03 00 90 80 70 60 50 40 30 0 0 0 Dserso OEX (retur o $00) févr-0 août-0 mars-03 Results of Back-testg sgal realzed ul-0 déc-00 $-retur
Smulato for QQQ grou $0MM wth % target daly stdev sgal >threshold trades 96 QQQ 00 00 003 00-003 turover tme 76 aualzed retur -$,369,46 $,078,54 $5,339,45 $,533,4 ercetage -3.69 0.79 53.39 5.33 Share Rato -0.9 0.7 3.56.0
QQQ, retur o $00 0 00 80 $ -re tu r 60 40 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
0 8 6 4 0 8 6 4 0 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
Smulato for QQQ+OEX $0MM wth % daly stdev QQQ + OEX 00 00 003 00-003 turover tme 65 aualzed retur $3 054 673 $ 878 56 $ 64 803 $ 67 645 ercetage 30.5 8.8.6 6.7 Share Rato.9.8.4.7
OEX + QQQ, retur o $00 0.00 00.00 80.00 $-retur 60.00 40.00 0.00 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
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