American basket and spread options. with a simple binomial tree

Transcription

1 Amercan baske and spread opons wh a smple bnomal ree Svelana orovkova Vre Unverse Amserdam Jon work wh Ferry Permana acheler congress, Torono, June 22-26,

2 Movaon Commody, currency baskes conss of wo or more asses, conan shor posons (e.g., crack or crush spreads) aske opons are ofen Amercan-syle (or Asan-syle) The valuaon and hedgng of baske opons s challengng even n he lack-scholes framework, because he sum of lognormal r.v. s s no lognormal. Spreads can have negave values and negavely skewed dsrbuon, so lognormal dsrbuon canno be used even n approxmaon lognormal dsrbuon canno be used, even n approxmaon. Exsng approaches can only deal wh baskes wh posve weghs or spreads beween wo asses. umercal and Mone Carlo mehods are slow and do no provde closed formulae for opon prce or greeks. 2

3 Amercan baske opons Addonal dffcules for Amercan opons: o closed form soluon, ruly pah-dependen opon Mos common mehod: bnomal ree (Cox Ross & Rubnsen 79) Mone-Carlo: no feasble, soluon: Longsaff & Schwarz 01 Mul-asse suaon: nomal ree replaced by bnomal pyramd compuaonally no feasble for #asses > 2 Impled bnomal ree (Rubnsen): only posve weghs Longsaff & Schwarz: nvolves complcaed Herme polynomals, can be slow 3

4 GL approach (orovkova e al. 07) Essenally a momen-machng mehod, n he spr of he Wakeman mehod for Asan opons (Turnbull and Wakeman (1991)). ased on old and fundamenal research on he good qualy of approxmaon he sum of lognormal r.v. s by a lognormal dsrbuon (e.g., Mchell (1968)). aske dsrbuon s approxmaed usng a generalzed famly of lognormal dsrbuons: shfed and negave shfed lognormals The man aracons: applcable o baskes wh several asses and negave weghs Easly exended o Asan-syle opons (and now also o Amercan opons!) Allows o drecly apply lack-scholes formula (n European case) Provdes closed form formulae for he opon prce and he greeks (approxmae, European and Asan cases) Provdes he way o buld one bnomal ree for he whole baske prce process 4

5 Lognormal vs shfed and negave lognormal: wo-parameer vs. hree-parameer dsrbuon 5

6 Assumpons aske of fuures on relaed commodes or currences. The baske value a me of maury T ( T) 1 a F T where F a T : he wegh of asse (fuures conrac) : he number of asses n he porfolo, : he fuures prce a he me of maury., a or 0 The fuures n he baske and he baske opon maure on he same dae. 6

7 Indvdual asses dynamcs Under he rsk adused probably measure Q, he fuures prces are marngales. The sochasc dfferenal equaons for s where W, W F df dw, 1,2,3,..., F :he fuures prce a me :he volaly of asse :he rownan moons drvng asses and wh correlaon ( dw dw, d), F 7

8 Examples of baske value dsrbuon: Fo Shfed lognormal, 0 [ 100;90]; [0.2;0.3]; a [ 1;1]; X 10; r 3%; T 1 year ; 0.9 Fo egave shfed lognormal, 0 [ 105;100]; [0.3;0.2]; a [ 1;1]; X 5; r 3%; T 1 year; 0.9 8

9 Momen machng The frs hree momens and he skewness of baske dsrbuon: a F M E F F a a M E 1 1, 2 2 exp 0 0 M E 3 3 k k k k k k k F F F a a a exp where : sandard devaon of baske a he me 3 ) ( 3 ) ( E E 9

10 If we assume he dsrbuon of a baske () s shfed lognormal wh parameers m m ( ), s s ( ), ( ) he parameers should sasfy non-lnear equaon sysem : M M 1 M 1 exp ( m s s expm s exp 2m (we omed dependence d of he momens and parameers on ) m 2s exp3m expm s 3 exp s 2 2 If we assume he dsrbuon of a baske s negave shfed lognormal, he parameers should sasfy non-lnear equaon sysem above by changng o M and o M M M : For baske of fuures, we have an mporan smplfcaon: zero drf mples wo unknown parameers and no hree sysem of wo equaons can be solved explcly much beer dsrbuon machng! 10

11 Approxmang dsrbuon b Skewness 0 0 Approxmang dsrbuon Shfed lognormal egave shfed lognormal oe: we never use regular lognormal dsrbuon as shfed lognormal always provdes a beer approxmaon (mprovemen over Wakeman mehod for Asan and baske opons) Skewness sgn remans he same hroughou opon s lfeme, ncreases n absolue value as T 11

12 GL approach for Amercan opons eed o approxmae no us he ermnal baske dsrbuon by GL, bu he enre baske value process from me 0 o maury by a GM ( ) ( ) For hs, we replace our orgnal baske by or ( ) dependng d on he sgn of skewness. () oe: ( ) s lognormal wh parameers ( ( ), s( )) f () s GL (Ths s smlar o dsplaced dffusons (M.Rubnsen, JF 1983)) () So we can assume ha () follows GM wh parameers (, ) gven by 2 s 2 ( ) m ( m( ), s( ), ( )) m( ) log (0) 1 2 ( 0 2 and for fuures) Parameers (, ) are deermned va machng he second and hrd cenral momens O numercal soluon s requred, us solve smple wo equaons, wo unknowns sysem Skewness near zero: acheler model (normal dsrbuon raher han lognormal) 12

13 Parameers, : an example aske : for all : ( ) 0 Fo [100;120]; [0.2;0.3]; () 0, shfed lognormal approxmaon a [ 1;1] oe: boh parameers almos consan 0.9; T 1 year er esmaes volaly parame r esmaes shf parameer me (days) me (days) 13

14 Algorhm for buldng he bnomal ree uld he bnomal ree for he value ( ) whch follows GM: A each me sep on he ree, he value moves eher up o q 1 q wh probably or down o d wh probably, where u d exp(( exp(( q (exp( 1 / 2 1 / ) ) d )) /( u ) ) Translae he obaned bnomal ree for no he ree for usng ( ) for shfed lognormal case, or ( for negave shfed lognormal case. ) Ths ree can be used now for valung an Amercan (or e.g. ermudan) opon on, compung opon s dela a each node and decdng on early exercse. d ) u 14

15 Smplfcaon: consan shf parameer For consan shf parameer, can drecly use he ree bul for : For shfed lognormal approxmaon, recall ha ( ) ( ) The payoff of an Amercan call opon on wh he srke prce K s he payoff of a Amercan call opon on baske wh he srke K For negave shfed lognormal approxmaon, recall ha ( ) ( ) The payoff of an Amercan call opon on wh srke K s he payoff of a Amercan pu opon on baske wh he srke K oh can be evaluaed usng he ree for 15

16 aske 1 aske 2 aske 3 aske 4 aske 5 Fuures prce (Fo) Volaly ( ) Weghs (a) [100;100] [100;120] [150;100] [95;90;105] [100;90;95] [0.3;0.2] [0.2;0.3] [0.3;0.2] [0.2;0.3;0.25] [0.25;0.3;0.2] [0.3;0.7] [-1;1] [-1;1] [1; -0.8; -0.5] [0.6;0.8; -1] Correlaon () ,2 2, 3 1, ,2 2, 3 1, Srke prce (X) skewness 0 () T=1 year; r = 5 % 16

17 aske ATM pu n. Pyramd (=150) Smulaon resuls Impl. T (=150) GL (=250) OTM pu (ATM-10) n. Pyramd (=150) Impl.T (=150) GL (=150)

18 Oher ssues and furher research Dela-hedgng performance: analyzed on he bass of Mone Carlo smulaons hedge error s n he order of 5-10% of he opon prce almos he same as he hedge error for a sngle asse case wh machng vol Oher greeks (especally vegas) Qualy of generalzed lognormal approxmaon need heorems smlar o hose of Mchell Oher applcaons: - physcs: a wave propagang hrough a urbulen medum - wreless communcaon: aenuaon due o shadowng n wreless channels - healh scences: ncubaon perods of dseases, e.g., anhrax nhalaon - analyss of handwrng: drecon conrol ~ weghed dfference of lognormals - ecology and chemsry: parcle sze dsrbuon, polluon 18