Stochastic Volatility and Black Scholes Model Evidence of Amman Stock Exchange

Transcription

1 Internatonal Journal of Appled Scence and Technology Vol. 6, No. 3; September 06 Stochastc Volatlty and Black Scholes Model Evdence of Amman Stock Exchange DR. Mohammad. M. Alalaya Prof. Sulman A.Alkattab DR. Ahmad AlMouhtaseb DR.Jehad Alfarajat Department of Economcs College of Busness Al-Hussen Bn Talal Unversty Ma an, Jordan Abstract Ths paper s an attempt to decompose the Black Scholes nto components n Garch opton mode, and path dependence of the termnal stock prce dstrbuton of Amman Stock Exchange (ASE),as Black Scholes the leverage effect on ths paper result of analyss s mportant to determne the drecton of the model bas, a tme varyng rsk, may gve a frutful help n explanng the under prcng of trade stock shares and traded optons n ASE. Generally, ths study consdered varous prcng bases related to warrant of strke prces, tme to tme maturty. The Garch opton prce does not seem overly senstve to (a,b ) parameters, or the tme rsk premum, varance persstence parameter, Ω = a +B heavng on the magntude of the Black Scholes bas of the result of analyss, where the condtonal varance bas doesn't Keyword: Black-Scholes, Stochastc volatlty, ASE, Garch model, Opton prces, Strke prces. Jll classfcaton: C0, C30, G3. Secton One: Introducton:. Defnton terms of the study: We should clarfy some terms whch used n ths study, such as: 04 An opton: t s the rght for the dealer to buy or sell goods at a strke prce, but not oblgated. Exercse prce or strke prce: t ndcates to a prce that whch good n partcular may be bought or sold nurng a specfed perod or before, In the money: ths related to mmedate prce, whch has postve ntrnsc value. In addton, t gans from the mmedate exercse. At the money: n ths expresson, there s on equalzaton state between the exercse prce and share prce.. Objectves of the study: Ths paper ams to nvestgate the ASE stock and strke prces and the other sub objectves are: We can use the modfcaton of Black Scholes equaton and opton prcng also can use n ths feld. Wth respect of market value when testng the Black Scholes equaton on the bass of ASE ndex and market value stochastc volatlty present strongly..3 Research queston The man queston rased s, whether the prces that the dealer has pad s n the ASE market to buy or sell s far prces, and whether these prces can be valued by Black Scholes equaton Then other questons are as stated as:

2 ISSN (Prnt), -004 (Onlne) Center for Promotng Ideas, USA : Is Black Scholes model reasonably accurate for n the money prcng of stock shares n ASE? : Does the Black Scholes model approprate for prcng the traded warrants on the ASE. : Can the nvestor who have lttle grasped and understandng of the nature of opton warrants rarely make good proft?.4 Bref notes of ASE ASM (Amman stock market) or some named t Amman stock exchange (ASE ), ASE has proposed to have an ncreased of companes whch are lsted throughout the study perod,whch can gve us assgns that there s a confdental trust of ASE regulatons,and trust of economc growth of Jordan, n addton to these ths let us have a postve sgns of economc drectonal growth and stablty on a country the mprovement of ASE s demonstrated lke any emergng market on ts characterstc, the rato low turnover rato, low lqudty, low transparency, and the non exstence of market decson makers, the turnover rato for the perod of the study was 7.53%, and the daly average of turnover was %, ths rato s too few, and these ratos are consdered to vary per tme., The major actons that may affect shares buy and sell and tradng actvty manly s prces. And the average daly turnover s ownershp of ndvdual nvestors and nsttutonal, and government. The ownershp structure The ASE has wtnessed an ncreased n the number of lsted companes throughout the years, whch gves an ndcaton of economc growth n Jordan, and stablty durng the perod of n ASE. The tradng volume ncreased year to year durng the perod of study, the results of vsblty of ASE s superor than other stock markets n mddle east regon, t has undergone accelerated growth, especally durng the last 6 years due to stablty and the Arab shares such Iraq an nvestor, also Jordan government represented the board of nternatonal accountng standard. Some ndcators of ASE, It establshed 976 and t s emergng stock market, the captalzaton s 9.765m us, and the change n s 8.4%, whle t n s 7.3%, where the captalzaton rato to GNI IS 58.9%, where the turnover s 3.54 mus, and the turnover (lqudty)s 8.7%). (M.Alaya, 03). 5: Prevous studes of the subject: (Whaley et al. 987), they have studed, and analyzed the CBOE data for lmted perod extended from 975 to 978, they have used tme of expraton date, and the strke prce, ther results ndcate that the volatlty elmnated, and concluded the dvdend dvded nduced probablty ntegrated n Amercan call opton prcng. Brown has observed a random behavor of opton prcng, he notced the moton of pollen floatng n water s randomly, and dd not follow any dstnct pattern, ths observaton and t s subsequent s known as Brown moton..fscher black and Myron scholar employed these tools n ther research to model to prce dervatve of securtes. The Black-Scholes formula used to prce European call and put optons based on asset prce of dynamc stochastc model, whch depend on a stochastc dfferental equatons, therefore we can solve ths model of geometrc Brown moton, where the certan two real parameters of volatlty are drftng. The use of statstcal tests to solve the problem assocated wth the calbratng stochastc dynamcal system. (Fsher Black and Myron Schools, 973). We can defne optons as: securty has rghts of buy or sell n certan condtons, wth a lmted and specfed perod. The smple type of opton s one that gves the rght to buy or sell a sngle share of stock, ths called call opton. (Black-Scholes,973)., They notced that f an opton has a hgher prce daly traded value n the stock market, then the opton has a hgh traded value, then the opton should be exercsed, Also when expraton date s near or closed to, there s an equlbrum between the value of the opton or stock share and the prce of the stock.(dumas et al. 996,998), they have studed s & p 500 ndex opton data, through the perod 988- to 993. They have evaluated the volatlty, also they have found that performance s worse when we compare to the Black Scholes model. Were (Sorn-Staraga, 004), used a sample of s & p 500 ndex opton prces, utlzed of opton prces and mpled of determnstc volatlty approach to hs data, results ndcate that a parsmonous model s sutable for hs analyss through Akake crteron, and the model predcts the errors whch grows largely as volatlty functon, and hedge rato determned volatlty by Black Scholes equaton. The Black Scholes model has been stll used n opton prcng, where the nput s mportant requred for the call opton prcng, on a non dvdend opton. Hull and Whte (987 referred to as How, provde throng a power seres, whch compared and ths How model can provde two models wth stochastc volatlty. Rubnsten (994), he proved the volatlty of asset returns and opton, whch can be appear as a determnstc functon, then he dscussed the behavor system of changes of. Brande, et al. (00). 05

3 Internatonal Journal of Appled Scence and Technology Vol. 6, No. 3; September 06 Fnd that the mpled bnomal model poses the Amercan style optons. Were Cox. (979). Appled a tree model of volatlty. Some authors usng n ther studes ad- hoc procedure of Black-Scholes mpled volatlty. The volatlty ncreases the probablty that the stock prce wll rse or fall ncreases. Ths study s arranged as follows: Secton One: wll nclude objectve of ths study. And Assumptons of Black-Scholes model, then the hypothess of study. Where secton two ntroduced the background and theoretcal revew of black-scholes model, t wll also dscuss the model that ths study follows. Sectons three: contan the prcng equaton dervaton. Secton four: ncluded an overvew of data sources and methodology. secton fve ncluded the results and emprcal results of ths paper, fnally we concluded the fnal results of the study. Secton two: Lterature revew: : Assumptons of Black-Scholes model: The black-scholes model based on the followng assumptons: The nterest rate s known and constant through tme n a short term perod. 3 The stock pays on dvdend or other dstrbutons. 4 The stock prce follows a random walk n contnuous tme wth a co- varance rate proportonal to the square of the stock prce. 5 There are no transacton costs n buyng or sellng n the stock or optons. 6 Then are no penaltes for short sellng and the seller who does not own securty wll smply accept the prce of the securty from a buyer (Black & Scholes, 973). 7 The underlyng asset follows a long normal random walk. 8 The Arbtrage arguments allow us to use a rsk-neural valuaton approach (Cox-Rabnsten's Proof). Black-Scholes n ther paper derved on the opton prancng model, of whch one of man assumpton was that underlyng stock follows a Geometrc Brownan moton. Black-Sholes n ther paper exsts many ways to drve stock opton prces, many approaches can be used such as a martngale approach whch depends on rsk-neural valuaton formula as: 06 V op η asset Ec V pay η asset...() t Where V op : opton payoff at maturng,v pay =η (K, ST) s some determnstc functon and V op s opton value as of tme t = 0,, η s a so called numerc asset used the relatve prce assets t V pay η asset () t The Black - Scholes model assumes there s no arbtrage opportunty, whch emphaszes that there s an opportunty to gan profts, wthout any rsk nvolved. The arbtrageurs themselves be aware of t by buyng more there n one stock market, whch can be causes the prce of the stocks to rse there, and sellng n other stock market, whch causes the prce to declne, there such as buyng more stock n the New York stock market, and when there s a movement to rase up prces sellng there n London stock market. The Black - Scholes model for opton prcng dvde wth the dea of delta hedgng n mnd. The Black-Scholes equaton s wdely soon to have paved the way for an nflux of mathematcs of fnance and fnancal studes, the prcng formula can be expressed as Black-Scholes for put opton as: rt f ( S, T, K, r, ) Ke N( d ) SN( d )..(3) Where: S s the current stock prce, T s tme untl expraton n years, and s the annualzed volatlty of the stock, N s the cumulatve standard normal dstrbuton functon, r s the current rsk-free rate of return. The above functon nputs requred for the prcng of a call opton on a non-dvdend payng stock wth current stock prce strke prce, nterest rate, volatlty and tme to maturty. We can observe all these parameters. Through above dscusson the followng alternatve hypotheses are expressed as: H(): The market value for n the money warrants aganst Black-Scholes value for n the money warrants also for out-of-money warrants.

4 ISSN (Prnt), -004 (Onlne) Center for Promotng Ideas, USA H(): The market value of of for near-maturty warrants. H(4): The market value on all daly prces observatons are also volatle tme to tme. H(5): Tradng n stock takes place contnuously and market are always open; H(6): The stock pays no dvdend on any type of underlyng assets or securty; H(7): The stock prce follows a Geometrc Brownan moton process, as constant. H(8): The assets are completely dvsble n nature; H(9): There s no penaltes on short sellng of shares and nvestor also get full use of short-sell Procedure; and Stock prce opton follows an explct type of stochastc process called dffuson process; Wth the excepton of volatlty n the market we can express d and d n the (4) equaton as: t log r ( T T ) K d..(4) d d ( T t) ( T t) : s the varance of courteously compounded return. Stochastc processes have many propertes that appear n numerous applcaton, such as: - The of Xt h Xt dstrbuton depends only on h, then X t Is have statonary ncrements. - f t t t3, Xt Xt, Xt3 Xt, are ndependence, that sgn off X t, ths gves us assgn that the process wth ndecent ncrements. 3- As we dscussed n ths revew X t s sad to be the Markov property due to the ndependent state of varables. 4- We can see that the transton probabltes should be normally dstrbuted n functon (5),whch as follows: N N 6 P K f ( K, N, P ) P K ( P ) j K j j Where K (,,..., N) and f s the probablt y mass Functon for the bnomal dstrbuton further more one can restrct hs analyss by lookng at statonary market chans. : Hedgng portfolo through Black-Scholes equaton: If we start, our analyss one mportant note should be consdered of drvng the black-scholes equaton; va a bnomal prcng process, whch s based on the bnomal prce model descrbed n the above secton. Frst, we assumed one rsk-free rate at whch we can borrow or lend money and that there s one perod left call opton on the stock. Therefore the far value of the opton whch we have determned to be by requrng that t be the value that equals t s expected payoff. Ths can be determned merely by knowng some nformaton about nterest rates, underlng stock, the strke prce, and the range of the values that the stock can take on after a perod and we should know the possble values that the stock could take over ths perod, whch has a relatve to what the volatlty n a way n the Black- Scholes equaton volatlty of stock can be defned to be the standard devaton of the logarthm of the returns of the stock. A Black-Scholes model assumes a constant volatlty and one way can be used to estmate s to use hstorcal volatlty. The equaton of estmaton can be gven as: n n ( u u) ^ (6) t 07

5 Internatonal Journal of Appled Scence and Technology Vol. 6, No. 3; September 06 Where: ( n ) prce data from ( n ) and u s ln, u s the sample average of all u, s the stock prce n perod t and T s the total length n each perod n the year. T equal /5 or /365.The numerator represents the log-returns of the stock, and the denomnator s a scalng factor to make the estmates be equal one of the year volatlty. (Hull, et. al.,987) wrote an artcle the prcng opton on assets wth stochastc volatltes to have a soluton to the problem wth a soluton to the problem of prcng European earl optons. They determned that the prce depends upon some stock prce S, and t s nstantaneous varance, Functon (7) can be wrtten as: d Where d s v S d d t vt σ s v d d w : for thestock prce may depend on S, σ ant t. U : and the dfluson coeffcent. : for the varance whchmay depend only on σ and t. V σ. The two processes are correlated. Hull and Whte, they determned some assumpton accordng to ther work, such as: - The volatlty (V) s not correlated wth the stock prce. - The volatlty V s not correlated wth aggregate consumpton. 3- S and are only the two varable whch are effectng the prce of dervatve functon, therefore the rsk free rate must be constant or determnstc. The expected returns of the stock for example, are not ndependent of rsk preferences. Rsk n stock market adverse nvestors n how he would ask for hgher expected returns for ncreasng rsk level and rsk seekng nvestor d would ask for lower expected returns for decreasng rsk levels. Secton three: Volatlty Models: In ths model we can depend on hstorcal returns from stock prces f we use the movng Average model the formula s: S S t u.(8) S Where: u percentage changes of returns of the day - and of day - and the end day, S: represents value of S assets at day., The value of the day before of u beng u ln the unbased estmate of one S volatlty as: m U U (9) n m n - The Stochastc model proposed some propertes: - Impled volatlty has a random behavor at tme, where the smooth dependency s avalable.. - The Impled volatlty s more senstve accordng to the Wener processes, where the dsturbng nose s normally dstrbuted. 3- The market volatlty s reduced, 4- Stochastc volatlty model can be used smply to evaluate any stock market or portfolo. 5- The share and opton or stock prces can be determned or gven through the appled of Black-Sholes equaton. 08

6 ISSN (Prnt), -004 (Onlne) Center for Promotng Ideas, USA 3. Opton prcng model Black-Scholes one of ther attrbutes the lognormal dstrbuton has that stock prce can never fall by more than 00 percent, n other hand, there s a small chme to rase more than 00 percent, accordng to these the Black- Scholes model n one of ther assumpton weather the optons and stock prces are determned through Black- Scholes equaton lagged and normally dstrbuted. Prcng dervatve wll provde a payoff at one exact tme n the future smply because t takes a dscounted vat s rsk-free nterest rate (Er t ), from the underlyng assets and dscount vat s a rsk free nterest rate (r)..recent studes fndng shows that volatlty tends to vary over tme accordng to ths stuaton the assumpton of constant volatlty s unrealstc. The Garch opton model assumed that the condtonal volatlty of stock prces depends on the past prcng errors. Therefore, one phase of ths study s Garch models, and the study based on hstorcal return from ASE stock prces. Regardng to the nature of random varables whch dstrbuted as log-normal as a Markova process s a certan stochastc that may randomly vary, and on the other hand that hstory of varable prces rrelevant and the only present value s used to predct the future values. Brownan moton s a partal equaton of Markov process that posse s varance.00 and zero mean as (John. C. Hull, 006). Ths method, known as a Wener process varable (x) follows ths method when: - Δ X : represents two short ntervals at tme t, where Δ t ndcates to the ndependent varable of the seres. If we put equaton n a smplfed approach, so that: m U In ths method we should be aware of the data too old mght be unrelated to predct the m m n - future, then we should carefully about more data whch lead to better precson usually we used the chasng prces from daly data over "the last 90, 80 or even 5 days are often used. - Garch model: ths method usually used by many models that follows the mpled volatlty as (Engle, 98). Accordng to ths process, we should make the uncondtonal mpled volatlty as constant,where on the other hand the condtonal mpled volatlty can be changed and vary over tme(t), ths model s known as an arch model (q) s: 9 a U ( 0 ) n n Where s equal.vl and s calculatng by usng the followng equaton: S S S u or u Ln..( ) S S And s the long-term average of Co- varance and varance rate and s weghted assgnment to V varable, and S - : s the order of dependency to past returns. An ArCh model as a generalzed approach model was proposed n a paper of (Boterstev, 986), whch know Grach model (P, q), can be as: q p a U B.( ) n n n Where: > 0 a 0 And. B The changes whch happen on (Δ X) durng a short perod (Δ t) s Δ X t (where dsrbuted normally (0,). But n a long tme from (0) to maturty T, x (T)- X( ) s normally dstrbuted wth mean zero, standard devaton t and varance T the Wener process for a varable C add an expected drft rate and volatlty, whch can be wrtten as: d d d. c t x The varable C s normally dstrbuted n any tme of the nterval T, mean change of C s( T), standard devaton T and varance T) (. 09

7 Internatonal Journal of Appled Scence and Technology Vol. 6, No. 3; September 06 The stochastc mpled volatlty of ASE assets as a emergng market assets are randomly dstrbuted, we can measure the volatlty changes n a certan tme of unexpected changes of assets and other fnancal assets, when the fluctuaton can't be captured and unknown, therefore the mpled volatlty can measure the rsk of optons and assets. Wth dfferent exercse and strke prces the volatlty s skewed, and shows a skewed or smle sometmes nstead of lnear curve, n ths case the smle s reflectng a hgher mpled volatltes for cash flow or deep n- or out of the money optons. When the market prce has used the mpled volatlty model and the stock prce follows a stochastc process and as Arch model, varables (p,q) s the order of dependency the smplest model Garch model s Garch (, ) model, whch can specfy as: Q.U B..(3),and B must sum to one. - - (Engle and Bollerstev, 986) have ntroduced the I. Garch (ntegrated Garch), n ths model a,b s equal to I. when we lke to estmate the parameters of the model we should have n our results the mportant results of maxmum log lkelhood estmaton. The lkelhood of the (m) observaton as: m U exp [, v v v..( 4 ) Whch estmated varance for the day I and are the number of observatons? The best parameters are the therefore the logarthm of the expresson s equvalent, thus the log lkelhood expressed as: U e ln ( v).( 5 ) v m The parameters of Garch (, ) can be defned as a functon of tme, t means that the parameters at day () are estmated by means of maxmum lkelhood, and subsequently these parameters are used to forecast volatlty at day ( ), then we can adapt equaton (3) as: a U B, (6 ) -, a, b are changng over tme V, V, U and change over tme also, and the model estmated L - - through daly maxmzaton of the parameters, t's expected, whch known as a geometrc Brownan moton C (John, Hull, 006) d d d where the (d t ), and( d w ),are correlated to be before to estmated future v v t v w volatlty on daly bass. Secton four : Data sources and methodology: 4 - : data sources and collecton : The ASE database and stock prces database for the perod from January consdered n ths paper analyss as hstorcal data of volume of traded shares, prce of opton, prce of underlyng stock, strke prce, expraton data and a tme stamp that reveals at whch tme. The collected data studed were 04,daly value of the ASE ndex analyzed as the fgure() below, and the specfed ndex data consdered as the daly closng value of the ndex and the bd prces of these optons durng the perod of the study. Fgure(): shows that the data on the ASE ndex from 005 up to 04 that volatlty s clearly appears. 0

8 ISSN (Prnt), -004 (Onlne) Center for Promotng Ideas, USA Dagram (): Daly Returns of ASE Index from B: Methodology: There are two methods of analyss the huge data amount frst step I have used the (OLS), then mpled volatlty by Grach (, ). Secton Fve: emprcal results : Table () : Descrptve statc for hstorcal volatlty results N : 45 Mnmum Maxmum Mean St/devaton Volatlty Resdual * Square resdual * Square relatve resdual * Squared relatve devaton: M(P) P P P ( P) ; Represents the prce of Black-Scholes equaton, and P s the actual stock prce. We can summarze that the stocks have stochastc mpled volatlty, we have notced that when we appled the measurement to the hstorcal data, dfferent types of volatltes are mpled, ths happens when stocks are over evaluated as (5.49)$, were the lowest strke prce opton s over valuated (.64)$. These results persstent the bas whch appears when we plot the volatlty versus tme, the average of ASE optons overprcng by 3% wth a low 7% under prcng. The (OLS) results of the analyss are shown n the followng tables. Table () Model Summary Model R(correlaton) R squred Adjusted R St/Error D-Watson Dependent Varable Square Relatve resduals Table (3): Coeffcent of OLS Unstandardzed Model Stand Coeffcent Coeffcents t-test Prob.Level Constant Beta B Std. / Error Tme Dependent Varable: Square relatve resduals The effect of tme s well appear n the dependent varable chosen, ths effect s postve sgnfcant at the 5% level, snce the coeffcent stll postve. Ths can let us to conclude that ncreasng of tme, whch dealng wth data, ncreasng results becomes as the prcng bas, nsofar the short- run the optons and stocks should mnmze the prcng bas, ths can be shown through Black Sholes equaton appled results whch predcts the lnear functon of volatlty and the resduals, then we can predct the lnear functon of prces.

9 Internatonal Journal of Appled Scence and Technology Vol. 6, No. 3; September 06 Fnally, t seems to be a common pattern therefore we can summarze that there s spkes whch observed, ths can be nterpreted due to sudden changes n the opton and stock prces. We can conclude that there s a postve auto correlaton present n the analyss, ths means that values of mpled volatltes are correlated to each so we can conclude that n the absence of unusual spkes, gve assgn that best estmator of hstorcal volatlty are presented n our analyss. The maxmum lkelhood Estmaton Garch (, ) model: In table (4) the estmated represents and the log of Garch (, ) model of ASE ndex. Table (4): results of Garch(,_)of ASE ndex V L ω γ a B a + B 6.53E E Due to results of table (3 ) parameter α s , B s and γ s 0.96 ths means that past returns gve us a sgn that t proved a some nformaton to estmate present and future volatlty of ASE. Adaptve Garch model results, the weghts of α, B and γ begn wth (0.076), (0.89) and (0.44) respectvely, and ended by ( ), (0.7934) and (0.0073) correspondng γ evoluton of the parameters are gven n fgure ( ) for Amman stock exchange daly returns durng the study perod the sum of α, B and γ s always around one, from fgure ( ) we can notce that the parameter s shfted up. Fgure (): Evoluton of the Estmated Parameters α,b and γ for the ASE Beta The Garch opton prce does not seem overly senstve to ( α and B) parameters or the unt rsk premum, ( λ ) varance persstence parameter, Ω = α +B, havng magntude of the equaton of Black Scholes model bas, where the uncondtonal varance bas s not mportantly accurate to justfy the model to ASE data. Fgure (4 ) declare the opton prce dfference tmes.

10 ISSN (Prnt), -004 (Onlne) Center for Promotng Ideas, USA Fgure 3: Stock Prce of ASE Dfference wth Volatlty of Black Scholes and Stochastc Volatlty S.V B.S Opton prces are dfferent tmes, n ths fgure (4): the comparson between the stochastc volatlty and the Black schools, the dagram shows as the dfference from pont to another ths happen due to subtractng the Black Scholes prce of the two models, consdered the opton prce as dummy or default. Also, we can see ths volatlty, n fgure (4). Fgure (4): Estmated Volatlty of the ASE Index by 3 Models Garch ( ()) Movng Average 0 The Black Scholes n upper fgure s set an opton prce of ASE to whch other models are compared of volatlty Fgure (4 )llustrate the compare between models wth a slght volatlty,but the volatlty become more obvous. 3

11 Internatonal Journal of Appled Scence and Technology Vol. 6, No. 3; September 06 Black - Scholes overprce near and at the money call opton of the stock,whle t produces a lower prces for a deep out-of the money stock prce of movng average model, the pattern for mpled volatlty dfference s very small,the reason behnd ths may be that the mpled model generates a volatlty, due to the fgure we can conclude that Black - Scholes volatlty s less than other models. Concluded remarks: Ths paper uses hstorcal data to estmate the volatlty, whch gve us through analyss a sgnfcant bas toward overprcng of the opton when we use Black-Scholes equaton. Many reasons are behnd why we attend to use the model of volatlty to be a random process. One of these reasons s t could be smply represent estmaton uncertanty, or sometmes t can arse as a fracton of transacton costs, the thrd reason s t could be, t has a thck (heavy taled) returns dstrbuton,forth reason has ether been as a far reason s related to any extended model must also specfy what type of data can be calbrated wth. We can summarze that the calbraton procedure as follows: The ft near the money mpled volatltes for several maturtes as n ASE,so straght lne n the composte varable called Log moneyness -to maturty rato (LMMR). The estmate of the equaton gves us that the slope and the ntercept (a) snce LMMR= 0 when stock prce equal strke prce, where b s exactly,that atthe money mpled volatlty. In addton, we can estmate the hstorcal volatlty of stock prce returns References Berkowtz, J. Forecastng Opton Values Wth False Models. Bauer College of Busness, Unversty of Houston Workng Paper; (June 6) 003. Black, F.; Scholes, M. The Prcng Of Optons And Corporate Labltes. Journal of Poltcal Economy, 8 (May - June), ; 973. Brandt, M. W.; Wu, T. Cross - Sectonal Tests Of Determnstc Volatlty Functons. Journal of Emprcal Fnance 9: ; 00. Bouchand, J.P., Potters, M. (003), Theory of fnancal rsk and dervatve prcng: from Statstcal Physcs to Rsk Management, Second edton, Unted Kngdom, Cambrdge Unversty press, Cox, J. C.; Ross, S. A.; Rubnsten, M. Opton Prcng: A Smplfed Approach. Journal of Fnancal Economcs 7: 9-64; 979. Dumas,B.Flemng, J.Whaely,R,E (996), " Impled volatlty functon,emprcal test, " Bureau of economc research (NBER ) WORKING PAPER No 5500 march,996. Dumas,B.Flemng, J.Whaely,R,E (996)" Impled volatlty functon,emprcal test " the journal of fnance Vol 53 No 6, pp Eberlen, E. (007). Jump-type Levy processes. In T. G. Andersen, R. A. Davs, J.-P. Kre_, and T. Mkosch (Eds.), Handbook of Fnancal Tme Seres. Sprnger. (Forthcomng). Fsher Black.Myron Scholes (973): " The prcng of optons and corporate labltes " The journal of poltcal economy, Vol 8,No 3. PP Khan, M.U., Gupta, A., Sraj, S., Ravchandran, N. (0), Dervaton and Suggested Modfcaton In Black- Scholes Opton prcng Model, IME Journal,Vol 6(),pp Khan, M.U., Gupta, A., Sraj, S., Ravchandran, N. (0), The Overvew of Fnancal Dervatve and Its Products, Internatonal Journal of Fnance & Marketng, Vol,(3), pp John Hull and Alan Whte, The Prcng of Optons on Assets wth Stochastc Volatltes,The Journal of Fnance, Vol. 4, No. (June, 987), pp John C. Hull, Optons, Futures, And Other Dervatves, 6th Edton, Pearson Educaton, New Jersey, 006. pp 65, 69, 93. Jochen Wlhelm ( 008) " opton prces wth stochastc nterest Black -Scholes and Ho / LEE unfed Unversty Passua, Herausageber,Betrebswlschaftlch Rehe ISSN : Lorella.Fatone, Fransesca Maran, Mara Crstna,and Francesco Zrll (0) : "The use of statstcal tests to calbrate the Black optons wth uncertan volatlty " The journal of probablty and statstc, unpublshed paper Lu, B. (00), Uncertanty theory: A Branch of Mathematcs for Modelng Human Uncertanty, Second edton, Sprnger-Verlag Berln, Hedelberg,

12 ISSN (Prnt), -004 (Onlne) Center for Promotng Ideas, USA Heru.Sataputera (003): " Black Scholes opton prcng usng three volatlty models: movng average,garch (,) and Adaptve Garch " Bachelor thess,erasmus unversty, Netherlands. Hong Boon Kyun (004): " Emprcal study of Black Scholes warrant prcng model on the stock exchange of Malaysa. Master thess,unpublshed thess. Rable, S. (000). Levy processes n _nance: theory, numerc's, and emprcal facts. PH.. Thess, Unversty of Freburg Rubnsten, M. Impled Bnomal Trees The Journal of Fnance 49: 77-88; 994. Tavella, D. (00), Quanttatve Method n Dervatves Prcng: An Introducton to Computatonal Fnance, John Wley and Sons Prntng press, Sorn,Straga (004)" volatlty term structure " Montgomery nvestment technology.co, unpublshed paper. Matache, A.-M., C. Schwab, and T. P. Whler (005). Fast numercal soluton of parabolc ntegro-d_erental equatons wth applcatons n _nance. SIAM J. SCI. Comput. 7, Matache, A.-M., T. v. Petersdor_, and C. Schwab (004). Fast determnstc prcng of optons on L_evy drven assets. MAN Math. Model. Numer. Anal. 38, M.Chaudhury,Jason,Z, We (996) " A comparatve study of Garch (.) and Black-Scholes opton prces " unpublshed paper. Mltersen, K.R., Schwartz, E.S (998).," Prcng of Optons on Commodty Futures wth Stochastc Term Structures of Convenence Yelds and Interest Rates", n: Journal of Fnancal and Quanttatve Analyss, 33 (998), Mchel Kalaverzos. And Mcheal Wennerno (007) " stochastc volatlty models n opton prcng ", Master thess n appled mathematcs,malardalen unversty, Sweden. Merton, R. C. (973)." Theory of ratonal opton prcng". Bell, J. Econ. Manag. SCI.Vol 4, pp Merton, R. C. (976). Opton prcng wth dscontnuous returns. Bell, J. Fnance. Econ.Vol 3, pp Mohammad,Alalaya.(04 )" A case study :study of Amman stock Exchange volatlty durng (994-04)" nternatonal busness research,vol 7 ssue 5,pp.80-90,May 04. Mohammad,Alalaya,(04)" volatlty of ASM(Jordan) through a comparable model and approaches (996-00)" European journal of busness and management,vol 5 ssue 3,pp Papapantoleon, A. (007). Applcatons of sem martngales and Levy processes n dualty and valuaton. PH. D, Thess, Unversty of Freburg.. Peter Cross,(006) : " parameter estmaton for Black Scholes equaton ", advsor Dr.Jalng Da.Ura,sprng 006. Protte, P. (004). Stochastc Integraton and D_erental Equatons (3rd Ed.). Sprnger. Zhu, J. (009), Applcaton of Foures Transformaton to Smle Modellng: Theory and Implementaton, New York, Second Edton, Hedelberg Dordvecht London, Sprnger publcaton, 5-. 5