Equity Correlation Trading. Silverio Foresi and Adrien Vesval Goldman Sachs NYU, April 2006

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1 Equiy Correlaion Trading Silverio Foresi and Adrien Vesval Goldman Sachs NYU, April 2006

2 Ouline Equiy Correlaion: Definiions, Producs and Trade Srucures Raionale: Evidence and Models Opporuniies: an Hisorical Perspecive

3 Correlaion Producs

4 Building Blocks: Vol Producs Realized variance: RV = 1 n T (ln( S S = 1 1 )) 2 OTC producs o rade realized variance: Dela-hedged opions (sraddles) Volailiy swap Variance swap Lised Producs Fuures on realized variance

5 Implied Correlaion From index and single-sock implied vols, one can exrac he average pairwise Implied Correlaion (= IC) embedded in opion prices by he marke. Le FVV = Fair Value of Variance, hen IC is IC n 2 FVVIndex w i= 1 i = n 2 n ( w i= i FVV 1 i ) i= 1 FVV w 2 i i FVV i

6 Basic Trade Idea Mechanics: a dispersion rade consiss of selling vol on he index, while simulaneously buying vols on he componen Appeal: hisorically index volailiy has raded rich, while individual sock volailiy has been fairly priced implied correlaion has hisorically been above realized

7 Correlaion Marke Anomaly Index = Eurosoxx 80% 70% 60% 50% 40% 30% 20% 10% 0% Rolling 3-monh realized correlaion (forward looking) 1YR implied correlaion Rolling 1-year realized correlaion (forward looking) 12/1/92 12/1/93 12/1/94 12/1/95 12/1/96 12/1/97 12/1/98 12/1/99 12/1/00 12/1/01 12/1/02 12/1/03 12/1/04 12/1/05

8 Correlaion Marke Anomaly Index = Dow Jones 140% 120% 100% Rolling 1-year realized correlaion (frwd looking) 1-YR implied correlaion 80% 60% 40% 20% 0% 10/6/97 2/6/98 6/6/98 10/6/98 2/6/99 6/6/99 10/6/99 2/6/00 6/6/00 10/6/00 2/6/01 6/6/01 10/6/01 2/6/02 6/6/02 10/6/02 2/6/03 6/6/03

9 Correlaion Trading: Producs Correlaion swaps: pay he difference beween an implied correlaion srike and he average pairwise correlaion in a baske of socks. Correl-swaps are no a naural hedge for dealers or srucurers books, as heses books are mosly exposed o covariance risk. Dela-hedged sraddles: sell index sraddles, buy single-sock sraddles. Dela-hedging a book of opions is expensive and complicaed for a hedge fund. Index var-swaps agains single-sock var-swaps: i is he mos popular way o srucure he rade over he las 2/3 years has been o rade. This srucure fis broker-dealer books relaively well and is manageable from a hedge fund poin of view as no dela-hedging is necessary.

10 Dispersion Trading: Var-swaps Sell a var-swap on an index, buy variance swaps on he individual componens of he index. On he single sock side, vega noionals are ypically proporional o index weighs. By adjusing he raio of index vega noional o sock vega noional, one can modify he reurn disribuion profile of he porfolio. Mos people like he rade vega neural (sum of single sock vega noional = - index vega) or premium neural (sum of variance noional * variance srikes on he index side = index variance noional * index variance srike). As he nex 2 slides will show, a premium neural rade is a good way o replicae a covariance exposure.

11 Vega-Neural rade Payoff Marix (in $M, for a $1M index vega dispersion rade on DJIA,43% implied correlaion) VOL RATIO Correlaion (4.09) (3.74) (3.15) (2.33) (1.28) (4.10) (3.77) (3.23) (2.48) (1.50) (0.32) (4.10) (3.81) (3.31) (2.62) (1.73) (0.64) (4.11) (3.84) (3.39) (2.76) (1.95) (0.97) (4.12) (3.88) (3.48) (2.91) (2.18) (1.29) (0.24) (4.13) (3.92) (3.56) (3.05) (2.41) (1.62) (0.68) (4.14) (3.95) (3.64) (3.20) (2.63) (1.94) (1.12) (0.18) (4.15) (3.99) (3.72) (3.34) (2.86) (2.27) (1.57) (0.76) (4.16) (4.02) (3.80) (3.49) (3.08) (2.59) (2.01) (1.34) (0.57) (4.17) (4.06) (3.88) (3.63) (3.31) (2.92) (2.45) (1.91) (1.31) (0.63) (4.18) (4.10) (3.96) (3.78) (3.53) (3.24) (2.89) (2.49) (2.04) (1.53) (0.35) (4.19) (4.13) (4.04) (3.92) (3.76) (3.57) (3.33) (3.07) (2.77) (2.43) (1.65) (0.73) (4.19) (4.17) (4.13) (4.06) (3.99) (3.89) (3.78) (3.65) (3.50) (3.33) (2.95) (2.50) (1.98) (1.38) (0.72) (4.20) (4.20) (4.21) (4.21) (4.21) (4.22) (4.22) (4.22) (4.23) (4.24) (4.25) (4.27) (4.29) (4.31) (4.33) (4.21) (4.24) (4.29) (4.35) (4.44) (4.54) (4.66) (4.80) (4.96) (5.14) (5.55) (6.04) (6.60) (7.23) (5.90) (4.22) (4.28) (4.37) (4.50) (4.66) (4.87) (5.10) (5.38) (5.69) (6.04) (6.85) (7.81) (8.91) (10.16) (5.90) (4.23) (4.31) (4.45) (4.64) (4.89) (5.19) (5.55) (5.96) (6.42) (6.94) (8.15) (9.58) (11.22) (13.08) (5.90) (4.24) (4.35) (4.53) (4.79) (5.11) (5.52) (5.99) (6.54) (7.16) (7.85) (9.45) (11.35) (13.53) (15.34) (5.90) (4.25) (4.39) (4.61) (4.93) (5.34) (5.84) (6.43) (7.11) (7.89) (8.75) (10.75) (13.11) (15.84) (15.34) (5.90) (4.26) (4.42) (4.69) (5.08) (5.57) (6.17) (6.87) (7.69) (8.62) (9.65) (12.05) (14.88) (18.15) (15.34) (5.90) (4.27) (4.46) (4.78) (5.22) (5.79) (6.49) (7.32) (8.27) (9.35) (10.56) (13.35) (16.65) (20.46) (15.34) (5.90) A vega neural rade can loose money even if realized correlaion is below implied correlaion, in case realized vols are very low Simulaed performance resuls do no reflec acual rading and have inheren limiaions. Please see addiional disclosures.

12 Premium-Neural rade Payoff Marix (in $M, for a $1M index vega dispersion rade on DJIA,43% implied correlaion) VOL RATIO Correlaion (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 0.45 (0.01) (0.03) (0.06) (0.10) (0.16) (0.23) (0.31) (0.40) (0.51) (0.63) (0.90) (1.23) (1.60) (2.03) (2.50) (3.60) (0.00) (0.00) (0.00) (0.00) 0.5 (0.02) (0.06) (0.14) (0.24) (0.38) (0.55) (0.75) (0.98) (1.24) (1.53) (2.20) (2.99) (3.91) (4.95) (6.11) (4.03) (0.00) (0.00) (0.00) (0.00) 0.55 (0.02) (0.10) (0.22) (0.39) (0.61) (0.88) (1.19) (1.56) (1.97) (2.43) (3.50) (4.76) (6.22) (7.88) (9.72) (4.03) (0.00) (0.00) (0.00) (0.00) 0.6 (0.03) (0.13) (0.30) (0.53) (0.83) (1.20) (1.63) (2.13) (2.70) (3.33) (4.80) (6.53) (8.53) (10.80) (13.33) (4.03) (0.00) (0.00) (0.00) (0.00) 0.65 (0.04) (0.17) (0.38) (0.68) (1.06) (1.53) (2.08) (2.71) (3.43) (4.24) (6.10) (8.30) (10.84) (13.73) (16.95) (4.03) (0.00) (0.00) (0.00) (0.00) 0.7 (0.05) (0.21) (0.46) (0.82) (1.28) (1.85) (2.52) (3.29) (4.16) (5.14) (7.40) (10.07) (13.16) (16.65) (18.51) (4.03) (0.00) (0.00) (0.00) (0.00) 0.75 (0.06) (0.24) (0.54) (0.97) (1.51) (2.18) (2.96) (3.87) (4.89) (6.04) (8.70) (11.84) (15.47) (19.58) (18.51) (4.03) (0.00) (0.00) (0.00) (0.00) 0.8 (0.07) (0.28) (0.63) (1.11) (1.74) (2.50) (3.40) (4.44) (5.63) (6.94) (10.00) (13.61) (17.78) (22.50) (18.51) (4.03) (0.00) (0.00) (0.00) (0.00) 0.85 (0.08) (0.31) (0.71) (1.26) (1.96) (2.83) (3.85) (5.02) (6.36) (7.85) (11.30) (15.38) (20.09) (24.76) (18.51) (4.03) (0.00) (0.00) (0.00) (0.00) 0.9 (0.09) (0.35) (0.79) (1.40) (2.19) (3.15) (4.29) (5.60) (7.09) (8.75) (12.60) (17.15) (22.40) (24.76) (18.51) (4.03) (0.00) (0.00) (0.00) (0.00) 0.95 (0.10) (0.39) (0.87) (1.54) (2.41) (3.48) (4.73) (6.18) (7.82) (9.65) (13.90) (18.92) (24.71) (24.76) (18.51) (4.03) (0.00) (0.00) (0.00) (0.00) 1 (0.11) (0.42) (0.95) (1.69) (2.64) (3.80) (5.17) (6.76) (8.55) (10.56) (15.20) (20.69) (27.02) (24.76) (18.51) (4.03) (0.00) (0.00) (0.00) (0.00) The sign of he P&L only depends on wheher realized correlaion is above or below implied correlaion. The magniude of he P&L increase wih realized vol levels. The rade is essenially a covariance play Simulaed performance resuls do no reflec acual rading and have inheren limiaions. Please see addiional disclosures.

13 Mark-o-Marke For longer-daed rades P&L will come more from mainly from remarking of implied correlaion han from differences beween implied and acual correlaion. For a var swap, he P&L beween 1 and 2 is PNL = Var Var 2, where 1 Var = RV + FVV Therefore 2 1 T 2 PNL = RV FVV T T Similarly, for a correlaion rade, we have PNL 2 T ( ) ( ) 2 FVV FVV 1 T ( ) ( ) 2 2 RC IC + IC IC 1 1 T T Where IC is implied correlaion and RC is realized correlaion T T

14 Puzzle(1): Long-Daed Implied Correlaion Too Low? Mark-o marke risk for long-daed volailiy srucures, including correlaion rades, is possibly no compensaed... enough Marke segmenaion: here is no demand for shor-daed correlaion (srucurers use long-daed)

15 Puzzle(2): Long-daed Index- Volailiy Skews Too High Index-volailiy skews do no flaen wih longer mauriies True for all markes (world-wide Crash-o-Phopia, see Foresi-Wu, JOD 2005): pu opions are more expensive han he corresponding call opions index reurns have a risk-neural reurn disribuion ha, unlike empirical disribuion, is asymmeric This is likely o be consisen wih sysemaic risk, in he form of bad correlaion, or marke (world-wide marke) crash risk, an eminenly un-diversifiable equiy-marke risk Is he size of he premium reasonable, when one considers ha he marke is much more han jus he equiy-marke?

16 Correlaion Trading: Moivaion Why rade correlaion? Is i a be on correlaion being meanrevering or a premium for bea, possibly exoic bea? A reasonable model of correlaion has correlaion ime varying (Driessen e al, 2005) dρ κ ( ρ ρ ) d + α (1 ρ ) ρ dw = The equivalen risk-neural expression embeds a correlaion premium. The daa suggess ha his premium is large which is reasonable if marke crash-risk is no diversifiable * * * dρ = κ ( ρ ρ ) d + α (1 ρ ) ρ dw

17 Correlaion Modeling There is a relaion beween marke-wide realized vols and realized pairwise correlaion This model is shor-hand for a more complee model of crash risk which arguably should conain common asymmeric jump-risk, a more sensible way o produce increases boh in correlaions as well as in measured volailiy dw s d h d dw f d g d du S ds ), ( ), ( ), ( ), ( / σ ρ σ ρ ρ σ ρ σ ρ σ σ + = + = =

18 Correlaion Modeling 2 There is a more difficul relaion linking vols/correlaion o flows and posiions and he naure of he marke paricipans Feedback effec: i is a general principle in derivaives rading: If pary A sells and dela-hedge an opion o pary B who does no hedge, acual reurn volailiy will be dampened This is rue also for correlaion riss: he exisence of correlaion books, on he back of srucures placed wih reail invesors who do no hedge, impared downward pressures on realized correlaions A model wihou flow informaion is incomplee

19 Raionale for he Trade: A Demand & Supply Perspecive Why has index vol raded a a premium? Index vol is (relaively) rich: Porfolio insurance (makes pus expensive) Srucurers Individual vol is (relaively) cheap/fair Reverse converibles call-overwriing (indexers)

20 Opporuniies Equiy correlaion vs. credi correlaion Equiy bespoke correlaion Hybrid-baske bespoke correlaion: baskes of commodiies and equiies, or commodiies and FX, ec. Asse-class correlaion Pension plans exposure o fixed income and equiy Counerpary risk (bank s counerpary credi risk, by posiions)

21 General Noes This maerial is provided for educaional purposes only and should no be consrued as invesmen advice or an offer or soliciaion o buy or sell securiies. These examples are for illusraive purposes only and are no acual resuls. If any assumpions used do no prove o be rue, resuls may vary subsanially. Opinions expressed are curren opinions as of he dae appearing in his maerial only. No par of his maerial may, wihou GSAM s prior wrien consen, be (i) copied, phoocopied or duplicaed in any form, by any means, or (ii) disribued o any person ha is no an employee, officer, direcor, or auhorized agen of he recipien. Simulaed performance is hypoheical and may no ake ino accoun maerial economic and marke facors ha would impac he adviser s decision-making. Simulaed resuls are achieved by reroacively applying a model wih he benefi of hindsigh. The resuls reflec he reinvesmen of dividends and oher earnings, bu do no reflec fees, ransacion coss, and oher expenses, which would reduce reurns. Acual resuls will vary. Expeced reurn models apply saisical mehods and a series of fixed assumpions o derive esimaes of hypoheical average asse class performance. Reasonable people may disagree abou he appropriae saisical model and assumpions. These models have limiaions, as he assumpions may no be consensus views, or he model may no be updaed o reflec curren economic or marke condiions. These models should no be relied upon o make predicions of acual fuure accoun performance. GSAM has no obligaion o provide updaes or changes o such daa. Copyrigh 2006, Goldman, Sachs & Co. All righs reserved. Rev #

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