A New Approach For Modelling & Pricing Correlation Swaps in Equity Derivatives

Transcription

1 9 A MAY ew 006 Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 ICBI h Annual Meeng Pars Sebasen Bossu EQUITY STRUCTURIG GLOBAL DERIVATIVES DrKW LODO ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 0

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3 A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves Agenda. Fundamenals of ndex varance, consuen varance and correlaon. Toy model for dervaves on realzed varance 3. Quas-replcaon of correlaon swaps hrough varance dspersons ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006

4 A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves. Fundamenals of ndex varance, consuen varance and correlaon Realzed and Impled Correlaon Correlaon Proxy Applcaon: Varance Dspersons ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 3

5 Fundamenals of ndex varance, consuen varance and correlaon Realzed and Impled Correlaon Realzed Correlaon Par of socks: sascal coeffcen of correlaon beween he wo me seres of daly log-reurns Baske of socks: average of he (-)/ par-wse correlaon coeffcens Impled Correlaon Par of socks: usually unobservable Baske of socks: occasonally observable hrough quoes on baske calls or pus from exoc desks Lqud ndces: observable for lsed srkes and maures ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 4

6 5 ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 Fundamenals of ndex varance, consuen varance and correlaon Realzed and Impled Correlaon Realzed Correlaon Defnon (Equal Weghs Assumpon) Impled Correlaon Defnon (Equal Weghs Assumpon) < = j Impled j Impled Impled Impled Index Impled σ σ σ σ ρ ) ( ) ( < j Realzed j Realzed, ) ( ρ ρ = = Impled Impled ) ( σ σ

7 Fundamenals of ndex varance, consuen varance and correlaon Realzed and Impled Correlaon Realzed Calculaon Example: DAX 30 Consuen Socks ADS GY ALV GY ALT GY BAS GY BAY GY BMW GY VOW GY ADS GY 00% 9% -% 8% 3% 5% % ALV GY 9% 00% % 50% 44% 45% 36% ALT GY -% % 00% 9% 7% 5% 5% BAS GY 8% 50% 9% 00% 65% 5% 33% BAY GY 3% 44% 7% 65% 00% 39% 3% BMW GY 5% 45% 5% 5% 39% 00% 53% 00% VOW GY % 36% 5% 33% 3% 53% 00% ρ Realzed = 3.6% ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 6

8 Fundamenals of ndex varance, consuen varance and correlaon Realzed and Impled Correlaon Impled Calculaon Example: DAX Y ATM Index Vol: 6.88 Y ATM Consuen Vol: 3.76 Y ATM Resdual Vol : 4.38 Impled ρ = = 48.73% Y 90% Index Vol: 8.97 Y 90% Consuen Vol: 4.54 Y 90% Resdual Vol : 4.5 Impled ρ 8.97 = = 58.3% Resdual Volaly s gven as he square roo of Impled ( σ ) = ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 Here we see ha mpled correlaon for he 90% srke s hgher han a-he-money mpled correlaon. In oher words, mpled correlaon exhbs a skew owards he downsde. Ths can be explaned wh he emprcal observaon ha correlaon goes up when equy markes go down. 7

9 Fundamenals of ndex varance, consuen varance and correlaon Correlaon Proxy Proxy Formula Requremens Index Volaly ρ Average Consuen Volaly Large number of consuen socks (n pracce >0) Correlaon no oo low (n pracce >0.5) Well-behaved ndex weghs and volales ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 See Appendx B for he dervaon. 8

10 Fundamenals of ndex varance, consuen varance and correlaon Correlaon Proxy Realzed Calculaon Example Synhec DAX Volaly =.7% DAX Consuen Volaly = 0.66% Proxy for DAX Correlaon (.7/0.66) 3.3% Acual DAX Correlaon = 3.6% Impled Calculaon Example (ATM) DAX Volaly = 6.88% DAX Consuen Volaly = 3.76% Proxy for DAX Correlaon (6.88/3.76) 50.47% Acual DAX Correlaon = 48.73% Synhec consan wegh prce ndex sarng a 00 (.e. ndex daly log reurn s se equal o he arhmec average of he daly log reurns of he 30 consuens.) ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 oe ha f usng he acual DAX ndex, he proxy would gve 38.33%. Ths s because (a) acual ndex weghs are no equal and non-consan, (b) he number of DAX consuens (30) s n he lower range, (c) DAX s a oal reurn ndex. 9

11 Fundamenals of ndex varance, consuen varance and correlaon Correlaon Proxy Correlaon (Realzed and Impled) s hus close o he rao of ndex varance o he square of average consuen volaly ρ σ σ Index Consuens σ ( σ Index ) Consuens σ = = ( σ ) ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 In pracce he wo raos end o be very close for well-dversfed ndces 0

12 Fundamenals of ndex varance, consuen varance and correlaon Applcaon: Varance Dsperson Tradng Varance Dsperson Trades Offseng varance swap posons beween an ndex and s consuens, usually: Long Average Consuen Varance Shor Index Varance.e.: Payoff = σ σ Realzed Consuens Realzed Consuens ( σ Realzed Index ) [ ] Realzed ρ ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 oe ha because hs payoff s posve, he buyer mus pay a premum, whch usually akes place a maury n he form of a resdual srke equal o (Average Impled Consuen Varance) (Impled Index Varance). In hs forma, he sgn of he fnal P&L does OT exclusvely depend on he dfferenal beween mpled and realzed correlaons.

13 Fundamenals of ndex varance, consuen varance and correlaon Applcaon: Varance Dsperson Tradng By underweghng he consuens leg wh a facor β = ρ Impled <, several benefs are obaned: Vega-euraly On rade dae, f consuen varance goes up pon and mpled correlaon s unchanged, ndex varance would go up by ρ Impled pons and he P&L s: β xp ρ Impled ps = 0 Zero cos Cos = β x Impled Consuen Varance Impled Index Varance 0 Clean correlaon P&L P & L Realzed σ Consuens [ ] Ιmpled Realzed ρ ρ ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 oe ha hese resuls are vald for Varance oonals. For Vega oonals he bea adjusmen s approxmaely β = ρ Impled. oe ha n hs forma he sgn of he fnal P&L exclusvely depends on he dfferenal beween mpled and realzed correlaons.

14 A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves. Toy Model for Dervaves on Realzed Varance Realzed Varance: A Tradable Asse Toy Model for Realzed Varance Applcaon: Volaly Swap ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 3

15 Toy Model for Dervaves on Realzed Varance Realzed Varance: A Tradable Asse Varance Swap A expry wo pares exchange he realzed varance of e.g. DAX daly log-reurns, agans a srke ( mpled varance ) OTC marke has become very lqud on S&P 500 and DJ EuroSoxx 50, wh bd-offer spreads somemes as gh as ¼ vega. CBOE nroduced Three-Monh Varance Fuures on he S&P 500 n 004. ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 4

16 Toy Model for Dervaves on Realzed Varance Whch Model for Realzed Varance? Fscher Black: I sar wh he vew ha nohng s really consan. Volales hemselves are no consan, and we can wre down he process by whch he volales change wh any assurance ha he process self wll say fxed. We ll have o keep updang our descrpon of he process. Sudes of Sock Prce Volaly Changes, ced n Fscher Black and he Revoluonary Idea of Fnance, John Wley & Sons, 005 ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 5

17 Toy Model for Dervaves on Realzed Varance Toy Model for Realzed Varance Popular models (n parcular Heson) for volaly or varance focus on he nsananeous, non-radable volaly Oher approaches (Buehler) focus on he varance swap curve, whch s radable; or a fxed-erm varance asse (Duanmu, Carr-Sun) Toy Model Sraghforward modfcaon of Black-Scholes where he volaly of he varance asse v lnearly collapses as we approach s expry T: dv v ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 T = ω T dz Volaly of volaly Because we assume ha varance s radable, we can work under a rsk-neural measure. For ease of exposure raes are assumed o be zero. 6

18 Toy Model for Dervaves on Realzed Varance Toy Model for Realzed Varance v T s he prce of he varance asse a expry and concdes wh realzed varance over he nerval [0, T] v 0 s he far prce of he varance asse whch can be observed on he varance swap marke or calculaed hrough he replcang porfolo of pus and calls v 0 = E * ( v T ) The ermnal dsrbuon of v T s lognormal, makng closedform formulas for European dervaves on realzed varance easy o derve ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 Under R measure, zero rae assumpon. 7

19 Toy Model for Dervaves on Realzed Varance Applcaon: Volaly Swap Payoff = v T K vol Wh he Toy Model we fnd: = v 0 exp ω T 6 Kvol Convexy Adjusmen Varance Swap Srke umercal example: v 0 = 400 (0 ), T =, ω = 50% K vol 9. ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 8

20 A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves 3. Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Correlaon Swaps Arbrage Prcng Dynamc Hedgng Sraegy ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 9

21 Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Correlaon Swaps Correlaon Swap A maury wo pares exchange he average realzed correlaon beween e.g. he 30 DAX consuens, agans a srke. OTC marke, no very lqud. Inroduced n early 000 s as a means for equy exoc desks o recycle her correlaon paramerc rsk. Typcally correlaon swaps rade a a srke whch s 5 o 5 pons below mpled correlaon. Ths s because he correlaon swap marke s drven by supply and demand, wh no sraghforward arbrage ha would push he srke owards mpled correlaon levels ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 0

22 Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Correlaon Swaps Correlaon Swap Payoff: Payoff = ( ) Srke The prcng and dynamc hedgng of hs payoff s non-rval. However we can smplfy he problem usng he Proxy formulas: Realzed ( σ Index ) Payoff Srke Realzed ( σ ) = < j Realzed, j whch s he rao of wo radable asses: ndex varance and average consuen varance ρ ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006

23 Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Arbrage Prcng Defne a as he ndex varance asse, b as he average consuen varance asse, wh he followng rsk-neural dynamcs: Volaly of ndex volaly Volaly of consuen volaly da db T = ω a dw T T = ωb T [ ] χdw + χ dz Defne c T = a T /b T as he quas-payoff o replcae. Correlaon beween ndex and consuen vols ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006

24 Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Arbrage Prcng Afer calculaons we fnd ha he far srke of a correlaon swap should be close o: Impled Correlaon c 0 a0 4 = exp b a b T b 3 0 ( ) ω ω ω χ Model adjusmen For smlar vol of vol parameers and hgh correl of vols, he model adjusmen s close o. Ths means ha he far srke of a correlaon swap should be close o mpled correlaon. ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 3

25 Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Dynamc Hedgng Sraegy Hedgng coeffcens (delas): δ a c b = δ a = c b Hedgng porfolo: Long ndex varance Π a = δ a b + δ b = c a a c b b = 0 Shor consuen varance Zero cos Shor vega-neural varance dsperson [Wegh rao beween he consuen and ndex legs s equal o correlaon] ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 4

26 Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Key Resuls A correlaon swap on an equy ndex can be quasreplcaed by dynamcally radng vega-neural varance dspersons a zero cos Because he payoff of varance dspersons s essenally deermned by he dfference beween realzed and mpled correlaon, he no-arbrage srke of a correlaon swap should be close o mpled correlaon Ths resul s no heavly model-dependen n he sense ha he model adjusmen s close o for sensble parameer values ( vol of vol and correl of vols ) ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 5

27 Quas-Replcaon of Correlaon Swaps hrough Varance Dspersons Lmaons Praccal lmaons Transacon coss, parcularly for he consuen leg Varance caps: s marke pracce o cap he payoff of sngle sock varance swaps Cross-sensves: n case of a crash, boh volaly and correlaon go up Theorecal lmaons Approxmaons (-facor model nsead of + facors) Lognormal assumpon ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 Also: consan ndex weghs assumpon. 6

28 A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves References & Bblography ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 7

29 References & Bblography Conssen Varance Curve Models, H. Buehler, Deusche Bank AG, Workng paper (004) Raonal Prcng of Opons on Realzed Volaly, Z. Duanmu, Global Dervaves & Rsk Managemen Conference, Madrd (004) A ew Approach for Opon Prcng Under Sochasc Volaly, P. Carr and J. Sun, Bloomberg LP, Workng paper (005) Fscher Black and he Revoluonary Idea of Fnance, P. Mehrlng, Wley (005) Self-referencng: Fundamenal relaonshp beween an ndex s volaly and he average volaly and correlaon of s componens, wh Y. Gu, JPMorgan Equy Dervaves, Workng paper (004) Arbrage prcng of equy correlaon swaps, JPMorgan Equy Dervaves, Workng paper (005) ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 Also of neres s he sudy on Opon Impled Correlaons and he Prce of Correlaon Rsk by Dressen & al., Unversy of Amserdam & Insead, Workng paper (005) 8

30 A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves Appendces ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 9

31 Appendx A -Year DAX Impled Volaly Daa -Year DAX Impled Volaly Daa Tcker Prce 00% 90% Tcker Prce 00% 90%.GDAXI EOG.DE BMWG.DE SIEGn.DE DBGn.DE ALVG.DE TKAG.DE DBKGn.DE VOWG.DE DCXGn.DE MAG.DE SAPG.DE ADSG.DE DTEGn.DE HRXG.DE BASF.DE LHAG.DE RWEG.DE MEOG.DE BAYG.DE LIG.DE MUVGn.DE FMEG.DE CBKG.DE IFXGn.DE SCHG.DE HKG_p.DE DPWGn.DE TUIGn.DE COG.DE ALTG.DE ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET

32 Appendx B Dervaon of Impled Proxy Equal Weghs Assumpon Index Volaly O(/) ρ Impled = ( σ = Impled Index σ ) Impled = ( σ = Impled ( σ ) Impled ) Average Consuen Volaly σ σ Impled Index Impled Consuens ICBI GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 3