Lecture 12: Asymptotics

Transcription

1 Lecure : Asympocs Marco Avellaneda G Sprng Semeser 009

2 From SABR o Geodescs A sysemac approach for modelng volaly curves wh applcaons o opon marke-makng and prcng mul-asse equy dervaves Marco Avellaneda Couran Insue, New York Unversy Marco Avellaneda 005

3 Fng Volaly Skews SPX JUN04 (PRICING DATE MAY ) 50% 00% 50% 00% 50% 0% LN(K/S) Blue lne average mpled vol (pus/calls) Pnk lne fed parabola volaly

4 Zoom no he regon -0.<ln(K/F)<0. 40% 35% 30% 5% 0% 5% 0% 5% 0% If you zoom no he regon of neres, he parabolc f s seen as clearly nadequae Reason: he ou-of-he money opons ``lf he curve Parabolc fs are no conssen wh arbrage-free prcng

5 Parabolc fng requres Dela Truncaon! F only volales such ha -0.<<+0. 45% 40% 35% 30% 5% 0% 5% 0% 5% 0% parabolc ( ) ( )

6 0.3 Truncaed Parabolc F: a look a he full curve SPX JUN04 (PRICING DATE MAY ) % 500% 400% 300% 00% 00% 0% volaly LN(K/S) Ou of he money opons are no guaraneed o be well-fed

7 Usng a beer splne o f he daa (from SABR) ( ) ( ) 0 slope ln ln mp β γ β κ β κ κ β β + + F K e e Sgma, bea and kappa are adusable parameers Formula s derved from a sochasc volaly model so does no volae arbrage condons

8 Fng a SABR-lke splne o he SPX fron-monh curve SPX JUN06 SABR splne 80% 60% 40% 0% 00% 80% 60% 40% 0% 0% ln(k/) vol

9 Greenparabolc Red SABR Blue Md-Marke

10 Dfferenal Geomery and Impled Volaly Modelng

11 Facor Models and Dffuson Kernels w ( ) ( X ( ),..., X n ( ) ( ) ( W ( ),..., W ( ) ) m ) CIR-ype seng, X sae varables W m-dm Brownan moon dx m k k dw k + b d,,,3..., n π { } (, ; y, T ) Prob. ( T ) y ( ) Dffuson kernel E n { F( ( T )) ( ) } F( y) π (, ; y, T ) d y y R n

12 Fokker-Planck Equaon and Dmensonless Tme ( ) ( ) y y, ;, δ π π π π T T b a n n k m k k a Covarance mar of sae varables ( ) ( ) a n E n τ ``ypcal varance of Dmensonless me volaly of S&P0.5 yr. corresponds o au0.05<<

13 Varadhan Asympocs for he Dffuson Kernel lm τ 0 τ ln π L (, y) (,0; y, T ) ; τ ( ) T, L(, y) geodesc dsance beween and y L(, y) v γ γ nf 0) ( ) n g y 0 dγ d ( ) v v γ d, g ( ) ( a ) Dmensonless Remann ensor

14 Heurscally: Dffuson Kernels ``resemble Gaussan Kernels wh -y replaced by L(,y) π (,0; y, T ) c( τ ) e ( L(, y )) τ τ << n ( dl) g ( ) dd We shall use hs appromaon o compue opon prces and mpled volales assumng au s small

15 Eample : Local volaly model df F d ( F, ) dw (, ) dw + ln (...) d F F 0 ( ) ( dl) d d d ( ( )) ( ) ~,0,0 ( ) L (, y) y ~ du ( u) G ( y) G( ) -dmensonal dsances are always `rval

16 Specal solvable -D case: he CEV Model ( ) ( ) ( ) ( ) β β β β β β y e e y L e e F F F,,, ~ 0 Negave bea for Eques (leverage) Dsance area under he curve y ( ) β β e G

17 Sochasc Volaly Models E { dw dz } df F d ρd dw κdz β κρ Forward prce Sochasc vol. Leverage d df β F + ε Bea regresson coeffcen of vol on sock reurns

18 Equvalen Model wh Independen Brownan Moons (SABR) (0) ep ( β ) ln F F 0 ``Paramerc leverage SV for als d d (0) (0) + βd ``CEV wh sochasc ndependen volaly s equvalen o SV model wh correlaed volaly, from he Remann vewpon d e d ( 0) E ( 0) β ( 0) κdz ( dw dz ) 0 dw

19 Remann Merc for SV / SABR: The Poncare Upper Half-Space Model ( ) 0, η κ β κ η β d d dl e + η P Q Geodescs are half-crcles wh cener on he horzonal as ( ) Q P d Q P L θ θ θ θ κ sn,

20 Usng he asympocs o compue opon prces F T F ( ) P F( 0) T 0 < K CALL R c c c n R ma n ma { y: F ( y) > K} { y: F ( y) > K} ( F( y) K,0) π ( 0,0; y, T ) ( F( y) K,0) ( F( y) K ) e ln F y K ( ) e + L L e ( 0,y) ( 0,y) τ τ L ( 0,y) τ d d n n y d y n d y n y

21 Seepes-descen appromaon for compung mpled volales e ln { y: F (y) > K} F y K ( ) + L ( 0,y) ( 0,y) L mn ln + τ > n y: F ( y ) K F ( y) K τ d y e mn y: F ( y) > K ln F ( y) L + ( 0,y) τ τ mn y: F ( y) > K K τ ln F K ( y) + L ( 0,y) mn τ { } L y: F (y) > K, << ( 0,y) τ

22 Equae formulas for OTM calls wh Black-Scholes L * ( K ) mn L ( 0,y) { } y: F (y) > K Mnmum dsance from 0 o he regon {F(y)>0} ln CALL ( * L ( K )) τ Small-au asympocs (model) ln CALL ( ln( K / F0 )) ( K ) T mp ( ln( K / F0 )) ( K ) mp ( ) τ Small-au asympocs (Black-Scholes)

23 Appromaon for Impled Volaly for general dffuson model ( K ) mp mn ln( K / F0 ) ( 0, y) y : F( y) { L > K} mn ln( K / F0 ) ( 0, y) y : F( y) { L > K} L n ( y) mn a γ γ y 0 0) ( ) ( - ) γ ( ) ( )d, γ

24 Eample : Local Volaly Model mp ( K ) ln ln( K / F ) 0 du ( ) K / F0 du ( u,0) 0 ( u,0) 0 ln K F 0 Impled Volaly Harmonc Mean of Local Volaly Beresyck, Busca and Floren, 00

25 Eample : Consan Elascy of Varance (, ) e β 0 CEV mp β e ( ) 0 β Impled volaly volaly(%) ln(k/f)

26 Eample 3: Sochasc Volaly / SABR ( ) mp ln ln F K e e β κ β κ κ β β

27 η ( ) 0, 0 ( ) η, mpled ( ) * / * 0 sn L d L R R e + θ θ κ η β κ η π θ β R Mnmzng he dsance o he lne ea cons. n he Poncare plane

28 Eample 3bs: Sochasc Volaly / Hull-Whe ds dw, S d κdz, E ln( S ( dwdz ) / S 0 ) ρd dl κ ( ρ ) κ d ρκdd + d z κ ρ ρ dl κ dz + d Poncare plane afer change of varables

29 Geodesc Dsance sgma Q P R ρ ρ κ ρ,,0 P R z

30 Appromaon for Impled Volaly Curves ( ) ( ) ( ) ( ) ( ) ( ) ) ( ln ) ( ln sn, * * ρ ρ ρ κ ρ κ κ ρ ρ ρ κ ρ κ κ κ ϑ ϑ L u du Q P d L mp Q P

31 Auo-calbraon of SABR and Heson volsabr vol heson % β 4 κ κ sabr Heson 0.5 κ 0 sabr 0.

32 Eample 4: he Heson Model A varan of he Poncare Half-Space ds S V dw dv κ V dz E ( dw dz ) ρd dl κ dξ + V dv ξ κ ( β e ) β Noe: V, no V squared

33 Closed-form soluon for geodescs ( ) ( ) ( ) ( ) θ θ κ π θ θ θ ξ θ θ θ θ ξ κ ξ β κ ξ β d R dl R V R V dv d dl e sn 0 sn 0 cos sn + + Geodescs are cyclods

34 Impled volaly curve for Heson model s obaned as an algebrac sysem ξ sn 0 θ n π θ n + snθ n cosθ n ( ξ ) κ ξ sn 0 θ cosθ n n Gven, solve for hea_n, and subsue n he second equaon

35 Mul-Asse Dervaves

36 Mul-Asse Dervaves: Inde Opons, Ranbows Derve nde volaly skew from sngle-sock skews and correlaon mar d (, ) E dw, ( dw dw ) ρ d,,..., n N equaons for he nde componens I ln n n w S ws ( 0) e I 0 F I ( )

37 BBH: ETF of 0 Boechnology Socks ( Componens of IBH) Tcker Shares ATM ImVol Tcker Shares ATM ImVol ABI 8 55 GILD 8 46 AFFX 4 64 HGSI 8 84 ALKS 4 06 ICOS 4 64 AMGN IDPH 7 BGEN 3 4 MEDI 5 8 CHIR 6 37 MLNM 9 CRA 4 55 QLTI 5 64 DNA SEPR 6 84 ENZN 3 8 SHPGY GENZ 4 56 BBH - 3

38 Impled Volaly Skews Mulple Names, Mulple Epraons AMGN ImpledVol BGEN ImpledVol BdVol BdVol ImpledVol BdVol AskVol VarSw ap AskVol VarSwap ImpledVol BdVol AskVol VarSw ap AskVol VarSwap MEDI ImpledVol BdVol AskVol ImpledVol BdVol AskVol VarSw ap VarSwap

39 BBH March 003 Impled Vols Prcng Dae: Jan 03 0:4 AM vol bd vol ask vol srke Wha s he `far value of he nde volaly reconsruced from he componens?

40 Remannan merc for he mul-d local vol model dl n ( - ρ ) d (,0) (,0) d n ( - ρ ) dy dy, dy d (,0) If correlaons are consan, he merc s ``fla : s Eucldean merc afer makng he change of varables ->y. Geodescs are sragh lnes n he y-coordnaes

41 Seepes DescenMos Lkely Sock Prce Confguraon * F e I Replace condonal dsrbuon by Drac funcon a mos lkely confguraon

42 Eac soluon: Euler-Lagrange Equaons mpl. I ( ) n - ( ρ ) * * du 0 0 du ( u,0) ( u,0) n - ( ρ ) * mpl., * * * ( ) ( ) mpl., Euler - Lagrange equaons * 0 du u (,0) Λ n ρ p * * ( ) (,0),,... n

43 Appromae soluon: nroduce he sock beas * β + ε β Regresson relaon beween sock and nde reurns Appromae formula for he opmal sock confguraon mp, I ( ) n ( - ρ ) ( β ) ( β ) mp, mp, β β mp, I n ( ) ρ ( ) ( ) p p mp, β mp, β Performs well n he range -0.<<+0.

44 DJX: Dow Jones Indusral Average T monh DJX Nov 0 Prcng Dae: 0/5/0 40 Vol BdVol AskVol SDA Dela

45 T monhs DJX Dec 0 Prcng Dae: 0/5/0 Vol Dela BdVol AskVol SDA

46 T3 monhs DJX Jan 03 Prcng Dae: 0/5/0 Vol BdVol AskVol SDA Dela

47 T 5 monhs DJX Mar 03 Prcng Dae: 0/5/0 Vol Dela BdVol AskVol SDA

48 T7 monhs DJX June 03 Prcng Dae: 0/5/0 Vol BdVol AskVol SDA Dela

49 BBH: Boechnology HLDR T monh BBH Nov 0 Dae: Oc 5 0 Vol Dela BdVol AskVol WKB

50 T monhs BBH Dec 0 Dae: Oc 5 0 Vol Dela BdVol AskVol SDA

51 T 6 monhs BBH Apr 03 Dae: Oc 5 0 Vol BdVol AskVol SDA Dela Is dmensonless me s oo long? (Error bars: Juyoung Lm) Is correlaon causng he dscrepancy?

52 S&P 00 Inde Opons (Quoe dae: Aug 0, 00) Epraon: Sep mpled vol srke BdVol AskVol WMC vol Seepes Desc

53 S&P 00 Inde Opons (Quoe dae: Aug 0, 00) Epraon: Oc 0 40 mpled vol BdVol AskVol WMC vol Seepes Desc srke

54 S&P 00 Inde Opons (Quoe dae: Aug 0, 00) Epraon: Nov 0 mpled vol srke BdVol AskVol W MC vol Seepes Desc

55 S&P 00 Inde Opons (Quoe dae: Aug 0, 00) Epraon: Dec 0 mpled vol srke BdVol AskVol WMC vol Seepes Desc

56 Impled Correlaon: a sngle correlaon coeffcen conssen wh nde vol ( ) ( ) ( ) ( ) ( ) ( ) ( ) mpl mpl mpl mpl mpl mpl mpl mpl mpl mpl mpl mpl mpl mpl + N I N N N I N N I N N I p p p p p p p p p p ρ ρ ρ Impled correlaon can be defned for dfferen srkes, usng SDA Appromae formula:

57 Dow Jones Inde

58 Dow Jones Inde: Correlaon Skew

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60

61 A model for ``Correlaon skew : Sochasc Volaly Sysems ds S E dw ( dw dw ) ρ d E( dw dz ) d κ dz r d di I, ds S y d Look for mos lkely confguraon of socks and vols (,..., n, y,..., y n ) correspondng o a gven nde dsplacemen

62 Mos lkely confguraon for Sochasc Volaly Sysems n I I I I I e e p p r y ρ κ γ γ ρ β β γ γ ) (0, ) (0, ), (,loc * * Mos lkely confguraon for socks moves and volaly moves, gven he nde move SDA

63 Mehod I: Dupre & Mos Lkely Confguraon for Sock Moves (,) N-dmensonal Equy marke ( ), ( ) I,loc, ( ) n n, Sep : Local volaly for each sock conssen wh opons marke Sep : Fnd mos lkely confguraon for socks

64 Mehod II: Sochasc Volaly Sysem and on MLC for Socks and Volales N-dmensonal ( ) Equy marke I,loc, Only one sep: compue he mos lkely confguraon of socks and volales a he same me

65 Mehods I and II are no `equvalen Dupre local vol. for sngle names,loc (, ) ( 0, ) e ϖ ϖ κ r Inde vol., Mehod I (, ) p p ( 0, ) ( 0 ) I, loc, ρ e ϖ β e ϖ β Inde vol., Mehod II (, ) p p ( 0, ) ( 0 ) I, loc, ρ e γ e γ

66 Sochasc Volaly Sysems gve rse o Inde-dependen correlaons ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),, 0, 0, 0, 0,,,loc,loc loc, I e p p e e e e e e p p e e p p ϖ β β ϖ γ γ β ϖ β ϖ γ γ ϖ β β ϖ γ γ ρ ρ ρ β β ρ ρ + Mehod II

67 Equvalence holds only under addonal assumpons on sock-volaly correlaons ϖ β κ r I I Mehod I ρ I κ r ρ I γ κ r I I Mehod II r r I r r ρ ρ I Condons under whch boh mehods gve equvalen valuaons

68 Open (and very doable) problems Apply hs echnology for prcng swapons based on he volaly skew of LIBOR raes or forward raes If we use a Local Volaly model (e.g. BGM wh squareroo volaly), he answer s dencal o he prevous formula The ``full SABR mul-asse model gves rse o a complcaed Remannan merc dl n g dη dη + n ( d ) κ Cred defaul models for prcng CDOs are amenable o he same approach, especally copula-ype models. I am no aware of any soluons

69 Eplogue: Srucural Cred Model (,..., ) n vecor of frm values Frm defauls before me T f ( T ) < α Equal weghed CDO :loss of m dollars f ( T ) Ω { : < α } Solve nf m card U I ( I ) m { L( 0, ) : Ω } m I