Estimation of Stochastic Volatility with a Compensated Poisson Jump Using Quadratic Variation

Transcription

1 Applied Mahemaics, 07, 8, hp://wwwscirporg/journal/am ISSN Online: ISSN Prin: Esimaion of Sochasic Volailiy wih a Compensaed Poisson Jump Using Quadraic Variaion Perpeual Saah Andam, Joseph Ackora-Prah, Sure Maaramvura Deparmen of Mahemaics, Kwame Nkrumah Universiy of Science and Technology (KNUST), Kumasi, Ghana Deparmen of Acuarial Science, Universiy of Cape Town, Cape Town, Souh Africa How o cie his paper: Andam, PS, Ackora-Prah, J and Maaramvura, S (07) Esimaion of Sochasic Volailiy wih a Compensaed Poisson Jump Using Quadraic Variaion Applied Mahemaics, 8, hps://doiorg/0436/am Received: June 9, 07 Acceped: July 4, 07 Published: July 7, 07 Copyrigh 07 by auhors and Scienific Research Publishing Inc This work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY 40) hp://creaivecommonsorg/licenses/by/40/ Open Access Absrac The degree of variaion of rading prices wih respec o ime is volailiymeasured by he sandard deviaion of reurns We presen he esimaion of sochasic volailiy from he sochasic differenial equaion for evenly spaced daa We indicae ha, he price process is driven by a semi-maringale and he daa are evenly spaced The resuls of Malliavin and Mancino [] are exended by adding a compensaed poisson jump ha uses a quadraic variaion o calculae volailiy The volailiy is compued from a daily daa wihou assuming is funcional form Our resul is well suied for financial marke applicaions and in paricular he analysis of high frequency daa for he compuaion of volailiy Keywor Sochasic Volailiy, Compensaed Poisson Jump, Quadraic Variaion Inroducion Volailiy is he downward and upward movemens of he marke Naurally sock prices arac volailiy Economiss argue ha when demand is high, price goes up and producers are willing o sell more because a lo of people wan o buy he same goo Also higher supply lea o price fall and producers are willing o supply less because a lo of people do no wan o pay more for goo ha can be easily found in he marke The bigges driver of volailiy is a drop in he marke which laer goes up for a while and comes down again Mosly, when he economy is in a very precarious sae, invesors inervene o help solve he crisis which causes he sock marke o revive and recover from is decline [] DOI: 0436/am July 7, 07

2 P S Andam e al High volailiy brings panic o invesors and his fear has an impac on he financial marke Volailiy of a financial insrumen measures he uncerainy of he reurns I is an imporan parameer in pricing of an asse, porfolio managemen, invesmen analysis and risk managemen Volailiy influences cash flow from selling asse a a specific fuure dae Higher volailiy means a greaer chance of a fall in he price of an asse Invesors find i difficul o deermine he movemen of he prices of sock because of he sochasic naure of volailiy which makes i difficul o predic sock prices In 973, Fischer Black and Myron Scholes published a paper called The Pricing of Opions and Corporae Liabiliies [3] in which volailiy was assumed o be consan In realiy, real sock reurns have higher kurosis; here is a high chance of large price change Empirical evidence has shown ha, volailiy assumed consan is resricive, herefore esimaion of volailiy is he firs sep o adjus he Black-Scholes model o real daa Volailiy canno be direcly observed and mus herefore be esimaed from hisorical daa In mos recen financial economerics lieraure, volailiy is regarded as a high degree of persisence and ime-varying [4] The developmen of new meho for esimaing volailiy sill play acive role in research There is much evidence of volailiy affecing financial asses and so esimaing he ex-pos volailiy is an imporan ool for financial economics research The common way o esimae volailiy is by summing he squared reurns Alhough meho for volailiy esimae work under some assumpions, i runs ino a problem when he daa exhibi some kin of jumps in real life Mosly since volailiy canno be observed, mos researchers fi i in parameric economerics models like Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) and is family [5] They also compue implied volailiies using Black-Scholes for opion pricing or compue he absolue reurns or ex-pos squared [6] Daily daa available on prices of asse which are speculaive allows quadraic variaion mehod o be used o measure he aciviies of reurns in financial marke Mos researchers have advocaed for quadraic variaion as a nonparameric mehod o esimae volailiy of an asse Mos researches done on quadraic variaion use ick-by-ick inra daily daa, weekly or monhly daa [6] [7] [8] Quadraic variaion conrols micro-srucure effec, i is consisen and feasible [9] and he error of realized volailiy of is momen and asympoic disribuion has been sudied [7] For Pure Jump and mixed Jump-Diffusion Processes, i is approximaely free of measuremen error under general condiions [6] Esimaion of insananeous volailiy or spo volailiy for high frequency daa using quadraic variaion is being researched recenly and has been esed on some simulaions [0] [] and i has numerical derivaive involved in i [] [3] [4] Quadraic variaion have been used o esimae he insananeous volailiy of an asse price wihou jump diffusion process [] A proposed exension of a rolling sample variance wih a coninuous record asympoic analysis of he use of quadraic variaion for he esimaion of he asse reurns has been done [5] The ine- 988

3 P S Andam e al graed volailiy was reaed as a coninuous ime sochasic process using a high frequency daa and esablished he link beween inegraed volailiy esimaors and spo volailiies [5] Unlike many oher researches ha has been done on quadraic variaion, here has no been a consideraion of he addiion of compensaed poisson jump o he sochasic differenial equaion model In view of his, we are moivaed o propose a quadraic variaion as a nonparameric mehod for esimaing insananeous volailiy from a sochasic differenial equaion model by including compensaed poisson jump as an exension This jump process has an inensiy which is able o capure he seepness and skewness in volailiy smiles for shor-daed opions The presence of he jump will add more realiy o financial marke applicaions Secion reviews some mahemaical preliminaries used in esimaing volailiy Secion 3 presens he esimaion of sochasic volailiy using quadraic variaion Secion 4 presens he analysis of he resuls using daa Secion 5 concludes he work Mahemaical Preliminaries In his secion, we review some mahemaical heories which are used in esimaing volailiy Definiion A real valued process X defined on he filered probabiliy space ( Ω,,, ), 0 is called a semi-maringale if i can be decomposed in he form of X( ) = M( ) A ( ) where M is a local maringale and A is a càdlàg adaped process of locally bounded variaion [6] Definiion Suppose ( M, d ) is a meric space, and E, a funcion f : E M is a càdlàg funcion if f is righ-coninuous wih lef limis Tha is for every E, he lef limi f ( ) : lim f s ( s = ) exiss and he righ limi f ( ) : = lim s f ( s ) exiss and equals f ( ) [7] Definiion 3 A poisson process is a sochasic process N, which has a jump size of only and has a consan pah beween wo jumps A ime,, he value of N is N = I T j, j= ) ( ) where I Tj, ) ( ) if Tj = 0 if 0 < Tj 989

4 P S Andam e al j and where T j is he increasing family of jump imes of N : lim j Tj = N saisfies: Independen incremens: { 0 0 < < < n, n } he random variables N N,, N 0 N are independen n n Saionary incremens: N h Ns h has he same disribuion as N Ns, h > 0 and 0 s [8] Definiion 4 The compensaed poisson process M = N λ, is a maringale wih respec o is own filraion [8] Definiion 5 Le V be a real-valued sochasic process defined on a probabiliy space ( Ω,, ) and wih ime ha ranges over non-negaive real numbers hen he ph variaion is defined as, n V = lim V V 0 k Π k k = where Π [ 0,], and Π is he norm of he pariion = < < < < = such ha we have ( ) 0 0 n if he above sum converges [9] p { i i i n} Π = max, =,, This is rue under cerain condiions for example, p = defines he firs variaion or oal variaion process, for p =, he ph variaion equals he quadraic variaion if he sum converges Also, i is a bounded variaion if and only if for p =, V < For a generalized Iô processes, = 0 σ 0 s s µ 0 s X X db d, s where B is a Brownian moion, is quadraic variaion is given by [ X] σ = [0] 0 The quadraic variaion of a compensaed poisson process M = N λ is [ M ] = M N s = [] s 3 Sochasic Volailiy Esimaion A poisson process added o he sochasic differenial equaion wih Brownian moion dependen embodies disconinuiies in he reurns on sock This helps fi a beer marke daa wih regar o he reflecion of he realiy in he sock marke and also accuraely calculae volailiy because i incorporaes he effec of uncerainy over he jump size [] When here is a jump-diffusion process, i can describe he sock prices more accuraely a he expense of making he marke incomplee because jumps in sock prices canno be hedged in raded securiies When he marke is incomplee hen he payoffs of he opion canno be replicaed which will make i difficul o price he opion [] In order o describe he sock prices more accuraely, a compensaed poisson jump is incorporaed in a sochasic differenial equaion Suppose ha he price process p( ) of an asse follows he Iô process of a s 990

5 P S Andam e al semi-maringale form: d p ( ) = α( B, ) d σ ( B, ) db ( ) d M( ), () where M ( ) = N( ) λ and N( ) is a poisson process wih inensiy λ, α is he drif, σ is he volailiy, ime is represened by and he Brownian moion is B hen α and σ are adaped o a random process saisfying he condiions in (QA) below; (QA) ( α ( )) T E d 0 < ( σ ( )) T E d 0 < where α( ) = α( B, ), σ ( ) = σ ( B, ) and i s bounded by α σ c, where c is a consan The ime inerval can be spli ino subsequen equal inervals which conain a cerain number of observaions The volailiy can be expressed in erms of is mean quadraic variaion From equaion (), α = 0 implies an efficien marke [3] The drif erm makes no conribuion o he quadraic variaion process [4] The heorem below is he main resuls of his paper which exhibis he esimaion of volailiy wih a compensaed poisson jump Lemma 6 For he ime inerval [, ], where denoes he inerval beween he wo consecuive poins, he volailiy is consan in he inerval and is value of observed quoaions of he volailiy is he mean quadraic variaion: Proof σ ( ) σ ( s) lim = 0 σ ( s) lim = lim σ ( s) d s 0 0 Assuming σ is càdlàg adaped process, if is generaed by he full observaion of he price process, hen using Faou s Lemma we have, Hence, lim σ ( s) 0 d σ ( s) σ ( s) lim lim d = = 0 0 d ( ) d d = lim σ ( s) lim σ ( ) 0 d = 0 = σ ( ) = σ ( ) 99

6 P S Andam e al ( s) σ lim σ ( ) = 0 The expression in Lemma 6 shows he volailiy process inegraed over a given inerval [, ] σ ( ) On he oher hand, ( ) This expressions gives he general case for obaining σ can be esimaed from a given model, for insance equaion (), as shown in he heorem below If volailiy is esimaed, i will help idenify he ype of financial model he sock price follows; wheher i is geomeric Brownian moion or Ornsein- Uhlenbeck process and many more for he purpose of opion pricing This will also help invesors o evaluae he rae of heir invesmen risks The heorem below is he volailiy esimae based on Equaion () Theorem 7 Suppose ha he price process, p, is given as d p ( ) = α( B, ) d σ ( B, ) db ( ) dm( ), hen is volailiy, Vol ( p)( ) is; ( ( ) ( )) ) ( ) p p I T j, j= Vol ( p)( ) σ ( ) = lim 0 where denoes he condiional expecaion operaor wih respec o he σ-field Proof Given a price process d p ( ) = α( B, ) d σ ( B, ) db ( ) dm( ), we esimae he volailiy for evenly spaced ime inervals using quadraic variaion as follows: ( ) = α( ) σ ( ) ( ) ( ) d p B, d B, db dm ( p ( )) = ( α( B) σ ( B) B ( ) M( ) ) ( ) d, d, d d Since db( ) d, ( d) 0 and ( db( ) ) d [5], and ( ) ngale, i implies ha d M ( ),d = 0, d M ( ),db( ) = 0, If ( pm, ) :[, ] inerval gives: M is a mari- and we obain, ( dp( ) ) = σ ( ) d ( dm ( ) ) ( ( )) ( ( )) σ ( ) dp dm = d (), hen inegraing Equaion () wih respec o he ( ( )) ( ( )) = σ ( ) dp s dm s s ( ) ( ) ( ) ( ) = σ ( ) dp s dp s dm s dm s s d s Wih respec o p and M, we have heir quadraic variaion as dp s dp s d p, p d p dm s dm s = d M, M = d M, ( ) ( ) = [ ] = [ ] [ ] and ( ) ( ) [ ] [ ] s [ s ] [ s ] [ s ] his implies ha, Then we have, [ ] [ ] [ λ λ ] ( λ ) d M = d N s, N s = d N s = d N s s s 99

7 P S Andam e al [ ] [ ] [ ] [ ] ( ) s = s σ d p d M s d s [ ] [ ] s = ( ) s σ d p d N s d s [ ] [ ] [, ] σ ( ) = p N s s d, ( ) ( ) ( ) p p N N = σ s d s Dividing boh sides by, we have, ( p p ) N N σ ( s) = d s Taking he condiional expecaion of boh sides we have, ( p( ) p( ) ) ( N ) ( s) N σ = ( p( ) p( ) ) ( N ) ( s) N σ lim = lim 0 0 N N can be wrien as N, bu from Definiion (3), N = I, hen i follows ha, j= T j, ) ( ) lim σ ( s) lim = 0 0 From Lemma 6, we have, ( p( ) p( ) ) I ) ( ) σ ( ) T j, j= σ ( s) lim = 0 Therefore he volailiy of a price process which has an evenly spaced inerval is: ( ( ) ( )) ) ( ) p p I T j, j= Vol ( p)( ) σ ( ) = lim 0 This operaor ( ) is generaed by he full observaion of he price process unil ime where is he observaion of single pah of he marke evoluion wihin a given period This heorem is a link o Lemma 6, ha is: 993

8 P S Andam e al Vol ( p)( ) σ ( ) = lim 0 σ ( s) ( ( ) ( )) ) ( ) p p I T j, j= = lim 0 If he price process is independen of is pas informaion, hen ( )( ) σ ( ) Vol p will be, σ ( ) ( ( ) ( )) ) ( ) p p I T j, j= = lim = 0 0 since he maringale par of he price process will vanish and he jump is esimaed from he price process and hence he volailiy will be consan From Theorem 7, i can be seen ha he compensaed poisson jump has a negaive effec on he price process Also, Theorem 7 requires high frequency daa since i is he ime evoluion of he price of an asse in is semi-maringale form wih a compensaed poisson jump Theorem 7 can be furher simplified o ( ( ) ( )) ) ( ) I T p p j ( )( ) ( ), j= Vol p σ = lim lim 0 0 bu i canno be deermined numerically, because canno be deduced from he observaions bu i can be deermined when he disribuion of he price process is known For example, if he disribuion of he price process follows a normal disribuion hen, we can apply aniheic variae Mone Carlo simulaion o i Aniheic variae is defined as; ( p( s ) p( s) ) 0 = n n s = ( p( s ) p( s) ) ( p( s ) p( s) ) where ( p( s ) p( s) ) = ( p( s ) p( s) ), bu p( s ) = p( s, s,, sn ) and p( s ) = p( s, s,, s ) = p( s, s,, s ) n s, s,, sn are independen random numbers and are uniformly disribued on ( 0, ), s, s,, sn are also uniformly disribued on ( 0, ) and i p s p s p s p s makes ( ( ) ( )) has he same disribuion as ( ( ) ( )) since s, s,, sn are negaively correlaed wih s, s,, sn Then can be compued as: n 994

9 P S Andam e al ( p( ) p( ) ) = ( p( s ) p( s) ) (3) 0 The algorihm below esimaes using aniheic variae Mone Carlo simulaion When simulaing an evenly spaced daa, here is a need o have a benchmark in order o assess he good predicion of he model proposed To achieve his, we use he rue volailiy as he benchmark calculaed as: n ( σ ) = ( p( ) p( ) ) rue volailiy (4) Equaion (4) assiss in assessing he performance of differen esimaors used In his work, he esimaors used are; he rue volailiy, he insananeous volailiy wih a compensaed poisson jump and he insananeous volailiy proposed by Malliavin and Mancino [] Malliavin and Mancino [] proposed; σ ( s) Vol ( p)( ) σ ( ) lim = (5) 0 for calculaing he insananeous volailiy wihou a jump In order o apply daa o Equaion (5), we use aniheic variae Mone Carlo simulaion o evaluae i From he Algorihm above, o 8 is used o esimae he rue volailiy using Equaion (4) The algorihm numbered o 3 is used o generae he aniheic variae Mone Carlo simulaion using Equaion (5) and he numbered algorihm from o 9 is used o compue he insananeous volailiy wih a compensaed poisson jump using Theorem 7 4 Empirical Analysis of Theorem Using Daa Sock prices indicae he srengh and performance of he company Any daa can be used bu wih reference o his analysis we use daa from AngloGold = 995

10 P S Andam e al Ashani Ghana (AGA) A daily high frequency sock prices daa from AGA conaining he closing prices from 6 April 008 o 7 June 06 was ploed wih ime (in years) as shown in Figure Closer o he laer par of 05 o 06, he graph exhibis some sor of a jump A he op of bear marke, reurns are earned han he boom of a bull marke Increase in sock prices means invesors are buying socks raher han selling i and he fall in he price of socks means invesors are selling heir socks raher han buying Also, an increase in he sock price of a company means here is a profi making and a risk reducion of layoff of workers in he company When here is an increase in sock price, good reurns are generaed When he sock price of a company falls coninuously hen here is a higher chance of a akeover by anoher company, a merge or even close down of he company The volailiy is deermined from he sock price of AngloGold Ashani I is used o deermine he performance and srengh of he sock price I also help generae he invesors reurn made from he sock price The graphs below show he volailiy of he sock prices obained from AngloGold Ashani The graph in Figure indicaes he plo of he rue volailiy using Equaion (4) The graph in Figure 3 reveals he plo of he insananeous volailiy wihou a jump by Malliavin and Mancino (009) [] using Equaion (5) The graph in Figure 4 shows he plo of he insananeous volailiy wih a compensaed poisson jump using Theorem 7 wih he Algorihm saed above The sock price is assumed o be normally disribued hen aniheic variae Mone Carlo simulaion is applied o i The higher he volailiy, he greaer he reurns and he higher he risk associaed wih he ype of invesmen Comparing he graphs in he sock prices as in Figure and he volailiy graphs in Figures -4, i shows ha from 009 o he mid of 00, he sock prices increased sharply and is corresponding volailiy was high which implies ha invesors were buying socks raher han selling i Also when here was a drasic decrease in sock prices from 0 o 04, he volailiy also decreased sharply Closer o 05, he sock prices sho up again which made i highly volaile and migh be Figure Plo of sock prices from AGA 996

11 P S Andam e al Figure Plo of he True volailiy Figure 3 Plo of Insananeous volailiy wihou a jump Malliavin and Mancino (009) [] aribued o he merge ha ook place beween he hen Ashani Goldfiel Corporaion limied and AngloGold o form AngloGold Ashani [6] The rend of volailiy helps he invesor o re-balance is weighs on porfolio of socks and 997

12 P S Andam e al Figure 4 Plo of Insananeous volailiy wih a compensaed poisson jump also help he invesor knows he performance and srengh of he company Comparison of Resuls The resuls obained by Malliavin and Mancino [] was ( p( ) p( ) ) Vol ( p)( ) σ ( ) = lim 0 from he model d p ( ) = α( B, ) d σ ( B, ) d B ( ), and when compensaed poisson jump was added o i we obained ( ( ) ( )) ) ( ) p p I T j, j= Vol ( p)( ) σ ( ) lim = 0 From his, i can be seen ha he compensaed poisson jump has an effec on he price process Also, error analysis of he esimaes were done using he Roo Mean Square Error(RMSE) calculaed as; N RMSE = esimaed rue volailiy ( ) N i = The RMSE of he insananeous volailiy proposed by Malliavin and Mancino [] was 300 and he RMSE of he insananeous volailiy wih a compensaed poisson jump was 3084 This means ha, he inroducion of compensaed poisson jump reduced he error margin and is a good predicive model han he 998

13 P S Andam e al insananeous volailiy proposed by Malliavin and Mancino [] This means ha he sochasic differenial equaion wih a compensaed poisson jump is a beer represenaive model for sock prices han sochasic differenial equaion wihou a compensaed poisson jump 5 Conclusion We have esablished he heoreical basis for esimaing sochasic volailiy wih he presence of a compensaed poisson jump for univariae seings for evenly spaced observaions using quadraic variaion The condiional expecaion, was calculaed using aniheic variae Mone Carlo simulaion of he price process for implemenaion of daa o be possible We analyzed his using daa from AGA, Ghana Also, he RMSE of he insananeous volailiy wih a compensaed poisson jump gave a smaller value as compared o he RMSE of he insananeous volailiy proposed by Malliavin and Mancino [] I is herefore advisable for financial and economic analys o employ jump processes when using such daa for forecasing else; here will be error incorporaed This will help minimize he risk (in erms of loss) associaed in invesing in asses ha are highly volaile like gold I will be ineresing o invesigae he effec of oher ypes of jumps on evenly spaced daa Acknowledgemens This work was carried ou wih financial suppor from he governmen of Canada s Inernaional Developmen Research Cenre (IDRC), and wihin he framework of AIMS Research for Africa Projec and also our graiude o he Deparmen of Mahemaics, KNUST, Universiy of Cape Town and African Insiue for Mahemaical Science, Ghana References [] Malliavin, P and Mancino, ME (009) A Fourier Transform Mehod for Nonparameric Esimaion of Mulivariae Volailiy Annals of Saisics, 37, hps://doiorg/04/08-aos633 [] Ibboson, RG (0) Why Does Marke Volailiy Maer? Yale School of Managemen, Yale School of Managemen [3] Black, F and Scholes, M (973) Pricing of Opions and Corporae Liabiliies Journal of Poliical Economy, 8, hps://doiorg/0086/6006 [4] Andersen, TG, Bollerslev, T and Diebold, FX (00) Parameric and Nonparameric Volailiy Measuremen NBER Technical Working Paper, 79 [5] Barucci, E and Reno, R (00) On Measuring Volailiy and he GARCH Forecasing Performance Journal of Inernaional Financial Markes, Insiuions and Money,, [6] Andersen, TG, Bollerslev, T, Diebold, FX and Labys, P (00) The Disribuion of Realized Exchange Rae Volailiy Journal of he American Saisical Associaion, 96, 4-55 [7] Barndorff-Nielsen, OE and Shephard, N (00) Economeric Analysis of Realized Volailiy and Is Use in Esimaing Sochasic Volailiy Models Royal Saisical 999

14 P S Andam e al Sociey, 64, [8] Zhang, L, Mykland, PA and Ai-Sahalia, Y (005) A Tale of Two Time Scales: Deermining Inegraed Volailiy wih Noisy High-Frequency Daa American Saisical Associaion, Theory and Meho, 00, [9] Large, J (007) Esimaing Quadraic Variaion When Quoed Prices Change by a Consan Incremen Universiy of Oxford, Oxford [0] Alvarez, A, Panloup, F and Savy, N (00) Esimaion of he Insananeous Volailiy American Mahemaical Sociey hp://wwwarxivorg/abs/083538v4 [] Zu, Y and Boswijk, HP (03) Esimaing Spo Volailiy wih High-Frequency Financial Daa Discussion Paper Series [] Come, F and Renaul, E (998) Long Memory in Coninuous Time Sochasic Volailiy Mahemaical Finance, 8, [3] Foser, DP and Nelson, DB (996) Coninuous Record Asympoics for Rolling Sample Variance Esimaors Economerica, 64, hps://doiorg/0307/797 [4] Mykland, PA and Zhang, L (006) ANOVA for Diffusions and Io Processes The Annals of Saisics, 34, [5] Andreou, E and Ghysels, E (00) Rolling-Sample Volailiy Esimaors: Some New Theoreical, Simulaion, and Empirical Resuls Journal of Business and Economic Saisics, 0, hp://wwwjsororg/sable/393 hps://doiorg/098/ [6] He, S, Wang, J and Yan, J (99) Semimaringale Theory and Sochasic Calculus Science Press, CRC Press Inc [7] Billingsley, P (995) Probabiliy and Measure John Wiley and Sons, Hoboken [8] Privaul, N (06) Noes on Sochasic Finance: Sochasic Calculus for Jump Processes Technical Repor, Nanyang Technological Universiy, Singapore hp://wwwnuedusg/home/nprivaul/indexhml [9] Karandikar, RL and Rao, BV (007) On Quadraic Variaion of Maringales Proceedings of Indian Academy of Sciences, 4, [0] Proer, PE (004) Sochasic Inegraion and Differenial Equaions In: Sochasic Modeling and Applied Probabiliy, nd Ediion, Springer-Verlag, Berlin, Heidelberg [] Jeanblanc, M (007) Jump Processes Universie d Evry Val d Essonne, Cimpa School [] Navas, JF (003) Calculaion of Volailiy in a Jump-Diffusion Model The Journal of Derivaives,, 66-7 hps://doiorg/03905/jod [3] Kanaani, T (004) High Frequency Daa and Realized Volailiy PhD Thesis, Graduae School of Economics, Kyoo Universiy, Kyoo [4] Calderon, CP (009) Lecure III: Review of Classic Quadraic Variaion Resuls and Relevance o Saisical Inference in Finance PASI, Rice Universiy/Numerica Corporaion [5] Oksendal, B (003) Sochasic Differenial Equaions: An Inroducion wih Applicaions 6h Ediion, Springer-Verlag, New York hps://doiorg/0007/ [6] AngloGold Ashani (0) Hisory Johannesburg, Souh Africa hp://wwwanglogoldashanicom/en/abou-us/hisory/pages/defaulaspx 000

15 Submi or recommend nex manuscrip o SCIRP and we will provide bes service for you: Acceping pre-submission inquiries hrough , Facebook, LinkedIn, Twier, ec A wide selecion of journals (inclusive of 9 subjecs, more han 00 journals) Providing 4-hour high-qualiy service User-friendly online submission sysem Fair and swif peer-review sysem Efficien ypeseing and proofreading procedure Display of he resul of downloa and visis, as well as he number of cied aricles Maximum disseminaion of your research work Submi your manuscrip a: hp://papersubmissionscirporg/ Or conac am@scirporg