A general decomposition formula for derivative prices in stochastic volatility models

Transcription

1 A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, Barcelona Absrac We see ha he price of an european call opion in a sochasic volailiy framework can be decomposed in he sum of four erms, which idenify he main feaures of he marke ha affec o opion prices: he expeced fuure volailiy, he correlaion beween he volailiy and he noise driving he sock prices, he marke price of volailiy risk and he difference of he expeced fuure volailiy a differen imes. We also sudy some applicaions of his decomposiion. Keywords: coninuous-ime opion pricing model, sochasic volailiy, Iô s formula, incomplee markes. JEL code: G3 Inroducion Ever since he work of Black and Scholes [BS] and Meron [M] opion pricing heory has moived a grea deal of aenion from reseachers of various disciplines. he original work of Black and Scholes assumes ha he sock prices X saisfy a sochasic differenial equaion of he form dx = X d σx dw, where and σ are consans W is a sandard Brownian moion. he parameer σ is called he volailiy of he model. I is widely recognized ha his model provides a less-han-perfec descripion of he real world. In paricular, he consan volailiy assumpion is clearly no rue from empirical sudies. If opion prices in he marke were conformable wih his model, all he Black-Scholes implied volailiies corresponding o he same asse would coincide wih he volailiy parameer σ of he underlaying asse. In realiy his is no he case, and he Black-Scholes implied volailiy

2 heavily depends on he calendar ime, he ime o mauriy and he moneyness of he opion. he naural exension of he Black-Scholes model ha has been pursued in he lieraure and in pracice is o modify he specificaion of volailiy o make i a sochasic process. In mos cases, σ is repaced by σ () =f (Y ), where f is a given funcion and Y saisfies a sochasic differenial equaion driven by anoher (maybe correlaed) Brownian moion. Some examples of modelling are: Hull and Whie ([HW]), Sein and Sein ([SS]) and Heson ([H]). I is difficul in general o choose a specific sochasic volailiy model and o develop closed forms for opion pricing in his framework. Neverheless, one feaure ha mos models seem o like is mean reversion. he volailiy ends o flucuae a a high level for a while, hen a a low level for a similar period, hen high again, and so on. aking ino accoun his fac, in [FPS] he auhors developed a mehod for approximae derivaive prices ha i is valid for fas mean revering volailiies. hey idenify he imporan groupings of marke parameers, which oherwise are no obvious, and hey urn ou ha esimaion of hese composies from marke daa is exremely efficien and sable. Furhermore, he meodology is robus and i does no assume a specific volailiy model. he basicideaisoworkinlargeimeinervals,wherewecanassumehahemean reversion is fas and hen he consan-volailiy Black-Scholes model (wih a correcion o accoun for random volailiy) is a good approximaion. he main goal of his paper is o prove a general decomposiion formula for derivaive prices in he sochasic volailiy framework. By means of Iô s formula we see ha he opion price of an european call can be decomposed as he sum of several facors depending on he basic properies of he marke:. he expeced value (a presen ime) of he mean-square ime average of he volailiy process unil he expiraion ime,. he correlaion beween he volailiy process and he Brownian moion W, 3. he marke price of volailiy risk, and 4. he difference beween he expeced value of he mean-square ime average of he volailiy a differen imes. his formula gives us a way o undersan he influence of each one of he above facors o he final opion price. As a paricular applicaion we deduce an approximaion opion pricing formula for he case of fas mean revering volailiies ha coincides wih he one presened in [FPS]. Preliminaries on opion pricing We will consider he following model for sock prices dx = X d σ X dw, [,] ()

3 where is a consan, W is an sandard Brownian moion and σ is a square inegrable process adaped o he same filraion as W (ha we will denoe by F ). An European call opion is a conrac ha gives is holder he righ, bu no he obligaion, o buy one uni of an underlying asse for a predeermined srike price K on he mauriy ime. he asse is assumed no o pay dividends and here are no ransacion coss. I follows ha he value of his conrac a ime,ispayoff, is given by he quaniy h (X )=(X K ). A ime < his conrac has a value, known as he derivaive price V, which will vary wih and he observed sock prices unil ime. In he nex subsecion we presen he general mehodology for opion pricing based in he Girsanov ransformaion. We will denoe by r he ineres rae, ha we will assume consan.. Pricing wih equivalen measures Suppose ha here exiss a probabiliy disribuion P equivalen o he original one P under which he discouned sock price process X =e r X is a maringale. I is well-known ha if we price an european call by he formula V = e r( ) E (X K ) F, () where E denoes de expecaion wih respec o P, here is no arbirage oporuniy. hus V is a possible price for his derivaive. Le us now consruc equivalen maringale measures. As he process X saisfies he equaion d X =( r) X d σ X dw we need, in order o absorb he drif erm of X in is maringale erm, o se Z W = W ( r) ds. σ s Onheoherhand,ifweassumehaσ () depend on a second independen Brownian moion Z, any ransformaion of he form Z = Z Z γ s ds, where γ s is a process such ha he inegral R γ sds is well defined, will no change he drif of X. By Girsanov s heorem we know ha, if ( r) σ s and γ s are adaped and bounded processes here exiss a probabiliy disribuion P equivalen o he original one such ha W and Z are independen sandard Brownian moions. Noice ha any allowable choice of γ leads hen o an equivalen maringale measure and o a differen no arbirage derivaive price. his process γ is called he risk premium facor or he marke price of volailiy risk. 3

4 Much research has invesigaed he range of possible prices in general seings. he approach ha we will follow here is he same as used in [FPS], where i is assumed ha he marke selecs a unique equivalen maringale measure under which derivaive conracs are priced. Noice ha he value of he marke s price of he volailiy risk γ can be seen only in derivaive prices, since γ does no feaure in he real world for he sock price.. Iô s formula he main echnical used in his paper is he following classical Iô s formula for semimaringale processes. We refer o [KS] for a more complee explanaion on maringale inegral calculus. heorem (Iô s formula) Consider A = A,..., A n ª, [,] a n-dimensional semimaringale. ha is, A i = A i B i M i,where for all i =,..,n, B i is a bounded-variaion process and M i is a maringale. hen, for all f C,,..., b (R n ) and for all [,] f, A,..., A n = f Z, A,..., A n f s, A s s,..., A n s ds nx Z f s, A s,..., A n nx Z s db i f s s, A i s,...,a n s dm i s i i= nx i,j= Z f i i s, A s,..., A n s i= d M i,m j s, (3) where he differenial dm i s is inerpreed in he Iô sense and M i,m j s denoes he covariaion process of M i and M j. Using he characerizaion of opion prices as condiional expecaions and he above Iô s formula we will develop in he nex secion he main resul of his paper. 3 he general decomposiion formula he main goal of his secion is o decompose he price of an european opion in he sum of several erms, which allows us o idenify he influence of he basic properies of he marke on he opion price. We will use he following noaion: R v := E σ s F ds. ha is, v will denoe he expeced value (a presen ime) of he mean-square ime average of he volailiy process unil he expiraion ime. 4

5 M := R E σ s F ds. Noice ha v = ³ M R σ sds. C BS (, x; σ) will denoe he Black-Scholes funcion wih consan volailiy equal o σ, curren sock price x, ime o mauriy, srikepricek adn ineres rae r. ha is, C BS (, x, σ) =xn (d ) Ke r( ) N (d ), where and d = log (x/k) r ± σ ( ), σ ( ) N (z) = π Z z e y / dy. L BS (σ) will denoe he Black-Scholes differenial operaor wih volailiy σ : L BS (σ) = σ x r x. I is well-known ha L BS (σ) C BS (, ; σ) =. We will assume also he following hypohesis: (H) For all [,],M = M Λ where, under P,M Λ is a bounded variaion process. is a maringale and Now we are in a posiion o prove he main resul of his paper. heorem Assume he model (), where σ = {σ s,s [,]} is an adaped and square inegrable process such ha hypohesis (H) holds. hen, for all [,] V = C BS (, X, ν )E ( e r(s ) X s e r(s ) X s P e r(s ) P (s, X s,v s ) dλ s (s, X s,v s ) σ s d hw, Mi s ) P (s, X s,v s ) d hm,mi F s (4) Proof. Noice ha C BS (,X, ν )=V. As e r V is a P maringale we can wrie e r V = E e r V F = E e r C BS (,X, ν ) F (5) 5

6 Le us consider he process e r C BS (, X, ν ). Applying Iô s formula (3)and aking ino accoun ha we deduce ha C BS σ (s, X s,v s ) = X s ( s)v s (s, X s,v s ) e r C BS (,X, ν ) = e r C BS (, X, ν ) e rs L BS (v s ) σ s vs X s e rs CBS (s, X s,v s ) σ s X s dw s e rs X s e rs X s C BS C BS (s, X s,v s ) ν s σ s ds (s, X s,v s ) dm s e rs (s, X s,v s ) dλ s e rs X s (s, X s,v s ) e rs X s from where i follows ha e r C BS (,X, ν ) = e r C BS (, X, ν ) e rs CBS e rs X s C BS (s, X s,v s ) (s, X s,v s ) σ s X s dw s (s, X s,v s ) dm s e rs (s, X s,v s ) dλ s e rs X s (s, X s,v s ) e rs X s C BS (s, X s,v s ) ds σ s X s d hw, M s i (s, X s,v s ) d hm,mi s, (6) σ s X s d hw, M s i d hm,mi s. (7) 6

7 hen, aking condiional expecaion wih respec o F in boh sides of (7) we deduce ha E e r C BS (,X, ν ) F = e r C BS (, X, ν ) ( E e rs (s, X s,v s ) dλ s e rs X s e rs X s (s, X s,v s ) σ s d hw, Mi s ) C BS (s, X s,v s ) d hm,mi F s. Now, muliplying by e r and aking ino accoun(5)he resul follows. 4 Applicaions he pourpose of his secion is o commen how he above decomposiion formula can be applied o esimae opion prices in differen frameworks. 4. Approximae opion pricing formulas in he fas meanrevering case Suppose ha he volailiy model can be wrien as σ = f (Y ), being Y an Ornsein-Uhlenbeck process of he form dy = α (m Y ) d c αdb, (8) where α >,mand c are consans and B is a Brownian moion. he coefficien α is called he rae of mean reversion and m is he long-rum mean level of Y. he drif erm pulls Y oward m, so we sould expec ha σ is pulled oward he mean value of f (Y ) wih respec o he long-run disribuion of Y. he soluion of (8) can be explicily wrien in erms of is saring value y as Y = m (y m) e α c α Z e α( s) db s, (9) from where we deduce ha Y is a Gaussian process such ha E (Y )=m (y m) e α,var(y )=c e α and, for all s>, Cov(Y,Y s )=c e α(s ) e α. Under is invarian disribuion, Y N m, c and Cov (Y,Y s )=c e α s. For every real funcion f, we will denoe by f he expecaion of f under 7

8 his invarian disribuion. We noice ha he decorrelaion is of exponenial ype, ha is, he process Y has shor-memory. From his propery we can deduce he following resul Lemma 3 Consider Y he Ornsein-Uhlenbeck process defined by (8). Assume ha he volailiy process is given by σ = f (Y ), where f Cb (R). hen ν f C (α ( )), for some posiive and F -measurable random variable C. Proof. Noice ha Y s = m (Y m) e α(s ) c α e α(s r) db r. () hen can wrie E σ s F f = E ½f m (Y m) e α(s ) c α e α(s r) db r f m c ¾ α e α(s r) db r F E f m c α e α(s r) db r f () As f Cb (R) we can wrie f m (Y m) e α(s ) c α f m c α e α(s r) db r e α(s r) db r c (Y m) e α(s ). () On he oher hand, E f m c α = E f m c α f m c α e α(s r) db r f e α(s r) db r e α(s r) db r C Z αe Ã! e α(s r) db r ce α(s ) π. (3) 8

9 Now, aking ino accoun (), () and (3) i follows ha ν f Z = E σ s F f ds and now he proof is complee. C C α ( ), e α(s ) ds Remark 4 he above lemma proves ha he erm C BS (, X, ν ) can be approximaed by C BS (, X, p hf i), being his approximaion of order (α ( )). his means ha if he rae of mean reversion and he ime inerval are big enough, his approximaion is good. he advanage is ha he conan p hf i can be easily esimaed from hisorical volailiies, wihou assuming an specific volailiy model. Once we have found an easy-o-evaluae approximaion o C BS (, X, ν ) he naural quesions are if he remaining erms have an imporan influence in he opion price, and if yes, if i is possible o find a good approximaion for hem. In he nex lemma we find an answer o he firs quesion. Lemma 5 Assume ha for all [,] he volailiy process is given by σ = f (Y ), where Y is an Ornsein-Uhlenbeck process defined by (8) and f is a real funcion in Cb (R) such ha he funcion f is bounded. Assume also ha, under P,B = B R γ sds for some adaped and a.s. bounded process γ = {γ s,s [,]}. hen Hypohesis (H) holds and, for all [,] Ã Z! E e r(s ) (s, X s,v s ) dλ s F cα, (4) E Ã e r(s ) X s X! C BS s (s, X s,v s ) σ s d hw, Mi F s cα (5) and E Ã e r(s ) X s! C BS (s, X s,v s ) d hm,mi F s cα. (6) Proof. In order o simplify he proof we will assume ha m = y =. Now he proof will be decomposed ino several seps. 9

10 Sep. Leusseefirs ha Hypohesis (H) holds. For all s>we can wrie Y s = c α Z = : Z (s, )c α which implies ha e α(s r) db r c α e α(s r) db r e α(s r) db r where = E f (Y s ) F ds Z R f (x) g Z(s,), c e α(s ) (x) dx ds Ã! g m,σ (x) := σ π exp (x m) σ Z f (Y s ) ds, By classical Iô s formula (3) i is easy o deduce hen ha = = = E f (Y s ) F ds E f (Y s ) ds c α g Z(s,r), e α(s r)(x)dx E f (Y s ) ds 4c α g Z(s,r), e α(s r)(x)dx E f (Y s ) ds 4c α Z Ã Z r R e α(s r) ds db r Ã Z e α(s r) Z f (x)(x Z (s, r)) r R (ff )(x) ds db r Ã Z! e α(s r) E (((ff )(Y s ) F r ) ds B r, Z r which gives us ha Hypohesis (H) holds wih M = E f (Y s ) ds 4c Z Ã Z! α e α(s r) E (((ff )(Y s ) F r ) ds r B r and Λ =4c α Z Ã Z! e α(s r) E (((ff )(Y s ) F r ) ds γ (r) dr. r

11 Sep. Le us prove inequaliy (4). he proof of (5) and (6) would be similar. I is easy o check ha x C BS (s, x, y) is a bounded funcion. hen Ã E e r(s ) (s, X s,v s ) dλ s! F Ã Z! ce dλ s F = C Ã Z Ã Z!! αe e α(s r) E (((ff )(Y s ) F r ) ds γ (r) dr r F C Z Ã Z! α e α(s r) ds dr r Cα, and now he proof is complee. he above lemma shows ha C BS (, X, ν ) (and hen C BS (, X, p hf i)) is an approximaion of order α / for he opion price. he nex proposion will show us how o find an approximaion of order α. Proposiion 6 Assume ha for all [,] he volailiy process is given by σ = f (Y ), where Y is an Ornsein-Uhlenbeck process defined by (8). Assume also ha, under P,B = B R h (Y s) ds, for some bounded funcion h. hen here exiss wo consans V and V 3 such ha p C BS (, X, hf i) ( ) ½V X C BS p (, X, hf i) V 3 X C BS (, X, f )X 3 3 C ³ BS 3, X, p i ¾ hf (7) is an approximaion of order α for he opion price V. Proof. By going on he developemen of Formula (4) we can wrie V = C BS (, X, ν ) X X C BS (, X,v ) E Ã X (, X,v ) dλ s F! E Ã σ s d hw, Mi s F! R, (8)

12 where R : = E ( X e r(s ) e r(s ) X s X e r(s ) X s (, X,v ) (s, X s,v s ) X (s, X s,v s ) σ s d hw, Mi s (s, X s,v s ) C BS (, X,v ) dλ s ) d hm,mi F s.(9) Applying again Iô s formula and following he same seps as in he proof of Lemma5wecanseehaR C α. On he oher hand, Expression (8) can be wrien as C BS (, X, ν ) ( ( ) X X 3 C BS 3 C BS 3 Ã C BS (, X,v ) E Ã (, X,v ) E Ã (, X,v ) E X dλ s F! σ s d hw, Mi s F!!) σ s d hw, Mi F s. () As γ (r) =h (Y r ), using he same kind of argumens as in he proof of Lemma 3wecanseeha Ã Z! E dλ F s V C (α ( )) and Ã Z! E σ s d hw, Mi F s V C 3 (α ( )). his allows us o complee he proof. Remark 7 his approximaion formula coincides wih he one obained by Fouque, Papanicolau and Sircar in [FPS]. he consans V and V 3 can be callibraed from implied volailiy, as i is sudied in [FPS]. Remark 8 A naural quesion is: Why no proceed wih he above asympoic expansion, obaining more accurae approximaions? he answer is he following: C BS (, X,σ) approximaes C BS (, X, ν ) up o he order α, and i is necessary o fix aspecific volailiy model o find beer esimaes for his erm. ha is. i is no possible o obain higher-order approximaion formulas wihou choosing a concree volailiy model.

13 Remark 9 In order o capure he long-memory properies of he volailiy observed in some financial markes, some auhors have proposed o exend he Black-Scholes model o he case where he volailiy is modelled as a fracional Ornsein-Uhlenbeck process driven by a fbm (see for example [CR] and [Hu]). I is inuiive ha in his case he approximaion formula obained in he las secion is no going o be as accuraed as in he shor-memory case, because here he mean reversion is no so fas and he pricing formula will depend no only on he presen sock and volailiy levels bu also in all he pas. I becomes ineresing o sudy how Formula (4) can be used in his case. his is he subjec ofhework[a]. 5 Bibliography References [A] E. Alòs: Approximaion opion pricing formulas for long-memory sochasic volailiy models. Work in progresss. [BS] Black, F. and Scholes, M.: he pricing of opions and corporae liabiliies. Journal of Poliical Economy 3, (973). [CR] Come, F. and Renaul, E.: Long-memory in coninuous-ime sochasic volailiy models. Mahemaical Finance 8, 9-33 (998). [FPS] Fouque J-P., Papanicolau G. and Sircar, K. R.:Derivaives in Finanial markes wih Sochasic Volailiy. Cambridge (). [H] [Hu] Heson, S. L. : A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions. he Review of Finanial Sudies, 6, (993). Hu, Y.: Opion pricing in a marke where he volailiy is driven by fracional Brownian moions. Preprin. [HW] Hull J. C. and Whie, A.: he pricing of opions on asses wih sochasic volailiies. Journal of Finance 4, 8-3 (987). [KS] [M] [SS] Karazas, I. and Shreve, S. E. Brownian Moion and Sochasic Calculus. Springer-Verlag (99). Meron, R. C.: heory of raional opion pricing. Bell.J.Econom.and Managemen Sci (973) Sein, E. M. and Sein, J. C.: Sock price disribuions wih sochasic volailiy: an analyic approach. he Review of Finanial Sudies, 4, (99). 3