1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index, a fuues conac. Non-Abiage Picing a. Makes ae ficion-fee, as in Black-Scholes seing. b. Fom a iskless pofolio which hus mus ean he isk-fee ae, as in he deivaion of he Black-Scholes fomula. c. PDE: f S, f S, 1 f S, S S, S f S, 0 S S d. he ype of deivaive will be deemined by he specific final/iniial/bounday condiions chosen. 3. In Case he Asse Pays Dividends: a. Assume ha he asse pays a coninuous dividend yield δ, which means ha beween ime and +d he asse pays δ Sd. Bice Dupoye FIN 781 Semina in Opions
he iskless pofolio buil will hus povide a eun made up of a capial gain and a dividend, hence: d Sd d capial gain dividend oal eun b. he esuling PDE in his case is: f S, f S, 1 f S, S S, S f S, 0 S S Noice ha even hough he dif of he undelying asse depends on he isk avesion of invesos (high isk avesion implies a highe dif), he dif does no appea in he PDE. Hence deivaive pices do NO depend on he level of isk avesion of invesos. B. Non-aded Undelying Asse o Faco (such as inees aes, commodiies, weahe ) 1. Dynamics of he Faco Given by: a. df m F, Fd s F, FdZ b. Conside wo deivaives boh wien on faco F having pices f 1 (F,) and f (F,) especively. By Io s lemma and using j=1 and j= fo noaion, we have: Bice Dupoye FIN 781 Semina in Opions
3 f j F, f j F, 1 f j F, F F f j df F, m F, F s F, F d j f d j j f dz j j F, s F, F dz F c. In ode o fom a iskless pofolio, ecall ha we used o inves in he asse and he coesponding deivaive. Hee, he asse is a non-aded faco and i is heefoe impossible o adop he usual saegy. he new appoach is o fom a iskless pofolio by invesing (σ f ) in f 1 and (-σ 1 f 1 ) in f. d. he value of he esuling pofolio is hus given by: Π = (σ f ) f 1 - (σ 1 f 1 ) f while he change in value of he pofolio hus becomes: d f f f f d 1 1 1 1 Since he andom componen is gone, he pofolio mus heefoe eun he isk-fee inees ae, and so: d d. Make Pice of Risk Fom above, we can conclude ha: 1 f1f 1f1f f f f f! "! 1 1 1 And heefoe ha: 1 1 1 # $ # $ & & 1 Which can finally be esaed as: % 1 Bice Dupoye FIN 781 Semina in Opions
4 In ohe wods, he aio is he same fo evey deivaive wien on he same undelying faco F. 1 he aio 1 Bice Dupoye FIN 781 Semina in Opions j j is called he make pice of isk of faco F, and since boh µ and σ ae funcions of F and, so is he make pice of isk λ. 3. he Picing Paial Diffeenial Equaion Fo any deivaive of pice f, we mus have he following:, whee µ(f,) and σ(f,) ae deemined by: f F, f F, 1 f F, F F f F, F, f m F, F s F, F F, f s F, F F heefoe:!# # # "! ( # f F, $ f F, $ 1 f F, ) % & ' ( f F, + * # # #,* + # F F F - # # # f F, $ f F, % ' $ 1 f F, # # # % & m F, F s F, F f F, s F, F m F, F, s F, F s F, F f 0 F F Eliminaing he (F,) noaion fo claiy yields: 0 0 0 f f 1 f 0 1 0. 3 / 1 0 4 m s F s F f 0 F F Finally, o obain he pice of he deivaive by solving his PDE, ecall ha final/iniial/bounday condiions specific o he deivaive mus be saed.
5 4. Case Whee he Deivaive Iself Pays Income Some deivaives, such as bonds (bonds ae deivaive secuiies since hey deive hei values fom he value of an undelying faco, he inees ae), pay a seam of income (coupons in he case of bonds). Assume ha deivaive f j (F,) pays income h j (F,)d duing ineval of ime d. he pofolio Π = (σ f ) f 1 - (σ 1 f 1 ) f emains iskless bu we now have o ake he income ino accoun: 1 11 d f h f h d d capial gain Since 1 1 1 1 income oal eun d f f f f d, we heefoe have: 1 1 1 h f h f 1 And he following PDE esuls:!!! f " f # $ " 1 f!!! " # % m s F s F h f 0 F F Bice Dupoye FIN 781 Semina in Opions
6 II. Equivalen Maingale Measue, Risk- Neual Densiy (RND) A. Expeced Discouned Payoffs unde RND 1. he Feynman-Kac fomula says ha f f 1 f a S, b S, c S, f 0 S S wih final condiion f(s,)=g(s) having fo soluion: { } f S, E g S exp c S,u du S S u Noe ha he expecaion sign is wih espec o he densiy S S obained fom he dynamics ds a S, d bs, dz. he densiy S S is called he isk-neual densiy o sae pice densiy. If he picing PDE happens o be f f 1 f a F,! b F, h F, c F, f 0 F F hen Feynman-Kac ells us ha he soluion is: " # ) $ % * & ' { + 3,- u } / 3 & ' exp + 3 ( c F,u du h F (, 0 d0 F * F1 f F, E g F exp c F,u du { } & ' u Noe ha he main diffeence inside he expecaion sign is ha he sum of he discouned paymens (h) was added.. Example wih a aded Undelying Asse a. Acual Dynamics: ds 6 7 4S, 5 Sd 8 9 4S,5 SdZ Bice Dupoye FIN 781 Semina in Opions
7 Risk-Neual Dynamics: ds Sd S, SdZ he ems needed fo he PDE ae: a(s,)={ -δ }S, b(s,)=σ(s,)s and c(s,)= and so we ge: f f 1 f () S S f 0 S S b. Noice ha he acual dif ae is eplaced by he isk-fee ae (adjused fo he dividend yield) bu ha he volailiy is unchanged. Also noe ha he final condiion f(s,)=g(s) is he deivaive payoff funcion a eminal ime. { $ " } #! E " g S exp S S# " # c. If he inees ae is consan, he deivaive pice is: f S, E g S exp du S S e E g S S S d. And if he inees ae is sochasic: Bice Dupoye FIN 781 Semina in Opions % & ) ' ( * + -, 1 / + 0. { } f S, E g S exp du S S u 3. Example wih a Non-aded Undelying Faco a. Acual Dynamics: df 4 m F,3 Fd 5 s F,3 FdZ 8 6 7 6 7 6 79 6 7 Risk-Neual Dynamics: df : m F, ; < F, s F, Fd = s F, FdZ he ems needed fo he PDE ae: a(f,)={m(f,)- λ(f,)s(f,)}f, b(s,)=s(f,)f and c(s,)=, and so we ge: @ @ @ f f 1 f (N) @ A @ > B C? A @ A B D m s F s F h f 0 F F
8 b. Noice ha he acual dif ae (mf) is eplaced by he isk-fee ae ( {m-λs}f ) bu ha he volailiy is unchanged. Also noe ha he final condiion f(f,)=g(f) is he deivaive payoff funcion a eminal ime. c. When he inees ae is consan and when he deivaive does no yield any cash paymens (h=0), he deivaive pice is: { } E g F exp F F f F, E g F exp du F F e E g F F F d. In he mos geneal case, howeve, we have: # # #! " { } { } f F, E g F exp du exp du h F, d F F u u 4. Gisanov heoem he Gisanov heoem econciles he acual dynamics wih he isk-neual dynamics. If he acual dynamics ae descibed by: ds = µ ( S, ) Sd + σ ( S, ) SdZ he isk-neual pocess can hen be saed as: ds = ( δ ) Sd + σ ( S, ) SdZ he Gisanov saes ha if hee is a pocess λ saisfying ceain echnical condiions (Novikov condiion) and if hee is a Bownian moion Z unde he pobabiliy measue P, hen hee is anohe pobabiliy measue (Equivalen Maingale Measue) ha is equivalen o P in a way ha he pocess Bice Dupoye FIN 781 Semina in Opions
9 dz = dz + λ is a Bownian moion unde. Applying Gisanov o he undelying asse s pocess ds = µ ( S, ) Sd + σ ( S, ) SdZ, we mus have µ ( S, ) + δ λ = σ ( S, ) why? Check: µ ( S, ) + δ dz = dz + λ = dz + σ ( S, ) Sσ ( S, ) dz = Sσ ( S, ) dz S[ µ ( S, ) + δ ] ds = ( δ ) Sd + σ ( S, ) SdZ he inuiion fo his is ha a epesenaive inveso nomally equies he insananeous expeced eun µ in ode o hold he isky secuiy. Howeve, in a ficiious isk-neual wold govened by a diffeen pobabiliy measue - he equivalen Maingale measue -, ha same inveso would only equie an expeced ae of eun equal o he isk-fee ae. Unde some echnical condiions, non-abiage is EUIVALEN o he exisence of he equivalen maingale measue, Haison and Keps (1979) and Haison and Pliska (1981). Bice Dupoye FIN 781 Semina in Opions
10 B. Example wih Black-Scholes 1. Apply Gisanov o: ds Sd SdZ a. he isk-neual dynamics ae: ds Sd SdZ implied by he b. he isk-neual densiy S S dynamics ds Sd SdZ is known and given by: Ln S { } Ln S ~ N, heefoe, ( ) # $ % & 1 ' Ln S Ln S ( '! " ' ' & ' ) * ( { * + ' Ln S Ln S& ' } ),. exp -* *. By change of vaiable, we finally have: 5 6 1 5 S S Ln S Ln S S / 0 3 1 / 0 4. Picing a Euopean Call opion a. A expiaion, he opion payoff o value is: g(s)=max(0,s-k) 7 8 D 7 8 E? 9 : ; <? B F E CG E? 9 : ; <? B F e E max 0, S H K S E CG S E? 9 : @ A ; < = >? J H I E 0 E? 9 : @ A ; < = >? J H I E heefoe he opion pice is: C S,K,,, f S, e E g S S S e max 0, S K S S S ds K e S K S S S ds Bice Dupoye FIN 781 Semina in Opions