The Path Integral Approach to Financial Modeling and Options Pricing?

Compuaional Economics : 9 63, 998. 9 c 998 Kluwer Academic Publishers. Prined in he Neherlands. The Pah Inegral Approach o Financial Modeling and Opions Pricing? VADIM LINETSKY Financial Engineering Program, Deparmen of Indusrial and Operaions Engineering, Universiy of Michigan, 7 IOE Building, 05 Beal Avenue, Ann Arbor, MI 4809-7, U.S.A.; e-mail: linesky@engin.umich.edu (Acceped in final form: 0 March 997) Absrac. In his paper we review some applicaions of he pah inegral mehodology of quanum mechanics o financial modeling and opions pricing. A pah inegral is defined as a limi of he sequence of finie-dimensional inegrals, in a much he same way as he Riemannian inegral is defined as a limi of he sequence of finie sums. The risk-neural valuaion formula for pah-dependen opions coningen upon muliple underlying asses admis an elegan represenaion in erms of pah inegrals (Feynman Kac formula). The pah inegral represenaion of ransiion probabiliy densiy (Green s funcion) explicily saisfies he diffusion PDE. Gaussian pah inegrals admi a closed-form soluion given by he Van Vleck formula. Analyical approximaions are obained by means of he semiclassical (momens) expansion. Difficul pah inegrals are compued by numerical procedures, such as Mone Carlo simulaion or deerminisic discreizaion schemes. Several examples of pahdependen opions are reaed o illusrae he heory (weighed Asian opions, floaing barrier opions, and barrier opions wih ladder-like barriers). Key words: opions pricing, financial derivaives, pah inegrals, sochasic models. Inroducion In his paper we consider some applicaions of he pah inegral formalism of quanum mechanics o financial modeling. Pah inegrals consiue one of he basic ool of modern quanum physics. They were inroduced in physics by Richard Feynman in 94 in his Ph.D. hesis on pah inegral formulaion of quanum mechanics (Feynman, 94, 948; Feynman and Hibbs, 965; Kac, 949, 95, 980; Fradkin, 965; Simon, 979; Schulman, 98; Glimm and Jaffe, 98; Freidlin, 985; Dirich and Reuer, 994). In classical deerminisic physics, ime evoluion of dynamical sysems is governed by he Leas Acion Principle. Classical equaions of moion, such as Newon s equaions, can be viewed as he Euler Lagrange equaions for a minimum of a cerain acion funcional, a ime inegral of he Lagrangian funcion defining he dynamical sysem. Their deerminisic soluions, rajecories of he classical dynamical sysem, minimize he acion funcional (he leas acion principle). In quanum, i.e. probabilisic, physics, one alks abou probabiliies of differen pahs a quanum (sochasic) dynamical sysem can ake. One defines a? This research is parially suppored by he Naional Science Foundaion Gran DDM-95375.

30 VADIM LINETSKY measure on he se of all possible pahs from he iniial sae x i o he final sae x f of he quanum (sochasic) dynamical sysem, and expecaion values (averages) of various quaniies dependen on pahs are given by pah inegrals over all possible pahs from x i o x f (pah inegrals are also called sums over hisories, aswellas funcional inegrals, as he inegraion is performed over a se of coninuous funcions of ime (pahs)). The classical acion funcional is evaluaed o a real number on each pah, and he exponenial of he negaive of his number gives a weigh of he pah in he pah inegral. According o Feynman, a pah inegral is defined as a limi of he sequence of finie-dimensional muliple inegrals, inamuchhe same way as he Riemannian inegral is defined as a limi of he sequence of finie sums. The pah inegral represenaion of averages can also be obained direcly as he Feynman Kac soluion o he parial differenial equaion describing he ime evoluion of he quanum (sochasic) dynamical sysem (Schrodinger equaion in quanum mechanics or diffusion (Kolmogorov) equaion in he heory of sochasic processes). In finance, he fundamenal principle is he absence of arbirage (Ross, 976; Cox and Ross, 976; Harrison and Kreps, 979; Harrison and Pliska, 98; Meron, 990; Duffie, 996). In finance i plays a role similar o he leas acion principle and he energy conservaion law in naural sciences. Accordingly, similar o physical dynamical sysems, one can inroduce Lagrangian funcions and acion funcionals for financial models. Since financial models are sochasic, expecaions of various quaniies coningen upon price pahs (financial derivaives) are given by pah inegrals, where he acion funcional for he underlying risk-neural price process defines a risk-neural measure on he se of all pahs. Averages saisfy he Black Scholes parial differenial equaion, which is a finance counerpar of he Schrodinger equaion of quanum mechanics, and he risk-neural valuaion formula is inerpreed as he Feynman Kac represenaion of he PDE soluion. Thus, he pah-inegral formalism provides a naural bridge beween he risk-neural maringale pricing and he arbirage-free PDE-based pricing. To he bes of our knowledge, applicaions of pah inegrals and relaed echniques from quanum physics o finance were firs sysemaically developed in he eighies by Jan Dash (see Dash, 988, 989 and 993). His work influenced he auhor of he presen paper as well. See also Esmailzadeh (995) for applicaions o pah-dependen opions and Eydeland (994) for applicaions o fixed-income derivaives and ineresing numerical algorihms o compue pah inegrals. This approach is also very close o he semigroup pricing developed by Garman (985), as pah inegrals provide a naural represenaion for pricing semigroup kernels, as well as o he Green s funcions approach o he erm srucure modeling of Beaglehole and Tenney (99) and Jamshidian (99). See also Chapers 5 7 and in he monograph Duffie (996) for he Feynman Kac approach in finance and references herein. I is he purpose of his paper o give an inroducory overview of he pah inegral approach o financial modeling and opions pricing and demonsrae ha pah

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 3 inegrals and Green s funcions consiue boh a naural heoreical concep and a pracical compuaional ool in finance, especially for pah-dependen derivaives. The res of his paper is organized as follows. In Secion, we give an overview of he general framework of pah-inegral opions pricing. We sar by considering a single-asse Black Scholes model as an example. Then, we develop he pah inegral formalism for a muli-asse economy wih asse- and ime-dependen volailiies and correlaions. The cenral resul here is a general pah inegral represenaion (Feynman Kac formula) for a pah-dependen opion coningen upon a finie number of underlying asse prices. The pah inegraion measure is given by an exponenial of he negaive of he acion funcional for he risk-neural price process. This formula consiues a basis for pracical calculaions of pahdependen opions. In Secion 3, we give a brief overview of he main echniques o evaluae pah inegrals. Gaussian pah inegrals are calculaed analyically by means of he Van Vleck formula. Cerain iniially non-gaussian inegrals may be reduced o Gaussians by changes of variables, ime re-paramerizaions and projecions. Finally, essenially non-gaussian pah inegrals mus be evaluaed eiher numerically by Mone Carlo simulaion or a deerminisic discreizaion scheme, such as binomial or rinomial rees, or by analyical approximaions such as he semiclassical or WKB approximaion. In Secion 4, hree examples of pah-dependen opions are given o illusrae he heory (weighed Asian opions, floaing barrier opions and barrier opions wih ladder-like barriers).. Risk-Neural Valuaion and Wiener Feynman Pah Inegrals.. BLACK SCHOLES EXAMPLE We begin by reviewing he Black Scholes model (Black and Scholes (973) and Meron (973); see also Hull (996) and Duffie (996)). A pah-independen opion is defined by is payoff a expiraion a ime T O F (S T ;T)=F(S T ); (.) where F is a given funcion of he erminal asse price S T. We assume we live in he Black Scholes world wih coninuously compounded risk-free ineres rae r and a single risky asse following a sandard geomeric Brownian moion ds = m d + dz (.) S wih consan drif rae m and volailiy (for simpliciy we assume no dividends). Then he sandard absence of arbirage argumen leads us o consrucing a replicaing porfolio consising of he underlying asse and he risk-free bond and o he Black Scholes PDE for he presen value of he opion a ime preceeding expiraion S @ O F @S + rs @O F @S, ro F =, @O F @ (.3)

3 VADIM LINETSKY wih iniial condiion (.) (more precisely, erminal condiion since we solve backwards in ime). This is he backward Kolmogorov equaion for he risk-neural diffusion process (.) wih drif rae equal o he risk-free rae r. Inroducing a new variable x = ln S which follows a sandard arihmeic Brownian moion dx = m,! Equaions (.), (.3) reduce o d + dz; (.4) @ O F @x + @O F @x, ro F =, @O F @ ; (.5a) = r, ; (.5b) O F (e x T ;T)=F(e xt ): (.5c) A unique soluion o he Cauchy problem (.5) is given by he Feynman Kac formula (see, e.g., Duffie, 996; see also Io and McKean, 974; Durre, 984; Freidlin, 985; Karazas and Shreve, 99) O F (S; ) =e,r E (;S) F (ST ) ; = T, ; (.6) where E (;S) [:] denoes averaging over he risk-neural measure condiional on he iniial price S a ime. This average can be represened as an inegral over he se of all pahs originaing from (; S), pah inegral. Iisdefinedasalimi of he sequence of finie-dimensional muliple inegrals, in a much he same way as he sandard Riemannian inegral is defined as a limi of he sequence of finie sums (Feynman, 94 and 948; Feynman and Hibbs, 965). We will firs presen he final resul and hen give is derivaion. In Feynman s noaion, he average in (.6) is represened as follows (x = ln S; x T = ln S T ): O F (S; ) = e,r E (;S) F (e x T ) = e,r Z, Z x(t )=xt x()=x F, e x T e,a BS[x( 0 )] Dx( 0 )! dx T : (.7) A key objec appearing in his formula is he Black Scholes acion funcional A BS [x( 0 )] defined on pahs fx( 0 ); 0 Tg as a ime inegral of he Black Scholes Lagrangian funcion A BS [x( 0 )] = _x( 0 ):= dx d 0: Z T L BS d 0 ; L BS = (_x(0 ),) ; (.8)

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 33 This acion funcional defines he pah inegraion measure. The pah inegral in (.7) is defined as follows. Firs, pahs are discreized. Time o expiraion is discreized ino N equal ime seps bounded by N + equally spaced ime poins i = + i; i = 0; ;:::;N; = (T, )=N. Discree prices a hese ime poins are denoed by S i = S( i )(x i =x( i )for he logarihms). The discreized acion funcional becomes a funcion of N + variables x i (x 0 x; x N x T ) A BS (x i )=, N, (x T,x)+ (x i+,x i ) : (.9) This is obained direcly from he Definiion (.8) by firs noing ha X i=0 L = _x, _x + ; A BS [x( 0 )] =, (x T, x) +A 0 [x( 0 )]; A 0 [x( 0 )] = Z T L 0 d 0 ; (.0a) (.0b) (.0c) where L 0 is he Lagrangian for a zero-drif process dx = dz (maringale) L 0 = _x ; (.) and hen subsiuing Z T d 0! N, X i=0 :::; _x! x i+, x i : Now, he pah inegral over all pahs from he iniial sae x() o he final sae x T is defined as a limi of he sequence of finie-dimensional muliple inegrals: Z x(t )=xt x()=x := lim F (e xt ) e,a BS[x( 0 )] Dx( 0 ) Z Z N!,, {z } N, F (e xt ) e,a BS(x i ) dx p ::: dx N, p : (.) This definiion of pah inegrals is used in physics o describe quanum (probabilisic) phenomena. I can be shown (see, e.g., Kac, 95; Kac, 980; Glimm and Jaffe,

34 VADIM LINETSKY 98; Simon, 979; Freidlin, 985) ha his definiion is compleely rigorous and he limi does converge. In his paper, however, we will follow a heurisic approach o pah inegrals leaving ou he echnical deails. Since he payoff in Equaion (.7) depends only on he erminal sae x T,he payoff funcion F can be moved ouside of he pah inegral, and i can be re-wrien as follows: O F (S; ) Z = e,r F (e xt ) e (= )(x T,x),( = ) K(x T ;Tjx; ) dx T ; (.3), where K(x T ;Tjx; ) is he ransiion probabiliy densiy for zero-drif Brownian moion dx = dz (probabiliy densiy for he erminal sae x T a ime T condiional on he iniial sae x a ime ), or Green s funcion (also called propagaor in quanum physics) (see, e.g., Schulman (98)): K(x T ;Tjx; ) = Z x(t )=xt := lim x()=x Z N!,, {z } N, e,a 0[x( 0 )] Dx( 0 ) Z X exp, N, i=0 (x i+, x i )! dx p dx p N, : (.4) The muliple inegral here is Gaussian and is calculaed using he following ideniy Z, e,a(x,z),b(z,y) dz = r a + b exp, ab a + b (x, y) : (.5) This is proved by compleing he squares in he exponenial. Using (.5) consider he inegral on x in (.4) which equals Z, exp q exp (), ", (x, x 0 ) () (x, x ) +(x,x 0 ) # dx (.6a) : (.6b) Thus he effec of he x inegraion is o change o (boh in he square roo and in he exponenial) and o replace (x, x ) +(x,x 0 ) by (x, x 0 ).

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 35 The inegral over x changes o 3 (boh in he square roo and in he exponenial) and yields he erm (x 3, x 0 ). This procedure is coninued for all N, inegrals. Finally, becomes N, which is jus, and (x T, x 0 ) appears in he exponenial. Since here is no longer any dependence on N, he limi operaion is rivial and we finally obain he resul K(x T ;T jx; ) = p exp, (x T, x)! ; (.7) which is, as expeced, he normal densiy. This is he fundamenal soluion of he zero-drif diffusion equaion @ K @x =,@K @ (.8a) wih iniial condiion a = T K(x T ;T jx; T )=(x T,x); (.8b) where (x) is he Dirac dela funcion. Cerainly, in his simple case one can also solve he diffusion equaion direcly. Firs, a formal soluion o he Cauchy problem (.8) can be wrien as K(x T ;T jx; ) =exp! @ @x (x T, x): (.9) If we now represen he dela funcion as a Fourier inegral, we obain K(x T ;T jx; ) = exp = = Z, exp!z @ @x, e ip(x T,x) dp dp, p +ip(x T, x) p exp, (x T, x)! ; (.0) where we have used he sandard Gaussian inegral Z exp, a, y + by dy = p a exp! b : (.) a This proves ha he pah inegral (.4) indeed represens he fundamenal soluion of diffusion Equaion (.8).

36 VADIM LINETSKY I is useful o noe ha he Green s funcion for diffusion wih consan drif rae is obained by muliplying he zero-drif Green s funcion by he drif-dependen facor (see (.3)): K (x T ;T jx; ) = e (= )(x T,x),( = ) K(x T ;T jx; ) = p exp, (x T, x, )! : (.a) I is easy o check direcly ha K is he fundamenal soluion of diffusion equaion wih drif @ K @x + @K @x =,@K @ : (.b) The ransiion probabiliy densiy saisfies he fundamenal Chapman Kolmogorov semigroup propery (coninuous-ime Markov propery) (see Garman, 985, for semigroups in finance) Z K(x 3 ; 3 jx ; )= K(x 3 ; 3 jx ; )K(x ; jx ; ) dx : (.3), Now one can see ha he definiion of he pah inegral (.4) can be obained by repeaed use of he Chapman Kolmogorov equaion: K(x T ;T jx; ) = Z Z lim K(x T ;T jx N, ; N, ) N!,, {z } N, K(x ; jx; ) dx dx N, : (.4) Finally, subsiuing K inoequaion (.3)one obainshe Black Scholesformula for pah-independen opions Z O F (S; ) =e,r F (e xt ) p exp, (x T, x, ),! dx T : (.5) For a call opion wih he payoff Max(e x T, K; 0) one obains afer performing he inegraion C(S; ) =SN(d ), e,r KN(d ); (.6) ln S K + d = p ; d = d + p : (.7)

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 37 In he Black Scholes example of diffusion wih consan coefficiens and pahindependen payoffs here exiss a closed-form soluion for he ransiion probabiliy densiy as a normal densiy, and one cerainly does no need he pah-inegrals machinery in his simple case. However, he pah inegral poin of view becomes very useful for more complex models, especially for pah-dependenopions, general volailiies and drifs and derivaives coningen upon several underlying asses. American opions are valued in his framework by he procedure of Geske and Johnson (984) (see Dash (988))... THE FEYNMAN KAC APPROACH TO PRICING PATH-DEPENDENT OPTIONS Consider now a pah-dependen opion defined by is payoff a expiraion O F (T )=F[S( 0 )]; (.8) where F [S( 0 )] isagivenfuncional on price pahs fs( 0 ); 0 Tg, raher han a funcion dependen jus on he erminal asse price. We assume he risk-neural price process ds S = r d + dz; x = ln S; dx = d + dz; = r, : (.9) Then he presen value of his pah-dependen opion a he incepion of he conrac is given by he Feynman Kac formula O F (S; ) = e,r E (;S) F [S( 0 )] = e,r Z, Z x(t )=xt x()=x F h i e x(0 ) e,a BS[x( 0 )] Dx( 0 )! dx T ; (.30) where he average is over he risk-neural process. Since now F [e x(0) ] depends on he enire pah, i cannobe simply movedouside of he pahinegralas we didin he previous secion in he Black Scholes case. Le us firs consider a special case. Suppose he payoff funcional F can be represened in he form F = f (S T ) e,i[s(0 )] ; (.3) where f (S T ) depends only on he erminal asse price S T,andI is a funcional on price pahs from (; S) o (T;S T ) ha can be represened as a ime inegral I[S( 0 )] = Z T V (x( 0 ); 0 ) d 0 ; (.3)

38 VADIM LINETSKY of some poenial V (x; 0 )(x = ln S). Then he Feynman Kac formula (.30) reduces o O F (S; ) Z = e,r f (e xt ) e (= )(x T,x),( = ) K V (x T ;T jx; ) dx T ; (.33), where K V is he Green s funcion (ransiion probabiliy densiy) for zero-drif Brownian moion wih killing a rae V (x; 0 ) (see, e.g., Io and McKean, 974; Karlin and Taylor, 98; Durre, 984): K V (x T ;T jx; ) = Z x(t)=xt x()=x exp, Z T (L 0 + V ) d 0! Dx( 0 ): (.34) This is he Feynman Kac represenaion of he fundamenal soluion of zero-drif diffusion PDE wih poenial V @ K V @x, V (x; )K V =, @K V @ and iniial condiion K V (x T ;T jx; T )=(x T,x): (.35a) (.35b) I is easy o see ha he opion price (.33) saisfies he Black Scholes PDE wih poenial @ O F @x + @O F @x, (r + V (x; ))O F =, @O F @ (.36) and he erminal condiion O F (S T ;T) = f(s T ). I can be inerpreed as he Black Scholes equaion wih an effecive risk-free rae r + V (x; ) and coninuous dividend yield V (x; ). Now consider a more general pah-dependen payoff ha can be represened as a funcion of he erminal asse price as well as a se of n in some sense elemenary funcionals I i [S( 0 )] on price pahs F [e x(0) ]=F(e x T ;I i ); I i =I i [e x(0) ]: (.37) Some examples of such funcionals and corresponding pah-dependen opions are: weighed average price (weighed Asian opions), maximum or minimum prices (lookback and barrier opions) and occupaion imes (range noes, sep opions and more general occupaion ime derivaives, see Linesky, 996). We can employ he following rick o move he funcion F ouside of he pah inegral in Equaion

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 39 (.30) (Dash, 993). Firs, inroduce auxiliary variables i by insering he Dirac dela funcion as follows F (e x T ;I i )= Z R nn ( i,i i )F(e x T ; i )d n ; n ( i, I i ) (, I ) :::( n,i n ): Nex, he dela funcion is represened as a Fourier inegral F (e x T ;I i ) = Z T ; i )F, R nf(ex = () n Z R n Z " exp,i nx!# nx p i I i d n i= (.38) exp i p i ( i, I i ) F (e x T ; i )d n pd n : (.39) R n i= Finally, subsiuing his back ino Equaion (.30) we arrive a he pricing formula for pah-dependen opions Z Z O F (S; ) = e,r F (ex T ; i ) e (= )(x T,x),( = ) R n, P(x T ; i ;T jx; ) d n dx T ; (.40) where P is he join probabiliy densiy for he erminal sae x T and erminal values i of he Brownian funcionals I i a expiraion T condiional on he iniial sae x a incepion. I is given by he inverse Fourier ransform P(x T ; i ;T jx; ) = F, of he Green s funcion = () n Z K I;p (x T ;T jx; ) = KI;p (x T ;T jx; ) R n exp Z x(t)=xt x()=x i exp nx i=! p i i! K I;p (x T ;T jx; )d n p; (.4),A 0, i nx i= p i I i! Dx( 0 ) (.4) wih respec o he parameers p i. If he elemenary funcionals can be represened as ime inegrals Z T I i = v i (x( 0 ); 0 ) d 0 (.43) of some poenials v i (x; 0 ), hen he pah inegral (.4) akes he sandard form (.34) wih poenial V (x; ; p i )=i nx i= p i v i (x; ): (.44)

40 VADIM LINETSKY If he poenials v i are non-negaive funcions, hen Laplace ransform can be used in place of R he Fourier ransform. Consider a pah-dependen payoff F (e x T ;I), T where I = v(x( 0 ); 0 ) d 0 and v(x; 0 ) is non-negaive. Then h F (e x T ;I) = F(e x T ;)L, e,sii d Z Z (,I)F(e x T ;) d = 0 0 Z Z "+i = i 0 ",i e s(,i) F (e x T ;) ds d: (.45) Here he auxiliary variable akes only non-negaive values, and we represen he Dirac dela funcion as an inverse Laplace ransform. Then he pah-dependen pricing formula akes he form: Z Z O F (S; ) = e,r F (e x T ;) e (= )(x T,x),( = ), 0 P(x T ;;T jx; ) d dx T ; (.46) where P is he join probabiliy densiy for he final sae x T and he erminal value of he Brownian funcional I condiional on he iniial sae x a ime.iisgiven by he inverse Laplace ransform P(x T ;;T jx; ) =L, KV (x T ;T jx; ) (.47) of he Green s funcion for zero-drif Brownian moion wih killing a rae V (x; ) = sv(x; ) given by he pah inegral (.34). I is he Feynman Kac represenaion of he fundamenal soluion of zero-drif diffusion PDE wih poenial V (x; ) (.35). I is easy o see ha he densiy (.47) saisfies a hree-dimensional PDE: @ P, v(x; )@P @x @ =,@P @ : (.48) In summary, o price a pah-dependen claim wih he payoff coningen boh on he erminal asse price and he erminal value of some funcional I on price pahs ha can be represened as a ime inegral of non-negaive poenial v(x; ): () find he Green s funcion of Brownian moion wih killing a rae V = sv(x; ) by solving he PDE (.35) or calculaing he pah inegral (.34); () inver he Laplace ransform wih respec o s o find he join densiy for x T and (.47); and (3) calculae he discouned expecaion (.46). Equaions (.46,.47) ogeher wih (.34,.35) consiue he radiional form of he Feynman Kac approach. I allows one o compue join probabiliy densiies of Brownian funcionals and he erminal sae given he iniial sae. So far we considered pricing newly-wrien pah-dependenopions a he incepion of he conrac. Now consider a seasoned opion a some ime during

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 4 he life R of he conrac, T, wih he erminal payoff F (S; I), where T I = v(x( 0 ); 0 ) d 0,andv(x; 0 ) is non-negaive. The funcional I is addiive and can be represened as a sum I = I f +I u,wherei f is he value of he funcional on already fixed price R observaions on he ime inerval from he conrac incepion o dae ;I f = v(x( 0 ); 0 ) d; and I u is he funcional R on ye unknown segmen of he price pah from ime T o expiraion T;I u = v(x(0 ); 0 ) d 0. Then he seasoned opion price a ime is a funcion of he curren asse price S = S( ),hevaluei f of he funcional I accumulaed o dae, and curren ime : O F (S ;I f ; ) = e,r Z Z F (e x T ;I f +) e (= )(x T,x ),( = ), 0 P(x T ;;T jx ; ) d dx T ; (.49) where = T, and x = ln S. I is easy o see ha, under suiable echnical condiions, from (.48) i follows ha he seasoned opion price (.49) saisfies he following hree-dimensional PDE in variables x ;I f and (Wilmo, Dewyne and Howison, 993): @ O F @x + @O F @x, ro F + v(x ; ) @O F @I f =, @O F @ : (.50).3. VALUATION OF MULTI-ASSET DERIVATIVES WITH GENERAL PARAMETERS Consider a general D-dimensional diffusion process x ;=;;:::;D, dx =a d+ DX a= a dza ; (.5) a = a (x;); a = a (x;); where dz a ;a=;;:::;d, are sandard uncorrelaed Wiener processes E h dz a dz bi = ab d (.5) ( ab is he Kroeneker symbol, ab = ifa = b and zero oherwise). Suppose he risk-free rae is r(x;)andequaion (.5) describesa D-dimensionalrisk-neural price process wih he risk-neural drif (boldface leers x denoe D-dimensional vecors prices of D raded asses in our economy) a (x;)=r(x;)x,d (x;) (.53)

4 VADIM LINETSKY (D are dividends). Consider a pah-dependen opion wih he payoff a expiraion O F (T )=F[x( 0 )]: (.54) Then he presen value O F (x;) a he incepion of he conrac is given by he Feynman Kac formula O F (x;)= ZRD Z x(t)=xt x()=x F [x( 0 )] e,a[x(0 )] Dx( 0 )! d D x T : (.55) Here A is he acion funcional Z T A = L d 0 (.56) wih he Lagrangian funcion L for he process (.5) given by (see, e.g., Langouche, Roekaers and Tirapegui, 980 and 98; Freidlin, 985) L = DX ;= g (x; 0 )( _x ( 0 ), a (x; 0 ))( _x ( 0 ), a (x; 0 )) + r(x; 0 ); (.57) where g = g is an inverse of he variance-covariance marix g = g DX = g g = ; g = DX a= a a : (.58) Readers familiar wih he Riemannian geomery will recognize g as he Riemannian meric and a as componens of he local frame (vielbein). The general muli-asse pah-inegral (.55) is defined as a limi of he sequence of finie-dimensional muliple inegrals similar o he one-dimensional example. A discreized acion funcional is given by A(x i ) = N, X DX i=0 ;= X N, + i=0 g (x i ; i ) x i,a (x i ; i )!! x i,a (x i ; i ) r(x i ; i ); x i = x i+, x i ; (.59) and Z x(t )=xt F [x( 0 )] e,a[x(0)] Dx( 0 ) x()=x := lim N! N, Y i= ZRD Z R {z D } N, F (x i ) exp(,a(x i )) d D x q i () D de(g (x i ; i )) : (.60)

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 43 The deerminan de(g (x i ; i )) of he variance-covariance marix g (x i ; i ) appearing in he square roos defines he inegraion measure over inermediae poins x i. This discreizaion scheme is called pre-poin discreizaion and is consisen wih he Io s calculus. One could choose a differen discreizaion, such as mid-poin or symmeric discreizaion. The mid-poin discreizaion is consisen wih he Sraanovich calculus raher han Io s. Theoreically, differen discreizaion schemes are equivalen (see Langouche, Roekaers and Tirapegui, 980 and 98, for deailed discussions). However, in pracice differen discreizaions have differen numerical convergence properies and i may be advanageous o use one scheme over he oher for a paricular calculaion (see also Karlin and Taylor, 98, for a discussion of Io s vs. Sraanovich calculus). For pah-independen opions, when he payoff depends only on erminal saes x T ; O F (x T ;T)=F(x T ), he opion value saisfies he backward PDE HO F =, @O F @ ; (.6) where H is a second order differenial operaor (generaor of he diffusion process (.5) wih he killing erm r(x;)) H = DX ;= @ g (x;) @x @x + DX = a (x;) @, r(x;): (.6) @x The proof ha he pah inegral (.55) for pah-independen opions indeed solves he PDE (.6) is as follows. Consider he fundamenal soluion K(x T ;T jx;)of he PDE (.6) wih iniial condiion K(x T ;T jx;t)= D (x T,x): (.63) For a shor ime inerval =,, i can be represened up o he second order O( ) similar o he Black Scholes case (.0) (in conras o he case wih consan parameers, i is only valid up o he second order in in he general case): K(x ; jx ; ) = ( +H+O( )) D (x, x ) exp ( H) D (x, x ): (.64) Again inroducing he Fourier inegral represenaion of he dela funcion, we have K(x ; jx ; ) exp ( H) = Z R D exp 8 < Z :, R D exp 8 9 < X D = : i p (x, x ) ; dd p () = D DX ;= g (x ; )p p

44 VADIM LINETSKY + i DX =, x, x, a (x ; ) p 9 =,r(x ; ) ; dd p () D = q () D de(g (x ; )) 8 < exp :, DX ;= x,x g (x ; ) x,x,a (x ; ) =,a (x ; ),r(x ; ) ; : (.65) 9! To obain his resul we have used he following sandard muli-dimensional Gaussian inegral Z () D R D exp 0 @, DX ;= A y y + 0 = q exp@, () D de(a ) DX ;= DX = B y A d D y (A, ) B B A : (.66) Having a our disposal he shor-ime ransiion probabiliy densiy, we can obain he densiy for a finie ime inerval = T, in he coninuous ime limi by successively applying he Chapman Kolmogorov semigroup propery Z ZRD K(x T ;T jx;) = lim K(x T ;T jx N, ; N, ) N! R {z D } N, K(x ; jx;)d D x d D x N, : (.67) Subsiuing he expression (.65) for he shor-ime densiies and recognizing ha he individual exponenials of shor-ime densiies combine o form he expression (.59) for he discreized acion, we finally obain Z ZRD K(x T ;T jx;) = lim exp(,a(x i )) N! R {z D } N,

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 45 := N, Y i= Z x(t )=xt x()=x d D x q i () D de(g (x i ; i )) e,a[x(0 )] D x( 0 ): (.68) Thus, we have proved ha he pah inegral (.68) indeed represens he fundamenal soluion of diffusion PDE (.6). Then Equaion (.55) for pah-independen opions is simply Z O F (x;)= R DF(x T)K(x T ;T jx;)d D x T : (.69) This concludes he proof ha (.55) indeed solves he Cauchy problem (.6). For pah-dependen payoffs one mus employ he procedure oulined in he previous secion o move he funcional F ouside of he pah inegral. This will resul in he appearance of a non-rivial poenial V (x; ) in he exponenial in pah inegral (.68) and in he PDE (.6) for ransiion probabiliy densiy: HK V,V(x; )K V =, @K V @ : (.70) 3. Evaluaion of Pah Inegrals The Feynman Kac formula (.55) is a powerful and versaile ool for obaining boh closed-form and approximae soluions o financial derivaives valuaion problems. A number of echniques are available o evaluae pah inegrals (.60), (.68). They fall ino hree broad caegories: exac analyical soluions, analyical approximaions, and numerical approximaions. Analyical soluions are available for Gaussian pah inegrals and hose ha can be reduced o Gaussians by changes of variables, re-paramerizaions of ime and projecions. Suppose he Lagrangian funcion L (.57) is a mos quadraic in x and _x. Then he closed-form soluion for he Gaussian pah inegral (.68) is given by he Van Vleck formula: Z x( )=x x( )=x = e,a[x()] D x() vu u de,! @ A Cl (x ;x ) @x exp f,a Cl (x ;x )g : (3.) @x Here A Cl (x ;x ) is he acion funcional (.56) evaluaed along a classical soluion of he Euler Lagrange equaions A @L x = @x, d @L d @ _x = 0 (3.)

46 VADIM LINETSKY wih he boundary condiions x Cl ( )=x ; x Cl ( )=x : (3.3) The deerminan appearing in (3.) is called Van Vleck deerminan. Noe ha, in general, he explici evaluaion of A Cl (x ;x ) may be quie complex due o complicaed classical soluions x Cl () of he Euler Lagrange Equaions (3.). Models admiing closed-form soluions due o he Van Vleck formula include Gaussian models and models ha can be reduced o Gaussians by changes of variables, re-paramerizaions of ime and projecions. Examples of he former caegory include he Black Scholes model and mean reversion models (Ornsein Uhlenbeck, or harmonic oscillaor, processes). The laer caegory includes he Cox Ingersoll Ross model (Bessel process which is he radial par of he mulidimensional Brownian moion (projecion)). To illusrae he use of he Van Vleck formula, le us again consider he Black Scholes example. The Euler Lagrange Equaion (3.) for he Black Scholes acion A 0 = Z T _x d 0 (3.4) simply saes ha acceleraion vanishes in he absence of exernal forces (Newon s law) x = 0: (3.5) The soluion wih boundary condiions (3.3) is a classical rajecory from x o x a sraigh line connecing he wo poins: x Cl ( 0 )= x (T, 0 )+x ( 0,) : (3.6) T, The acion funcional evaluaes on his rajecory o A Cl = Z T (x, x ) d 0 = (x, x ) : (3.7) Subsiuing his resul ino he Van Vleck formula (3.) we again obain he normal densiy p exp, (x, x )! : (3.8) Furhermore, he Van Vleck formula serves as a saring poin for semiclassical (general momens) expansion. The firs erm in he semiclassical expansion is called

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 47 WKB (or semiclassical) approximaion. I approximaes he non-gaussian model by a suiable Gaussian. Finally, more complex pah inegrals can only be evaluaed numerically. Mone Carlo simulaion has long been one of he favorie echniques for compuing pah inegrals numerically (see, e.g., Meropolis e al. (953), Creuz e al. (983)). Mone Carlo simulaion simply approximaes he pah inegral by a sum over a finie number of sample pahs. Deerminisic low-discrepancy algorihms (quasi Mone Carlo) may be especially appropriae for simulaions in finance, as hey sample pahs more efficienly han unsrucured pseudo Mone Carlo (Birge, 995; Joy, Boyle and Tan, 995; Paskov and Traub, 995). Finally, differen finie-difference echniques for solving he backward PDE can also be alernaively viewed as discreizaion schemes for pah inegrals. For ineresing numerical algorihms for compuing pah inegrals in finance see Eydeland (994). 4. Examples 4.. WEIGHTED ASIAN OPTIONS Asian opions are opions wih he payoff dependen on he average price of he underlying asse over a specified period of ime. The average price over a ime period preceeding expiraion, raher han jus a erminal price, has wo main advanages. Firs, i smoohs he opion s payoff and prevens i from being deermined by he underlying price a a single insan in ime. A given erminal asse price may be unnaurally biased or manipulaed. The laer has been a concern in cerain commodiy markes dominaed by large insiuions whose acions migh emporarily disor prices. Anoher need of using he average price ofen arises in corporae hedging siuaions. For example, many corporaions exchange foreign currency for domesic currency a regular inervals over a period of ime. Asian-syle derivaives provide a cheaper alernaive o hedging each individual ransacion. They hedge only he average exchange rae over a period of ime, hus significanly reducing he hedge coss. Moreover, if individual ransacion daes are unknown in advance, i is impossible o hedge each individual ransfer precisely, bu i is sill possible o hedge he average exchange rae over ime. See, e.g., Kemna and Vors, 990; Levy and Turnbull, 99; Turnbull and Wakeman, 99; Chance and Rich, 995 and references herein for deails on usage and pricing of Asian opions. To accommodae hedging of cash flows ha may no be equal in amoun, bu raher follow a specific schedule, weighed or flexible Asian opions (WAOs) have been recenly inroduced (Dash, 993; Zhang, 994 and 995a). Specific weighed averaging schemes are used in hese opions. WAOs became quie popular in foreign exchange and energy markes in paricular.

48 VADIM LINETSKY In case when he weighed averaging is geomeric, since he geomeric average of a lognormal variae is iself lognormally disribued, a closed-form pricing formula can be easily obained. Consider a WAO wih he payoff a expiraion O F (T )=F(S T ;I); (4..) where S T is he erminal asse price, I is a weighed average of he logarihm of he asse price, x = ln S, over a specified ime period 0 0 T preceeding expiraion Z T I = w( 0 )x( 0 ) d 0 ; (4..) 0 w( 0 ) is a given weigh funcion specified in he conrac and normalized so ha Z T 0 w( 0 )=; (4..3) and F is a given funcion of S T and I. The weighed geomeric average is given by he exponenial of I. Some examples of he possible choices for he weigh funcion are: Sandard opion wih payoff dependen on S T only w( 0 )=(T, 0 ); (4..4a) Sandard (equally weighed) Asian opion, coninuous averaging w( 0 )= T, 0 ; (4..4b) Sandard (equally weighed) Asian opion, discree averaging w( 0 )= N+ NX i=0 ( 0, i ); (4..4c) Discree weighed averaging w( 0 )= NX i=0 w i ( 0, i ); NX i=0 w i =; (4..4d) where w i are specified weighs and i = + ih; h =(T, 0 )=N; i = 0;:::;N. Here N + is he oal number of price observaions o consruc he weighed

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 49 averageprice specifiedin he conracand h is he ime inerval beween wo observaions. By using Dirac s dela funcions boh coninuous and discree sampling can be reaed uniformly. Some examples of he payoff funcion F are: Weighed average price call F (I) =Max(e I, K; 0); (4..5a) Weighed average srike call F (S T ;I)=Max(S T, e I ; 0); (4..5b) Digial weighed average price call F (I) =D(e I,K); (4..5c) where is he Heavyside sep funcion ((x) =(0)for x 0 (x <0))andD is a fixed payoff amoun if he average price is above srike K a expiraion. Asian pus are defined similarly. In discree case (4..4d), a geomeric weighed average price e I is given by e I = NY i=0 S( i ) w i : (4..6) If he curren ime when we price he opion is inside he averaging inerval, 0 <<T(seasoned Asian opion), hen I = I f + I u ; (4..7a) where I f is he weighed average of already fixed price observaions (x f = ln S f ) Z I f = w( 0 )x f ( 0 ) d 0 ; 0 (4..7b) and I u is he average of ye uncerain prices Z T I u = w( 0 )x( 0 ) d 0 : (4..7c) If < 0 (forward-saring Asian opion), hen i is convenien o exend he definiion of he weigh funcion o he enire inerval 0 T by seing w( 0 ) 0 for 0 < 0 : (4..8)

50 VADIM LINETSKY Then Equaion (4..7a) is always rue (I f 0if< 0 ). The presen value a ime of a weighed Asian opion wih he payoff F (S T ;I) is given by he average: O F (S; ) =e,r E (;S) F (ST ;I f +I u ) : (4..9) According o he mehodology developed in Secion., his average reduces o (noe ha since x R, he linear poenial v(x; 0 )=!( 0 )xis unbounded, and we use he Fourier ransform raher han he Laplace, and inegrae from, o ): O F (S; ) = e,r Z, Z, F (e x T ;+I f )P (x T ;;T jx; ) d dx T ; (4..0) where P is he join probabiliy densiy for he logarihm of he erminal sae x T and he weighed average of he logarihm of he asse prices a expiraion T condiional on he iniial sae x a ime. I is given by he inverse Fourier ransform P (x T ;;T jx; ) = e (= )(x T,x),( = ) F, Z = e (= )(x T,x),( = ), KV (x T ;T jx; ) e ip K V (x T ;T jx; ) dp : (4..) Here K V is he Green s funcion for zero-drif Brownian moion wih poenial V (x; 0 )=ip!( 0 ) x (4..) given by he pah inegral (.34). Since he poenial is linear in x, he pah inegral is Gaussian and hus can be evaluaed in closed form: where K V (x T ;T jx; ) = ( ) p exp, (x T, x), ip( x T + x), p ; (4..3) = ; =, ; = = Z T Z 0 Z T w( 0 )( 0, ) d 0 ; w( 0 )w( 00 )(T, 0 )( 00, ) d 00 d 0 : (4..4a) (4..4b)

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 5 I is a classic resul (see, e.g., Feynman and Hibbs, 965; Schulman, 98). In physics, his Green s funcion describes a quanum paricle in an exernal imedependen elecric field pw( 0 ). Now he inegral over p in (4..) is Gaussian and he inverse Fourier ransform can be evaluaed in closed form yielding he resul for he densiy: P (x T ;;T jx; ) = ( p exp, (x T, x, ), (, x T, x) 4 ) : (4..5) This is a bivariae normal densiy for wo random variables x T and (weighed average of he logarihm of he asse price) a ime T condiional on he sae x a ime. I can be re-wrien in he sandard form P (x T ;;T jx; ) = wih he means xt p, exp (, (, ) (x T, xt ) x + (, ) T ), (x T, xt )(, ) ; (4..6) xt xt = x + ; = x + ; (4..7a) sandard deviaions xt = p ; = p ; (4..7b) and he correlaion coefficien = p ; (4..7c) = + : (4..7d) Here p is he volailiy of he weighed average and is he correlaion coefficien beween he weighed average and x T. Formulas (4..0), (4..6) allow one o price any geomeric weighed Asian opions wih payoffs dependen boh on he erminal asse price and he geomeric average.

5 VADIM LINETSKY Le us consider a paricular case when he payoff funcion is independen of x T. Then we can perform he Gaussian inegraion over x T in (4..0), (4..5) and arrive a O F (S; ) Z ( ) = e,r F ( + I f ) p, exp (, x, ), d: (4..8) This coincides wih he Black Scholes formula (.5) wih re-scaled volailiy! p and drif rae!. All he informaion abou he weigh funcion w is encoded in he volailiy and drif rae mulipliers and. In paricular, for payoffs (4..5a) and (4..5c) we have: Digial weighed average price call C D (S; ) = D e,r Z ln K,I f p exp (, ) (, x, ) = D e,r N(d ); (4..9a) d where d = ln S K + I f + p ; (4..9b) Weighed average price call C(S; ) Z ( = e,r e +I f, K p ln K,I f exp, ) (, x, ) = e,q+i f SN(d ), e,r KN(d ); (4..0a) where d is given above and p d = d + ; q = r + (, ) =r,, : (4..0b) A he sar of he averaging period (I f = 0), his formula for he weighed geomeric average price call coincides wih he Black Scholes formula wih rescaled volailiy p and coninuous dividend yield q. The average srike opions are a paricular case of opions o exchange one asse for anoher; he erminal price and he weighed geomeric average wih volailiy d

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 53 and correlaion given by Equaions (4..7) are he wo underlying variables in his case. Now le us consider differen choices for he weigh funcion. For (4..4a) we have = ; = 0; = ; (4..) and Equaion (4..0) becomes he sandard Black Scholes formula. Consider he case of sandard (equally weighed) coninuously averaged geomeric Asian opions (4..4b). If = 0, i.e., he pricing ime coincides wih he sar of he averaging period, we have = ; = 4 ; = 3 : (4..) Thus, volailiy of he equally weighed geomeric average is = p 3 (he well-known p 3-rule). Now consider he case of discree weighed averaging (4..4d) and se = 0. The coefficiens and (4..4) reduce o = N = N NX k= NX k= kw k ; (4..3) k(n, k)w k + N NX k, X k= l= l(n, k)w k w l : (4..4) Subsiuing his ino Equaion (4..7d), we arrive a he discreized expression for he volailiy muliplier = N NX k= kw k + N NX k, X k= l= lw k w l : (4..5) In he case of arihmeic averaging, he second sae variable is a weighed average price Z T I = w( 0 ) e x(0) d 0 : (4..6) The poenial v(x; 0 )=!( 0 )e x is non-negaive and he join densiy for x T and I a ime T condiional on he sae x a ime is given by he inverse Laplace ransform P (x T ;;T jx; ) =e (= )(x T,x),( = ) L, KV (x T ;T jx; ) (4..7)

54 VADIM LINETSKY of he Green s funcion for Brownian moion wih killing a rae V (x; 0 )=sw( 0 ) e x (4..8) saisfying he PDE (.35a) wih poenial V and iniial condiion (.35b) (he inverse Laplace ransform is aken wih respec o he variable s). This PDE canno be solved in closed form for arbirary w( 0 ), and one mus resor o one of he approximaion procedures. However, in he special case of equally weighed coninuous averaging, an analyical soluion does exis. When w( 0 ) is independen of ime 0, poenial (4..8) defines he so-called Liouville model known in quanum physics. The corresponding closed-form expression for arihmeic Asian opions involves Bessel funcions (see Geman and Yor (993) and Geman and Eydeland (995)). 4.. FLOATING BARRIER OPTIONS Our second example are floaing barrier opions. This is an ineresing example of a wo-asse pah-dependen opion. Barrier opions have increasingly gained populariy over he recen years. A wide variey of barrier opions are currenly raded over he couner. Closed-form pricing formulas for barrier opions can be readily derived by employing he mehod of images. In addiion o he original eigh ypes of barrier opions priced by Rubinsein and Reiner, 99, double-barrier (Kuniomo and Ikeda, 99), parial barrier (Heynen and Ka, 994a; Zhang, 995a) and ouside barrier opions (Heynen and Ka, 994b; Rich, 996; Zhang, 995b) were sudied recenly. A key observaion is ha due o he reflecion principle he zero-drif ransiion probabiliy densiy for down-and-ou opions wih barrier B (Brownian moion wih absorbing barrier a he level B) is given by he difference of wo normal densiies (Meron, 973; Rubinsein and Reiner, 99) (b := ln B): K B (x T ;T jx; ) = p exp (, ) ( (x, y), exp, )! (x + y, b) : (4..) Floaing barrier opions (FBOs) areopionsonheunderlyingpayoff asse wih he barrier proporional o he price of he second barrier asse. To illusrae, consider a call on he underlying S. In a sandard down-and-ou call, he barrier is se a some consan pre-specified price level B (ou-srike) a he conrac incepion. The opion is exinguished (knocked ou) as soon as he barrier is hi. For a floaing down-and-ou call, he barrier B is se o be proporional o he price of he second asse S, B = S,where is a specified consan. Thus, a floaing knock-ou conrac is in effec as long as he price of he underlying payoff asse says above he price of he barrier asse imes ;S > S, and is exinguished as soon as S his he floaing barrier S.

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 55 Jus as sandard barrier opions, FBOs can be used o reduce premium paymens and as building blocks in more complex ransacions. For example, suppose a US-based invesor wishes o purchase a call on a foreign currency S, say Swiss Franc. Furhermore, suppose he holds a view ha anoher foreign currency S, which is correlaed wih S wih he correlaion coefficien, say Deusche Mark, is going o say below S = during he opion s lifeime, where is a given fixed hreshold cross-currency exchange rae. Then he may elec o add a floaing knockou provision, S > S, o he call o reduce his premium paymen. Similarly, S and S can be wo correlaed equiy indexes or a shor and long ineres rae. In he laer case, he opion can be srucured so ha i will knock ou if he yield curve invers and shor rae exceeds he long rae during he opion s life. The pricing of FBOs is somewha similar o quano opions (Babel and Eisenberg, 993; Derman, Karasinski and Wecker, 990; Reiner, 99). We assume we live in he Black Scholes world wih wo risky asses. The risk-neural price processes for hese wo asses are ds S = m d + dz ; ds S = m d + dz ; (4..) wih consan risk-neural drifs and volailiies. The dz and dz are wo sandard Wiener processes correlaed wih he correlaion coefficien.ifs and S are wo foreign currencies, hen he risk-neural drifs are m = r, r ; m = r, r ; where r;r and r are domesic and wo foreign risk-free raes, respecively. A floaing barrier call is defined by is payoff a expiraion fs ( 0 )>S ( 0 ); 0 T g Max(S T, K; 0); (4..3) where fs ( 0 )>S ( 0 ); 0 T g is he indicaor funcional on price pahs fs ( 0 ), S ( 0 ), 0 T g ha is equal o one if S ( 0 ) is greaer han S ( 0 );S ( 0 ) > S ( 0 ), a all imes 0 during he opion s life, and zero oherwise. To price hese opions, firs inroduce new variables x = ln S and x = ln S : dx = d + dz ; = m, ; (4..4a) dx = d + dz ; = m, : (4..4b) Now le us inroduce a cross-currency exchange rae S 3 = S =S : (4..5) Is logarihm x 3 = ln S 3 ;x 3 =x,x ;follows a process dx 3 = 3 d + 3 dz 3 ; (4..6)

56 VADIM LINETSKY wih drif rae and volailiy 3 =, ; 3 = q +, ; (4..7) and dz 3 is a sandard Wiener process correlaed wih he process dz wih he correlaion coefficien 0 0 =, q +, : (4..8) In he variables S and S 3, he problem reduces o pricing an ouside barrier opion wih he payoff asse S and he barrier asse S 3 wih he consan fixed barrier level and he payoff fs3 ( 0 )>; 0 T g Max(S T, K; 0): (4..9) Ouside barrier opions were sudied by Heynen and Ka (994), Zhang (995b) and Rich (996). A wo-asse Lagrangian for he wo-dimensional process is given by L = " _x, (, 0 ) _x3, # 3 +, 0 (_x, )( _x 3, 3 ) : (4..0) 3 3 If x and x 3 were uncorrelaed, he Lagrangian would reduce o he sum of wo independen Lagrangians L and L. The correlaion erm makes he problem more ineresing. The acion funcional can be re-wrien in he form Z T A = L d 0 = A 0 +, (x T, x ), 3 (x 3T, x 3 ); (4..) where Z T A 0 = L 0 d 0 ; L 0 = " # _x (, 0 ) + _x 3 3, 0 _x _x 3 ; (4..) 3 and = " # (, 0 ) + 3 3, 0 3 ; (4..3a) 3

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 57 = (, 0 ), 0 3 3 3 = 3 (, 3, 0 3 0 ) Now he presen value of a floaing barrier call C (S ;S ;)is given by Z Z e,r (e x T, K) e (x T,x )+ 3 (x 3T,x 3 ), ln K ln ; : (4..3b) K (x T x 3T ;T jx x 3 ;) dx 3T dx T : (4..4) A ransiion probabiliy densiy K is given by he wo-asse pah inegral of he ype (.68) over all pahs fx ( 0 );x 3 ( 0 ); 0 Tgsuch ha fx 3 ( 0 ) > ln ; 0 Tg. The pah inegraion measure is defined by he acion funcional (4..). I is calculaed by inroducing new uncorrelaed variables y = x, 0 x 3 ; y = x 3 ; (4..5) 3 so ha he Lagrangian L 0 reduces o he sum of wo independen erms _y L 0 = (, 0 ) + _y 3 : (4..6) Now, he pah inegral facorizes ino a produc of wo independen facors which yield a normal densiy for he variable y and a down-and-ou densiy of he form (4..) for he barrier variable y : K (x T x 3T ;T jx x 3 ;) =K(y T ;T jy ;)K (y T ;T jy ;) ( ) = p 3, exp, ( 3(x T, x ), 0 (x 3T, x 3 )) 0 3 (, 0 ) ( ) ( )! exp, (x 3T, x 3 ) 3, exp, (x 3T + x 3, ln) 3 : (4..7) Subsiuing his densiy back ino Equaion (4..4) and simplifying he inegrals, we arrive a he pricing formula for he floaing barrier call (recall ha S 3 = S =S, and 3 ; 3 and 0 are given by Equaions (4..7 8)): C (S ;S ;) = e,r S N[d ;d 4 ; 0 ],,e,r K S N[d6 ;d 8 ; 0 ] S N[d ;d 3 ;], 0 S N[d5 ;d 7 ; 0 ] S ; (4..8)

58 VADIM LINETSKY where we have inroduced he following noaions ln S K + p d = p ; d = d + ; d 3 = ln S S + 3 3 p ; d 4 = d 3 + 0 p ; S S 0 ln p d 5 = d + p ; d 6 = d 5 + ; 3 S S ln d 7 = d 3 + p ; d 8 = d 7 + 0 p ; 3 (4..9a) (4..9b) = 3 3 ; = + 0 ; (4..9c) 3 and N [a; b; ] is he sandard bivariae cumulaive normal disribuion funcion N [a; b; ] = p, Z a, Z b, exp, (, ) x + y, xy dy dx: (4..0) 4.3. LADDER-LIKE BARRIERS Our nex example illusraes he use of he Chapman Kolmogorov semigroup propery. Consider a down-and-ou call wih a ime-dependen ladder-like barrier B( 0 ); 0 T,definedby ( B B( 0 if 0 < )= B if 0 T ; (4.3.) and we assume ha B <B. A knock-ou provision of his ype can be included ino a long-erm opion or warran if he underlying is expeced o rise during he life of he conrac. The presen value of his pah-dependen opion a ime T is given by C B ;B (S; ) Z = e,r (e x T,K) e (= )(x T,x),( = ) K B ;B (x T ;T jx; ) dx T :(4.3.) ln K

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 59 The densiy K B ;B is obained from he Chapman Kolmogorov semigroup propery (.3) by convoluion of wo sandard down-and-ou densiies K B and K B of he form (.4.) for wo ime inervals 0 < and 0 T (he inegraion on x is from b o since K B is equal o zero for x b he conrac is already exinguished) Z K B ;B (x T ;T jx; ) = K B (x T ;T jx ; )K B (x ; jx; ) dx ; (4.3.3) b where we inroduced he following noaions =, ; = T, ; b = ln B ; b = ln B : (4.3.4) Subsiuing his back ino Equaion (4.3.), C B ;B (S; ) = Z e,r p exp (, exp, exp + exp ln K dx T Z b ), (x T, x ), (x, x) ( ( ( dx (e x T, K) e (= )(x T,x),( = ) ), (x T, x ), (x + x, b ) ), (x T + x, b ), (x, x), (x T + x, b ), (x + x, b ) )! : (4.3.5) Simplifying he inegrals, we arrive a he pricing formula for he down-and-ou call wih ladder-like barrier (4.3.): + N[d 6 ;d 8 ; C B ;B (S; ) = S N [d ;d 4 ;], B S, + B S + N[d 0 ;,d 8 ;,] B +! N [d ;,d 4 ;,] B

60 VADIM LINETSKY,e,r K B N[d ;d 3 ;], S N[d 5 ;d 7 ;] B, N[d 9 ;,d 7 ;,] S + B B N[d ;,d 3 ;,] where we have inroduced he following noaions = r ; = = r, ; ln S + K d = p ; d = d + p ; ln S B + d 3 = p ; d 4 = d 3 + p ; ln d 5 = d 7 = ln d 9 = B SK ln B S ln d = + p ; d 6 = d 5 + p ; + p ; d 8 = d 7 + p ; B SK + p ; d 0 = d 9 + p ; B S B K + p ; d = d 3 + p ; ; (4.3.6) (4.3.7a) (4.3.7b) (4.3.7c) (4.3.7d) If we se B = 0, his formula reduces o he pricing formula for parial barrier opions of Heynen and Ka (994). Seing B = B, i collapses o he sandard formula for he down-and-ou call. Using he same procedure one can obain pricing formulas for ladder-like barriers wih any finie number of seps hrough muli-variae normal probabiliies. 5. Conclusion In his paper we presened a brief overview of he pah inegral approach o opions pricing. The pah inegral formalism consiues a convenien and inuiive language for sochasic modeling in finance. I naurally brings ogeher probabiliy-based

THE PATH INTEGRAL APPROACH TO FINANCIAL MODELING AND OPTIONS PRICING 6 and PDE-based mehodologies and is especially useful for obaining closed-form soluions (when available) and analyical approximaions for pah-dependen problems. I also offers an ineresing numerical framework ha may yield some compuaional advanages for muli-dimensional models wih general parameers, such as muli-facor erm srucure models, as well as pah-dependen problems. In paricular, in Linesky (996) we apply he mehodology developed here o derived closed-form pricing formulas for a class of pah-dependen derivaives coningen on occupaion imes. To conclude, le us quoe Barry Simon (979): In par, he poin of funcional inegraion is a less cumbersome noaion, bu here is a larger poin: like any successful language, is exisence ends o lead us o differen and very special ways of hinking. Acknowledgemens The auhor hanks John Birge, Jim Bodurha, Vladimir Finkelsein and Bob Savi for useful discussions and encouragemen, Jan Dash for providing copies of his papers, useful discussions and suggesing some of he examples included in he presen paper, Rahim Esmailzadeh for providing a copy of his paper and useful discussion, Larry Eisenberg and Manfred Gilli for inviaion o presen a he Second Inernaional Conference on Compuing in Economics and Finance, June 6 8, 996, Universiy of Geneva, Swizerland, and Eric Reiner for drawing his aenion o he papers by Dash and Esmailzadeh. References Babel, D.F. and Eisenberg, L.K. (993). Quaniy adjusing opions and forward conracs, Journal of Financial Engineering (), 89 6. Beaglehole, D. and Tenney, M. (99). General soluions of some ineres rae coningen claim pricing equaions, Journal of Fixed Income, 69 84. Beilis, A. and Dash, J. (989a, b). A Mulivariae Yield-Curve Lognormal Model, CNRSPreprinCPT- 89/PE.334; A Srongly Mean-Revering Yield Curve Model, CNRS Preprin CPT-89/PE.337. Birge, J. (995). Quasi-Mone Carlo Approaches o Opions Pricing, Universiy of Michigan Technical Repor. Black, F. and Scholes, M. (973). The pricing of opions and corporae liabiliies, Journal of Poliical Economy 8, 637 659. Chance, D. and Rich, D. (995). Asse swaps wih Asian-syle payoffs, Journal of Derivaives, Summer, 64 77. Cox, J. and Ross, S. (976). The valuaion of opions for alernaive sochasic processes, Journal of Financial Economics 3, 45 66. Creuz, M., Jacobs, L. and Rebbi, C. (983). Mone Carlo compuaions in laice gauge heories, Physics Repors 95, 03 8. Dash, J. (988). Pah Inegrals and Opions, Par I, CNRS Preprin CPT-88/PE.06. Dash, J. (989). Pah Inegrals and Opions, Par II, CNRS Preprin CPT-89/PE.333. Dash, J. (993). Pah Inegrals and Opions, Invied Talk, SIAM Annual Conference, July. Derman, E., Karasinski, P. and Wecker, J.S. (990). Undersanding Guaraneed Exchange Rae Conracs in Foreign Sock Invesmens, Goldman Sachs Repor, June. Dirich, W. and Reuer, M. (994). Classical and Quanum Dynamics: From Classical Pahs o Pah Inegrals, Springer-Verlag, Berlin.

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