Available online a www.sciencedirec.com ScienceDirec Procedia Economics and Finance 8 ( 04 658 663 s Inernaional Conference 'Economic Scienific Research - Theoreical, Empirical and Pracical Approaches', ESPERA 03 Implied volailiy in black-scholes model wih GARCH volailiy Muhammad Sheraz a, *, Vasile Preda a,b a Faculy of Mahemaics and Compuer Science, Universiy of Buchares, Buchares, Romania b Naional Insiue of Economic Researcs, Calea 3 Sepebrie nr.3, Buchares, 0507, Romania, Romania Absrac The famous Black-Scholes opion pricing model is a mahemaical descripion of financial marke and derivaive invesmen insrumens. In his model volailiy is a consan funcion, where rading opion is indeed risky due o random componens such as volailiy. The noion of non-consan volailiy was inroduced in GARCH processes. Recenly a Black-Scholes model wih GARCH volailiy has been inroduced (Gong e al., 00.In his aricle we derive an implied volailiy formula for -Model wih GARCH volailiy. In his approach implied volailiy paerns are due o marke fricions and help us o suppor he evidence of fa-ailed reurn disribuions agains he dispued premise of lognormal reurns in Black-Scholes model (Black and Scholes, 973. 04 The Auhors. Published by by Elsevier B.V. Open access under CC BY-NC-ND license. Selecion and peer-review under responsibiliy of of he he Organizing Commiee of of ESPERA 03 03. eywords: Opion Pricing; Black-Scholes Model; GARCH Processes; Volailiy;. Inroducion Fischer Black and Myron Scholes published an opion valuaion formula in heir 973's aricle((black and Scholes, 973 ha oday is known as Black-Scholes model. The model has some resricions for example, a risk free ineres rae r (consan and a * Corresponding auhor. Tel.: +4-074870-58. E-mail address: muhammad_sheraz84@homail.com. -567 04 The Auhors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selecion and peer-review under responsibiliy of he Organizing Commiee of ESPERA 03 doi: 0.06/S-567(4004-5
Muhammad Sheraz and Vasile Preda / Procedia Economics and Finance 8 ( 04 658 663 659 consan volailiy σ (do no seem o be realisic.trading opion is risky due o possibly high random componens such as volailiy. The concep of non-consan volailiy has been inroduced by GARCH processes.the sudy of sock price models under hese processes is a new horizon in derivaive invesmen insrumens. Duan was he firs o provide a solid heoreical foundaion opion pricing in his framework (Duan, 99.Recenly a new exension of model ((Black and Scholes, 973 wih GARCH volailiy has been inroduced (Gong e al., 00.The volailiy measures, he variaion of price of financial insrumen over ime and implied volailiy can be derived from he marke price of a raded derivaive. In 986 he concep of implied volailiies was used for financial marke research(laane and Rendleman, 976. Taylor series approximaions have been frequenly followed in pricing opions. In Risk managemen paricularly firs and second order Taylor approximaions are crucial. Black-Scholes formula has been considered for Taylor approximaion for differen purposes (Buler and Schacher, 986, Laane and Rendleman, 976. In his aricle we consider he new model (Gong e al., 00.In Secion we provide fundamenal heory and ools. Secion 3 is consising of implied volailiy formula for - call opion of he model (Gong e al., 00 and we compare he formula wih original model (Black and Scholes, 973. Finally in secion 4 we presen some concluding remarks.. Preliminaries Le (,F, be he probabiliy space hen price of an asse S a ime is a Geomeric Brownian Moion (GBM. ds rs d S dw ( Here W is a sandard Brownian moion and is volailiy. We know ha according o Black and Scholes model (Black and Scholes, 973, A European call opion can be wrien as: r C S( d e ( d S log( ( r d, d d ( Where (. is a cumulaive disribuion funcion for sandardized normal random variable and T, S is he price of he asse, is a srike price, r is ineres rae and T is ime o expiry. Definiion (Black and Scholes, 973 If S is sock price, r is ineres rae (risk free, hen C is European call opion ha, gives is holder he righ, bu no he obligaion o buy he one uni of underlying asse for a predeermined price a he mauriy daet. Similarly a Pu opion P, gives is holder he righ, bu no he obligaion o sell he specified amoun of underlying asse for a predeermined price a he mauriy dae T. When variance of he log of sock reurns changes wih ime i.e. hen a new formula has recenly been presened (Gong e al., 00. The call opion for he model can be wrien as: rt C SE [ ( d ] e E [ ( d ] S log( rt (, ( d f d g d (3
660 Muhammad Sheraz and Vasile Preda / Procedia Economics and Finance 8 ( 04 658 663 Here, is a saionary GARCH process having mean and variance.opion pricing based on GARCH models has been sudied under he assumpion ha he innovaions are sandard normal (i.e. under normal GARCH. Definiion (Chrisian e al.,00 A process of he following form is called GARCH (p,q process p q i j j i j z z N i j, (0,, 0, 0, 0 (4 Proposiion If we consider equaion (3 hen we have following resuls. S S log( rt E( log( rt E( [ ( ] E d d E( E( S S log( rt E( log( rt E( E [ ( d ] d E( E( Then we have d d E( (5 Proof: The proof of he proposiion is immediae by using, d d. Definiion [8] Le C (, T be he marke price of sandard European Call opion wih srike price 0 and mauriy dae T a ime [0,T.The implied volailiy ( T, is hen defined as he value of he volailiy parameer which compare he marke price of he opion wih he price given by he formula. C (, T C (, S,, T, r, (, T (6 3. Implied Volailiy Formula in Black-Scholes Model wih GARCH Volailiy The Black-Scholes srucure relaes he price of an opion o he curren ime, sock price S, volailiy of he sock, he ineres rae r,he mauriy dae T and srike price. As we know he model inroduce ha volailiy is a consan funcion hroughou he life of an opion bu empirical research conflics wih he assumpion of he model.the implied volailiy by he marke price can be obained by invering he opion pricing formula. In his secion we use (Gong e al., 00, and obain implied volailiy, in addiion we use he same procedure for he original -model, and compare our resuled equaions. CASE-I: When and is a GARCH process. We know ha formula for call opion of new model wih GARCH volailiy process(gong e al., 00. rt CSE [ ( d ] e E [ ( d ]
Muhammad Sheraz and Vasile Preda / Procedia Economics and Finance 8 ( 04 658 663 66 We know he following expansion 3 5 x x Nx ( x... 6 40 (7 Using (7 for he call opion of Black-Scholes wih GARCH volailiy, also we follow he relaion dd E( and we suppose for simpliciy X e rt. C S ( d ( X d C S ( d X ( d E( ( S X S X X C ( d E( We wan o ge an equaion in erms of and we can simplify he above equaion by replacing d wih equivalen S expression. In addiion for simpliciy we suppose, ulog( rt C E( ( S X E( ( S X[ u ] XE( E( C E( ( S X E( u( S X ( S X E( XE( ( S X E( ( S X E( E( C u( S X 0 ( S X E( ( S X C E( u( S X 0 Here we have, ( S X equaion as follows:, ( S X C and ( us X, hen we may wrie he above E E( ( 0 (8 We suppose, E( x hen x E(,hen we may wrie equaion (8 as follows: x x 0 (9 The equaion (9 is a simple quadraic equaion and discriminae of he equaion can be wrien as 4
66 Muhammad Sheraz and Vasile Preda / Procedia Economics and Finance 8 ( 04 658 663 and roos of he equaion are : x. In addiion in he case of non-negaive roo of equaion (9 he signs of coefficiens, and are crucial. Furhermore if 0, 0, 0 hen discriminae, 0 always which implies here exis a leas one posiive real roo of he equaion (9. However he signs of coefficiens depend upon values of sock price, srike price. Here we sudy a special case when value of sock price S is equal o srike price i.e. S, hen opion is called a he money opion (ATM. If we consider r 0 hen in equaion (9 he coefficien 0 and we obained following equaion: x x 0 (0 Here he roos of he equaion (0 are: x0, x, where S and C. In oher words we can also say sum of he roos of equaion (9 is equal o he a he money opion. Example Consider he daa as used in(gong e al., 00. i.e. S 45.73, C 5.33635043, r 0, C hen a he money opion we have, x GARCH 0.49 S GARCH CASE-II: Using Black-Scholes assumpion of volailiy In his case we use he formula for call opion i.e. C and we know ha We know he following expansion r C S( d e ( d 3 5 x x Nx ( x... 6 40 Using (7 for he call opion of Black-Scholes wih GARCH volailiy, also we follow he relaion dd and we suppose for simpliciy r e X. C S ( d ( X d C S ( d X ( d ( S X S X X C ( d We wan o ge an equaion in erms of and we can simplify he above equaion by replacing d wih
Muhammad Sheraz and Vasile Preda / Procedia Economics and Finance 8 ( 04 658 663 663 S equivalen expression. In addiion for simpliciy we suppose, vlog( r. C ( S X ( S X[ v ] X ( S X ( S X C v( S X 0 Here we have, ( S X equaion as follows:, S X C ( and ( vs X, hen we may wrie he above 0 ( We suppose, y hen y,hen we may wrie equaion (8 as follows: y y 0 ( The equaion ( is similar o equaion (9 herefore we can use he same concep for roos of he equaion ( as discussed for (9. Conclusions In his aricle, some exensions of Black and Scholes model wih GARCH volailiy have been derived. We have used Taylor approximaion as discussed in (Gong e al., 00. However he value of implied volailiy migh depend upon naure of coefficiens of our resuled quadraic equaion. In addiion using our mehod in he Black- Scholes model wih GARCH volailiy, he implied volailiy of he sock, which varies over can be deermined.the underlying asse price process is coninuous and disribuion may urned ou o be asymmeric. References Black, F.,Scholes, M.,973. The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy 8, p.637-659. Buler, J.S., Schacher, B.,986. Unbiased Esimaion of he Black-Scholes Formula, Journal of Financial Economics 5 (3, p. 34-357. Chrisian, F., Jean, Z.M., 00. GARCH models, John Wiley and sons. Corrado, C.J., Miller,T.W., 996. A Noe on a Simple Accurae Formula o Compue Implied Sandard Deviaions, Journal of Banking and Finance 0, 595-603. Duan, J.C., 99. The GARCH Opion Pricing Model, Mahemaical Finance 5, p. 3-3. Gong, H.,Thavaneswaran, A., Sing,J.,00. A Black-Scholes Model wih GARCH Volailiy, The Mahemaical Scienis, 35, p. 37-4. Laane, H.A., Rendleman R.J., 976. Sandard Deviaion of Sock Price Raios Implied by Opion Premia, Journal of Finance 3, p.369-38. Reinhold, H., 004.Lecure Noes in Economics and Mahemaical Sysems: Sochasic Implied Volailiy, Springer-Verlag,Berlin Hidelberg.