The Investigation of the Mean Reversion Model Containing the G-Brownian Motion

Transcription

1 Applied Mahemaical Sciences, Vol. 13, 219, no. 3, HIKARI Ld, hps://doi.org/ /ams he Invesigaion of he Mean Reversion Model Conaining he G-Brownian Moion Zixin Yuan Dep. of Accounabiliy and Financial Managemen Business School King s College London WC2R2LS, UK Weipeng Yuan School of Economic Mahemaics Souhwesern Universiy of Finance and Economics Chengdu 61113, P.R. China Corresponding auhor Jixi Li he School of Accouning Jinjiang College of Sichuan Universiy Chengdu 6286, P.R. China Copyrigh c 219 Zixin Yuan, Weipeng Yuan and Jixi Li. his aricle is disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac Liquidiy financial markes, whose risks are caused by uncerain volailiies, are invesigaed. We suppose ha he price process of risky asse saisfies a mean reversion model which conains a G-Brownian moion insead of he classical Brownian moion. Under he assumpion of no arbirage, employing he concep of arbirage and he properies of G-expecaion, an inerval of no-arbirage price for he general European coningen claims is deduced. Mahemaics Subjec Classificaion: A69

2 126 Zixin Yuan, Weipeng Yuan and Jixi Li Keywords: Volailiy uncerainy, Mean reversion model, G-Brownian moion, Pricing of coningen claims 1 Inroducion In a real sock marke, invesors rade wih liquidiy risks caused by uncerain volailiies which play an imporan role in financial markes. Many researchers in he financial fields use muliple priors o discuss decisions under volailiy uncerainy. Gilboa and Schmeider 1] inroduce he decision heoreical seing of muliple priors. Maccheroni e al. 2] generalize muliple priors o preferences. Vorbrink 3] discusses a paricular example o illusrae an uncerain volailiy model. We consider he European coningen claim, whose characerisic is volailiy uncerainy. he price process of underlying asse is supposed o follow a mean reversion model. Mos exising resuls are obained by using daa from developed markes o es for mean reversion. Poerba and Summers 4] examine he evidence on he presence of mean reversion in sock prices. Chaudhuri and Wu 5] invesigae ha sock-price indexes of emerging markes are characerized by mean reversion processes. Akarim and Sevim 6] idenify he bes porfolio invesmen sraegy on he validiy of he mean reversion model. Relaxing he deerminisic marke liquidiy process o allow a mean reversion sochasic process, Feng e al. 7] develop a liquidiy-adjused opion pricing model. In our liquidiy financial markes wih volailiy uncerainy, he wealh is invesed in risk-free asse and risky asse. Denoe S as he underlying asse price a ime. he underlying asse price is assumed o follow a mean reversion model. Denoe he canonical process W = (W ) as a G-Brownian moion wih regard o a sublinear expecaion E G, called G-expecaion (see 8] for is consrucion). Under probabiliy measure, he model is wrien in he form ds = r(α S )d + σ dw, S = x >. (1) We aim o solve sup P P E P (B V ) and supe P ( B V ), (2) where V denoes he payoff of a coningen claims a mauriy, and B is a discouning. P is he probabiliy measure. We suppose ha he sock price process saisfies a mean reversion model. Namely, equaion (1) conains a G-Brownian moion raher han a classical Brownian moion. equaion (1) is differen from he geomeric G-Brownian moion used in 3]. In our model, he volailiy of S is a variable depending on, while he volailiy of S is consan in 3]. We ake advanage of he G- framework and he concep of arbirage o obain he inerval of no-arbirage, P P

3 he invesigaion of he mean reversion model 127 which is differen from ha in 6], in which he mean reversion model is employed o idenify he bes porfolio invesmen sraegy. he res of he paper is organized as follows. Secion 2 provides he mahemaical seing and he liquidiy financial markes. In Secion 3, he inerval of no-arbirage is derived. Secion 4 presens our main resuls. Conclusions are drawn a he end of his paper. 2 Model seings In he following liquidiy financial marke M, here are wo asses, a riskfree asse and a single risky asse. hey are raded coninuously over, ]. Assume ha risk-free asse is a bond wih price B a ime and is ineres rae is r. So he price process of he risk-free asse B is assumed o saisfy he following formula db = rb d, B = 1. (3) Assume ha risky asse is a sock whose price process S follows he mean reversion model: ds = r(α S )d + σ dw, S = x >. (4) In equaions (3) and (4), he consan r represens he riskless ineres rae. In (4), r is he mean reversion speed of S in he liquidiy financial marke. α is he mean level of S. σ is he level of volailiy of S. W = (W ) denoes he canonical process which is a G-Brownian moion under E G or P, wih parameers σ > σ >. In (4), he sock price process S is an elemen of M 2 G (, ) = H2 G (, ) which is a basic assumpion in marke M. We impose he condiion H :=B 1 θ + γ S = θ B 1 + γ S + θ u db 1 u + γ u ds u C, q.s., where H represens he value of rading sraegy a ime and C is a cumulaive consumpion process (see 9]). he meaning of equaion (5) is ha, saring wih an iniial amoun B 1 θ + γ S of wealh, all changes in wealh are due o capial gains (appreciaion of socks, and ineres from he bond), minus he consumed amoun. Remark 2.1. A porfolio process π represens proporions of a wealh X which is invesed in he sock. If we define (5) γ := X π S, θ := X B (1 π ),,

4 128 Zixin Yuan, Weipeng Yuan and Jixi Li hen X = H. As long as π consiues a porfolio process wih corresponding wealh process X, he (θ, γ) is a rading sraegy in he sense of equaion (5). Definiion 2.2. For a given iniial capial X, a porfolio process π, and a cumulaive consumpion process C, consider wealh equaion dx = X (1 π )B db 1 + X π ds S = X rd + X π σ dw dc dc wih iniial wealh X, which is equivalen o B X = X B u dc u + B u X u π u σ u dw u,. (6) If equaion (6) has a unique soluion X = (X ) := X X,π,C, hen i is called he wealh process corresponding o he riple (X, π, C). 3 A No Arbirage Inerval in he European coningen claim heorem 3.1. For he financial marke (M, V ), he following ideniies hold: v up = E G (B V ), v low = E G ( B V ). Proof. Le us begin wih he ideniy v up = E G (B V ). As seen in he proof of 1], for any X U, i has X E G (B V ). herefore, v up = inf{x X U} E G (B V ). o show he opposie inequaliy, we need o define he G-maringale M by M := E G (B V F ),. By he maringale represenaion heorem in 11], we know ha here exiss z H 1 G (, ) and a coninuous, increasing processes Γ = (Γ ) wih Γ L 1 G (Ω ) such ha for any M = E G (B V ) + z sdw s Γ q.s. For any, we se X = E G (B V ), X π σ = z B 1 HG 1 dγ s L 1 G (Ω ). hen he wealh process X X,π,C saisfies C = B 1 s B X X,π,C = X + B s X X,π,C s π s σ s dw s B s dc s = M. (, ), and As X X,π,C = B 1 M = V quasi-surely, we have X = E G (B V ) U. Due o he definiion of U, we conclude v up E G (B V ).

5 he invesigaion of he mean reversion model 129 he proof for he second ideniy is analogous. Employing he proof in 1], we have X E G ( B V ) for any X L. Hence v low E G ( B V ). In order o obain v low E G ( B V ), we define a G-maringale M by M = E G ( B V F ),. Using he maringale represenaion heorem in 11], here exis z HG 1 (, ), and a coninuous, increasing process Γ = (Γ ) wih Γ L 1 G (Ω ) such ha M = E G ( B V ) + z s dw s Γ q.s. For any, le and C = X = E G ( B V ), B 1 s X π σ = z B 1 H 1 G(, ) dγ s L 1 G (Ω ). hen he wealh process X X,π,C saisfies B X X,π,C = X + B s X X,π,C s π s σ s dw s B s dc s = M, where C obeys he condiion of a cumulaive consumpion process due o he properies of Γ (see 9]). Moreover, for any, we have Since X X,π,C B X X,π,C = E G ( B V F ) E G ( V F ). = B 1 M = V q.s., i has X = E G ( B V ) L. Owing o he definiion of L, we obain v low E G ( B V ). he proof is compleed. 4 he Markovian case Considering he liquidiy financial marke M, we have he form V = Φ(S ) for Lipschiz funcion Φ : R R. A nonlinear Feynman-Kac formula which is esablished by Peng 8] is employed o solve our issue. Le us rewrie he dynamics of S in Equaion 4 as ds,x u = r(α Su,x )du + σ u Su,x dw u, u, ], S,x = x >. A ime, ], we denoe ha he lower and upper arbirage prices are vlow (x) and v up(x), respecively. Using he variable x represens he sock price S a a considered ime. ha is, S = x. he following conclusion is uilized o esablish he connecion of vlow (x) and vup(x).

6 13 Zixin Yuan, Weipeng Yuan and Jixi Li Lemma 4.1. Given a European coningen claim V = Φ(S ), is upper arbirage price v up(x) is given by u(, x), where u :, ] R + R is he unique soluion of he PDE u + rx x u + G(σ 2 x 2 xx u) = ru, u(, x) = Φ(x). (7) A precise represenaion for he corresponding rading sraegy in he sock and he cumulaive consumpion process is given by γ = x u(, S ),, ], C = σ s 2 S s 2 xx u(s, S s )d W s σ s 2 S s 2 G( xx u(s, S s ))ds,, ]. he process of proving Lemma 4.1 is given in 1]. On he basis of he resul of Lemma 4.1, we know ha he unique soluion of equaion (7) is he funcion u(, x) = vup(x) and u(, x) = vlow (x). If Φ is a convex or concave funcion, equaion (7) will become very simple. heorem If Φ is concave, hen u(, ) is convex for any. 2. If Φ is convex, hen u(, ) is concave for any. Analogously, if Φ is convex, hen u(, ) is concave for any. If Φ is concave, hen u(, ) is convex for any. Proof. We only need o ake ino accoun he funcion of he upper arbirage price which is deermined by u(, x) = E G ( Φ(S,x )B ) = v up (x), (, x), ] R +.

7 he invesigaion of he mean reversion model 131 Le Φ be convex,, ], and x, y R +. For any λ, 1], i has u(, λx + (1 λ)y),λx+(1 λ)y =E G Φ(S r( )e )] =E G Φ ( (λx + (1 λ)y)e r( ) + α(1 e r( ) ) ) ] + e r( ) σ s e rs dw s e r( ) =E G Φ ( λ(xe r( ) + α(1 e r( ) ) + e r( ) σ s e rs dw s ) + (1 λ)(ye r( ) + α(1 e r( ) ) + e r( ) σ s e rs dw s ) ) ] e r( ) E G λφ ( xe r( ) + α(1 e r( ) ) + e r( ) + (1 λ)φ ( ye r( ) + α(1 e r( ) ) + e r( ) E G λφ ( xe r( ) + α(1 e r( ) ) + e r( ) + E G (1 λ)φ ( ye r( ) + α(1 e r( ) ) ) ] + e r( ) σ s e rs dw s e r( ) =λe G Φ(S,x )] )e r( + (1 λ)e G Φ(S,y =λu(, x) + (1 λ)u(, y), )e r( )] σ s e rs dw s ) ] σ s e rs r( ) dw s ) e ) ] σ s e rs dw s e r( ) where we have used he convexiy of Φ, he monooniciy of E G and he sublineariy of E G in he second inequaliy. Hence, u(, ) is convex for all, ]. Le Φ be concave. For any (, x), ] R +, we se where,x d S s = r h(, x) := E P,x Φ( S )] )e r(,,x,x S s ds + σ S s dw s, s, ], S,x = x. E P is linear which means ha h(, ) is concave for any, ]. herefore, h solves equaion (7). he uniqueness is used o conclude h = u. Hence, u(, ) is concave for any, ]. 5 Conclusion In our paper, he mean reversion model is employed o describe he sock price process. We resric our sudy in liquidiy financial marke which feaures

8 132 Zixin Yuan, Weipeng Yuan and Jixi Li volailiy uncerainy. he srucure of a G-Brownian moion is employed in our mahemaical seing. We find an inerval of no-arbirage price for he general European coningen claims. he parial differenial equaions are applied o esablish he connecion of he lower and upper arbirage prices. References 1] I. Gilboa, D. Schmeidler, Maxmin expeced uiliy wih non-unique prior, Journal of Mahemaical Economics, 18 (1989), hps://doi.org/1.116/34-468(89) ] F. Maccheroni, M. Marinacci, A. Rusichini, Dynamic variaional preferences, Journal of Economic heory, 128 (26), hps://doi.org/1.116/j.je ] J. Vorbrink, Financial markes wih volailiy uncerainy, Journal of Mahemaical Economics, 53 (214), hps://doi.org/1.116/j.jmaeco ] J.M. Poerba, L.M. Summers, Mean reversion in sock prices: Evidenceand implicaions, Journal of Economic Finance, 22 (1988), hps://doi.org/1.116/34-45x(88) ] K. Chaudhuri, Y. Wu, Mean reversion in sock prices: evidence from emerging markes, Managerial Finance, 29 (23), hps://doi.org/1.118/ ] Y.D. Akarim, S. Sevim, he impac of mean reversion model on porfolio invesmen sraegies: Empirical evidence from emerging markes, Economic Modelling, 31 (213), hps://doi.org/1.116/j.econmod ] S. Feng, M. Hung, Y. Wang, Opion pricing wih sochasic liquidiy risk: heory and evidence, Journal of Financial Markes, 18 (214), hps://doi.org/1.116/j.finmar ] S. Peng, Nonlinear Expecaions and Sochasic Calculus under Uncerainy, Firs Ediion, Jinan, 21. 9] I. Karazas, S. Kou, On he pricing of coningen claims under consrains, Annals of Applied Probabiliy, 6 (1996), hps://doi.org/1.1214/aoap/

9 he invesigaion of he mean reversion model 133 1] W. Yuan, S. Lai, he CEV model and is applicaion o financial markes wih volailiy uncerainy, Journal of Compuaional and Applied Mahemaics, 344 (218), hps://doi.org/1.116/j.cam ] Y. Song, Properies of hiing imes for g-maringales and heir applicaions, Sochasic Processes and heir Applicaions, 121 (211), hps://doi.org/1.116/j.spa Received: January 17, 219; Published: January 3, 219