A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

Transcription

1 A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical properies of a generalized bivariae Ornsein-Uhlenbeck model for financial asses. Originally inroduced by Lo and Wang, his model possesses a sochasic drif erm which influences he saisical properies of he asse in he real observable world. Furhermore, we generalize he model wih respec o a ime-dependen bu sill non-random volailiy funcion. Alhough i is well-known, ha drif erms under weak regulariy condiions do no affec he behaviour of he asse in he risk-neural world and consequenly he Black-Scholes opion pricing formula holds rue, i makes sense o poin ou ha hese regulariy condiions are fulfilled in he presen model and ha opion pricing can be reaed in analogy o he Black-Scholes case. Keywords: generalized Ornsein-Uhlenbeck process, financial analysis, opion pricing, Black-Scholes formula, hedging MSC2 classificaion scheme numbers: 6G15, 6G44, 91B28

2 12 R. Krämer, M. Richer 1 Inroducion We consider he price P of a financial asse during he ime inerval [, T]. By p he logarihm of he asse price is denoed, p = ln P. The basis for he model which is analysed in his paper forms he Bivariae Trending Ornsein-Uhlenbeck model of Lo and Wang, inroduced in [3]. The logarihm of he asse price p is assumed o have a linear deerminisic rend µ. Then i is convenien o inroduce he process q := p µ, 1.1 and o consider he sochasic properies of he cenered derended log-price process q. Uncerainy is modelled by means of a complee filered probabiliy space Ω, F, P. Addiionally, we consider a filraion F [,T] saisfying he usual condiions see for insance [4]. The Bivariae Trending Ornsein-Uhlenbeck model of Lo and Wang assumes ha q saisfies he following pair of sochasic differenial equaions, dq = γq λx d + σ dw q dx = βx d + σ X dw X, 1.2 where γ, λ, β, σ >, µ R and σ X > are real-valued parameers, he iniial condiions q = c q, X = c X hold and W q and W X are correlaed Wiener processes wih correlaion coefficien κ, i. e. E W q W X = κ. As a moivaion for considering his model Lo and Wang argue ha empirical observaions have indicaed ha he reurns r τ P = ln P τ show cerain correlaion paerns, which means ha he classical Black-Scholes model is inappropriae for describing he price process of hese asses. For a deailed discussion and furher properies i is referred o [3]. In his paper, he aim consiss only in he descripion of he mahemaical properies, namely in he explici soluion of he defining sochasic differenial equaions and in he problem of pricing European call opions wrien on a corresponding asse. The process X which influences he sochasic drif componen of q is some underlying process, which may also be relevan for oher asses. I should be noed, ha if one is no able o observe i, one could also consider a scaled version ˆX := 1 σ X X. This process ˆX saisfies he sochasic differenial equaion d ˆX = β ˆX d + 1 dw X wih iniial condiion ˆX = ĉ X := 1 σ X c X. Thus, seing ˆλ = λσ X he process q could also be described by dq = γq ˆλ ˆX d + σ dw q d ˆX = β ˆX d + 1 dw X wih iniial condiions q = c q and ˆX = ĉ X. 1.3

3 A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses 13 However, in his paper we consider he model in he form 1.2. We resric our consideraions o he case of independen Wiener processes W q and W X, i.e. κ =. On he oher hand wih respec o several effecs in opion pricing we are ineresed in a more general behaviour of he asse prices wih respec o he risk neural measure as he consan volailiy coefficien σ would admi. Therefore we generalise his model inasmuch as we allow he volailiy σ o be ime-dependen bu sill non-random. Furhermore, in order o allow a scaling of he prices which plays he role of adjusing he moneary uni we inroduce an addiive consan a o q. Obviously, his leads o a muliplicaion of he asse prices by expa. Summarising, he model which is in he focus of his paper is described by Model 1.1 We assume ha he derended log-price process of a radable financial asse saisfies q := ln P µ a 1.4 dq = γq λx d + σ dw q dx = βx d + σ X dw X. 1.5 wih µ, a R, γ, λ, β and a ime-dependen, coninuous volailiy funcion σ wih σ >, T. The iniial values q = c q and X = c X are assumed o be sochasic variables wih finie second order momens, i.e., Ec 2 q < and Ec2 X <. Furhermore, we assume he vecor c q c X T, which conains he iniial values of he processes q and X, and he Wiener processes W q, W X o be muually independen. Using Iôs Lemma i can be easily shown ha under he assumpions of Model 1.1 he price process iself saisfies he sochasic differenial equaion dp = γ lnp + γ µ + γa + λ X + µ + σ2 P d + σp dw q, wih iniial condiion P = expa + c q. 2 Soluion of he sochasic differenial equaion As model 1.1 leads o a sysem of linear sochasic differenial equaions, he soluion can easily be derived. This will be done in his secion, we will prove he exisence and uniqueness of he soluions q and X, [, T] of 1.5. To do his, we combine he processes q and X ino an R 2 -dimensional random process Y and he independen Wiener processes W q and W X ino a 2-dimensional Wiener process W, i.e., we se Y := q X and W = W q W X. Workshop Sochasische Analysis

4 14 R. Krämer, M. Richer Thus, he sysem 1.5 aains he form γ λ dy = Y d + β σ σ X dw 2.1 wih iniial value Y = c := c q c X T, which is a linear sochasic equaion in he narrow sense cf. [1][p. 128 ff.]. Noe, ha he marix-valued funcion B : [, T] R 2 2, defined by σ B :=, σ X is measurable and bounded on [, T]. Thus, Theorem of [1] implies ha for every iniial value c here exiss a unique soluion Y of 2.1. Furhermore, Corollary of [1] saes ha his soluion is given by Y = e A c + where we have inroduced he noaion A := γ e sa Bs dw s, 2.2 λ β In order o compue he exponenial of he marix A i is necessary o consider he wo cases γ β and γ = β separaely. 1. For γ β he marix A can be diagonalised, i.e., here exiss an inverible marix P such ha D = P 1 A P is a diagonal marix. Indeed, he marix and i holds P = 1 λ γ β has he inverse P 1 = D = P 1 A P =. γ β Thus, e A can be compued by e A = P e D P 1 and we obain e A e γ λ = γ β e β e γ. e β Therefore, he soluion Y of 2.1 is given by e γ λ Y = γ β e β e γ c e β. 1 λ β γ 1 γ β

5 A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses 15 + γ s e λ γ β e β s e γ s σs dw s. e β s σ X Thus, he processes q and X are given by q =e γ c q + + λ e β e γ c X + λσ X γ β γ β σse γ s dw q s X =e β c X + σ X e β s dws X. [ e β s e γ s] dw X s 2. For he siuaion γ = β he marix A is a muliple of a 2 2 Jordan block. Thus, he exponenial can be easily compued and i holds e A e = γ λ e γ. e γ Therefore, in his siuaion he soluion Y of 2.1 is given by e Y = γ λe γ e c + γ s λ se γ s σs dw e γ e γ s s σ X Thus, he processes q and X are given by q =e γ c q + λe γ c X + λσ X se γ s dws X + X =e γ c X + σ X e γ s dws X. σs e γ s dw q s Clearly, if c = c q c X T is normally disribued or non-random he vecor process q X T is Gaussian. 3 Pricing of European Call Opions We consider now an European vanilla call opion C wih srike K and expiry T, i. e., C := max{p T K, }. We are ineresed in he fair price of his opion a he ime poin [, T]. As in he Black-Scholes model, besides he radable asse of Model 1.1 a bond, whose price process is given by B = expr shall exis. We inroduce he Black-Scholes funcion U BS as follows. Workshop Sochasische Analysis

6 16 R. Krämer, M. Richer Definiion 3.1 For parameers P >, K >, r, τ and s he Black-Scholes funcion is defined as PΦd U BS P, 1 Ke rτ Φd 2 if s > K, r, τ, s := 3.1 max P Ke rτ, if s = wih d 1 := ln P + rτ + s K 2, d 2 := d 1 s. s In 3.1 Φ denoes he disribuion funcion of he sandard normal disribuion. Furhermore we se S := σ 2 u du [, T]. I is well-known ha he opion price formula derived from he Black-Scholes model is unaffeced by he drif erm of he underlying asse. As long as he logarihm of he price process of he underlying saisfies he sochasic differenial equaion d lnp = µ d + σ dw, he fair price of a European call opion wih payoff max{p T K, } a mauriy T a a ime poin [, T] is given by C, P = U BS P, K, r, T, ST S. 3.2 While in he Black-Scholes model µ is assumed o be a consan i is well-known ha µ can be a sochasic process, depending on P iself as well as on oher sochasic influences, which fulfil mild regulariy condiions. In he remaining par of his secion we show ha in he considered Model 1.1 he fair opion price is indeed given by 3.2. In general, he iniial values q = c q and X = c X are assumed o be sochasic variables. For he sake of simpliciy here we resric o he case where c q and c X are deerminisic quaniies, which are chosen o be zero. I should be menioned, ha as X is in general no observable he assumpion abou X is a sligh resricion of he model. However, he following consideraions can be generalised sraighforwardly o he case of sochasic iniial condiions. Moreover, we concenrae o he fair opion price a ime poin =. The generalizaion o imes [, T] is also sraighforward. In a firs sep, we show ha here exiss an admissible self-financing sraegy duplicaing he call opion. The exisence of such a sraegy can be shown under very general assumpions, for insance as long as he asse price is modelled by dp = σp dw q + P dz

7 A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses 17 where σ is a coninuous funcion and Z is a coninuous random process of zero square variaion, possibly dependen on P, fulfilling weak regulariy condiions. For deails see [5]. For our Model 1.1 we consider he funcion C C [, T], C 1,2 [, T, defined by x Φ C, x := ln x K +rt TR +1 σ 2 2 s ds s TR maxx K, σ 2 s ds Ke rt Φ ln x K +rt TR 1 σ 2 2 s ds s TR σ 2 s ds < T = T = U BS x, K, r, T, ST S. We consider he rading sraegy consising a ime of shares of he asse and a := C x, P b := e r C, P P C x, P unis of he bond. Then he processes a and b possess coninuous realisaions, which implies T b d < a.s. and T a P 2 d < a.s. Concerning he wealh process V, P, defined by i yields V, P = a P + b expr V, P = P C x, P + C, P P C x, P = C, P. I can be shown by elemenary consideraions ha V, P holds. Obviously, he rading sraegy has a.s. he same final value V T, P T as he call opion which is o valuae. Applying Iô s Lemma we obain V, P =V, P C s, P s ds + C xx s, P s P 2 s σ2 s ds. C x s, P s dp s Workshop Sochasische Analysis

8 18 R. Krämer, M. Richer On he oher hand i can be derived easily ha C, x fulfils he Black-Scholes differenial equaion and consequenly C x2 σ 2 C xx + rxc x rc = V, P =V, P + + =V, P + C x s, P s dp s r Cs, P s P s C x s, P s ds a s dp s + r b s exprs ds, which proves ha he considered rading sraegy is self-financing. I is imporan o remark ha he hedging sraegy a, b [,T] is even adaped o he filraion F P he augmened filraion generaed by he process P. From his [,T] fac i follows ha in order o perform he hedging sraegy a he ime poin i is no necessary o know he value of X. By finding he hedging sraegy, he main work for calculaing he fair opion price is done. From easy non-arbirage argumens one usually concludes ha his fair price is he value of he hedging sraegy, i.e., he fair opion price a ime is equal o C, P, which would prove our asserion. However, a more careful consideraion has o ake ino accoun he fac ha even in he classical Black-Scholes model here are sill some pahological sraegies namely he so-called suicide sraegies which make hings complicaed. To be precise, in many siuaions here exiss self-financing admissible sraegies wih arbirary saring value V > and V T a.s. For his reason, he fair opion price of an aainable claim a ime = has o be defined as { c := inf x : here exiss an admissible self-financing duplicaion sraegy ã, b for he opion wih ã P + b } 3.3 = x. In order o check ha our sraegy a, b considered above leads o he correc opion price, i.e., c = C, P 3.4 i is sufficien o show ha is discouned wealh process e r V, P follows a maringale wih respec o a maringale measure Q equivalen o P i.e., a measure Q under which he discouned asse price process e r P follows a maringale. To prove he laer saemen le us assume for he momen ha we have found such a measure. Clearly, Definiion 3.3 implies c a P + b. On he oher hand i is easy o show ha he discouned wealh process of any admissible self-financing duplicaion sraegy ã, b is a non-negaive local maringale and consequenly a supermaringale. This leads o ã P + b E Q e rt ã T P T + b T e rt

9 A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses 19 = E Q e rt C = E Q e rt a T P T + b T e rt = a P + b for all admissible self-financing sraegies ã, b and herefore c a P + b, which leads o c = a P + b. I remains o show ha here exiss an equivalen maringale measure and ha he discouned rading sraegy inroduced a he beginning of his secion is a maringale wih respec o his measure. For his we define he sochasic process Z := exp α s dw s 1 2 wih α 2 s ds α := γp + γµ + γa + λx + µ + σ2 r σ If i is possible o show ha Z is a maringale, hen by Girsanovs Theorem and W q :=W q + α s ds W X :=W X form a wo-dimensional Wiener process wih respec o he measure Q defined by QA = E1 A Z T A F T. To check ha Z is a maringale i is sufficien o show ha he condiion sup E exp δα 2 < 3.6 T is fulfilled for some δ > c. f. [2][Chaper 6.2]. Then, using he Novikov condiion he maringale propery of Z can be proven. In [2][Chaper 6.2] i is also shown ha in case of Gaussian processes α wih sup T E α < and sup D 2 α < T Condiion 3.6 holds rue. From he consideraions of Secion 2 i is clear ha he process γ defined in 3.5 is such a Gaussian process. Under he measure Q he asse price has he dynamics Workshop Sochasische Analysis

10 11 R. Krämer, M. Richer which implies dp = rp d + σp d W q d e r P = σe r P d W q and clearly Q is a no he unique one maringale measure. Finally, i remains o show ha he discouned value process e r V is a hones Q- maringale. From he self-financing propery i follows easily d e r V = re r V d + e r dv = re r a P d re r b e r d + e r a dp + re r b e r d = a d e r P = a σe r P d W q, which shows ha e r V is a local Q-maringale. A coninuous non-negaive local maringale is always a supermaringale. Performing an elemenary calculaion which is he same as done during he calculaion of he classical Black-Scholes formula using he expecaion in he risk neural world one ges E Q e rt V T = V, which proves ha e r V is a hones maringale. Thus, he fair opion price a ime = is given by 3.4. The proof of his propery was he aim of his secion. References [1] Arnold, L.: Sochasic Differenial Equaions: Theory and Applicaions. Krieger Publishing Company, Malabar, [2] Lipser, R. S.; Shiryaev, A. N.: Saisics of Random Processes. Springer-Verlag, Berlin, Heidelberg, New York, 21. [3] Lo, A. W.; Wang, J.: Implemening Opion Pricing Models when asse reurns are predicable. The Journal of Finance 5, , [4] Proer, P.: Sochasic Inegraion and Differenial Equaions. Springer-Verlag, Berlin, Heidelberg, New York, 199. [5] Schoenmakers, J. G. M.; Kloeden, P. E.: Robus opion replicaion for a Black-Scholes model exended wih nondeerminisic rends. Journal of Applied Mahemaics and Sochasic Analysis 12:2, , 1999.