Option Pricing with Constant & Time Varying Volatility

Transcription

1 Opion Pricing wih Consan & im arying olailiy Willi mmlr Cnr for Empirical Macroconomics Bilfld Grmany; h Brnard chwarz Cnr for Economic Policy Analysis Nw York NY UA and Economics Dparmn h Nw chool for ocial Rsarch 79 Fifh Avnu Room 4A Nw York NY 3 UA; mail: [email protected] Karim M. Youssf h Nw chool for ocial Rsarch Nw York NY UA and Financ & Risk Enginring Dparmn Polychnic Univrsiy 6 Mroch Cnr Brooklyn NY ; mail: [email protected]

2 Inroducion In March of 9 Louis Bachlir's dfns of his dissraion nild héori d la péculaion was prhaps h pivoal momn in h incpion of a cornrson in h modrn hory of financial conomics. Borrowing from ninnh cnury boanis Robr Brown Bachlir vnurd o pu oghr a horical foundaion for a phnomnon h winssd on h srs of Paris. Bachlir obsrvd ha sr cornr radrs wr in ffc placing a prsn valu on probabilisic fuur vns. adly Bachlir's achivmn rcivd a mixd racion in h Mahmaical communiy and h was rlgad o a lif of mdiocr acadmic sanding amongs his prs. Mor han a half a cnury passd whn h Mahmaician Edward horp pickd up on Balchlir's work and applid i o a mahmaical analysis of wagring. Following ha various amps by prnkl Bonnss and amulson wr mad o rviv h idas ha Bachlir bgan working on. Howvr i was no unil h arly 97s ha Fishr Black and Myron chols xpoundd upon Bachlir's hory o produc h Black chols modl for opion pricing. Evn hn hir work was no rcivd wih du prais or undrsanding. During ha im Robr Mron hn a Profssor a M.I. coninuously lobbid h acadmic journals on bhalf of Black and chols. h rsul was h publicaion of hir work in h Journal of Financ in 97 and again wih a modifid il and conn in h Journal of Poliical Economy in 973. I is saf o say ha Mron's suppor of Black and chols' work allowd h disciplin of Economics o ak h lad in shdding ligh upon on of h mos applicabl pics of horical work found oday. h imporanc of his horical work is qually rlad o h moivaions bhind is incpion as i is o h way i has ransformd our hinking abou financial marks. In rgards o is incpion hr is no doub ha Bachlir's moivaion was o xamin h raliy of randomnss and probabilisic vns which ar rood in our livs as social bings. Prhaps ha was don in an ffor o allow us o pl away h layrs of drminisic hinking o which w hav grown accusomd. As for h impac of h rsul as rndrd by h Black-chols work. W find ourslvs oday far from our goal of a firmly graspd undrsanding of h probabilisic naur of prics and marks y far away from our poin of dparur. As such significan advancs hav bn mad on h Black-chols modl. Wih ha in mind h following scions will prsn a drivaion of h Black- chols PDE a soluion for h PDE o arriv a h Black-chols formula and a criiqu of h wakr undrlying assumpions. Following ha a prsnaion of h Hson modl basd on im varying volailiy will highligh how som of h wll obsrvd financial facs can b comprhndd and incorporad in h probabilisic framwork which coninus o mrg from h pr-bachlirian ra unil our prsn im.

3 h Black-chols PDE Bfor driving h Black-chols PDE w bgin by xplaining som of h arlir work dscribd abov i's fauls and h rasons h Black-chols modl is diffrn. h Bachlir work gav ris o a normal disribuion for ass prics and hnc an arihmic Brownian Moion procss of h form d σ dw Whr is h undrlying ass pric a im ; σ is h volailiy; and dw is a Winr procss. h Problm wih Bachlir's procss is ha i has a normal disribuion ha allows for ngaiv valus which is inconsisn wih ass prics obsrvd in h mark. ubsqunly prnkl Bonnss and amulson all addd faurs in hir rvision of Bachlir's modl. prnkl bgan by swiching o a Gomric Brownian Moion procss for ass prics hrough h assumpion of a lognormal disribuion. Bonnss and amulson followd sui and rspcivly addd lmns which allowd for incrass in h ra of rurn of h undrlying ass as wll as of h opion islf. pon his foundaion Black and chols imagind a risklss porfolio consrucd from a numbr of socks and opions upon h sock wih h condiion ha h porfolio posiion rurns o h invsor h risk-fr inrs ra and h cava ha h opions ar fairly pricd. Our imaginary porfolio will conain a singl Opion and a variabl quaniy of socks such ha i rmains risk-nural rgardlss of h movmn in h undrlying sock's pric. his in urn will lad o a disinc rlaionship bwn h opion pric and h sock pric which w know up o h purchas of h opion bu no afrward as wll as h volailiy of h sock pric. Black and chols mad h assumpions ha h risk-fr inrs ra and h volailiy of h undrlying sock pric ar consans hrough h xpriaion of h Opion. ransacion coss axs dividnds and h possibiliy of arbirag ar all assumd away. rading is assumd o b coninuous and happns wih no im dlay. 3 h undrlying ass can b dscribd by a sochasic diffrnial quaion of h Gomric Brownian moion form d dw d μ σ Hr o h righ hand sid of h qualiy w hav a drif rm and a diffusion rm rspcivly from lf o righ wih μ bing h annual xpcd man rurn on h sock and σ h annual volailiy of h sock pric. Boh rms ar funcions of changs For a brif ovrviw of h works mniond s Haug 7 p.4 W will addrss hs rlaionships a a lar poin. 3 Hull 6 p.36 3

4 in h sock pric and in im and hnc hy ar random. Howvr h mr fac ha a mark paricipan obsrvs hm a im allows hm o bcom consans 4 and as such w hav a Gomric Brownian Moion procss of h form d μ d σ dw A paus is ncssary hr o rais on of h cnral issus in h Black-chols modl. h krnl of h ida is ha w ar rying o modl ass prics in such a way so as o isola ou h risk faur in h procss. As such a discrning look a quaion lls us ha dw is h sourc of h risk. Following his viw h ask hn would b o consruc a porfolio ha shilds us agains h risk xprssd in dw. L us assum ha an idal porfolio would conain a shor posiion in h Opions mark and a spcific numbr of shars of h undrlying so as o limina any risk of losss in h porfolio. Our posiion can hn b xprssd as Π Δ Whrby Π is h ovrall porfolio posiion Δ is h numbr of undrlying shars w hold a im and is h currn valu or pric of h opion. I is clar hr ha h movmn in our posion as a whol dpnds squarly on h procss xprssd in quaion. I bcoms impraiv hn o calcula dπ Δd d 3 in ordr o ascrain h valu for dla which allows us o hold a risk-nural porfolio fr of h risky ffcs of dw. As w know h procss for from quaion w simply apply Io's Lmma o in ordr o obain d μ σ d σ dw 4 ubsiuing h rsuls from quaions quaion 3 w g and 4 ino h righ hand sid of dπ Δ Δ μ μ d σ W μ μ σ d Δ σ σ d σ dw σ dw 4 Nfci 996. P.7-8 4

5 Mahmaically spaking if w s Δ w hav obaind wo hings. Firs w hav uncovrd a valu for h hdg raio or h Opion Dla which w shall addrss in furhr dail lar mor imporanly in h procss w hav achivd a porfolio posiion soluion which is nirly risk-nural as i dos no includ dw. Having don his w now hav a soluion for d Π dπ σ 5 Also l us no forg ha a risklss porfolio wih a no-arbirag condiion in ffc would ncssarily grow a h risk fr inrs ra and hus dπ rπd r d 6 Upon qualizing h rsuls of quaions 5 and 6 and wih a lil simplificaion w hav now drivd h Black-chols PDE r r σ 7 3 oluion Mhods On can rach a soluion o quaion 7 in on of wo wll known ways. Firs max k w can solv h PDE wih an appropria boundary condiion for a Europan syl Call Opion or max k for a Europan syl Pu Opion 5. cond w can ak on of hs boundary condiions and us hm o calcula an xplici xpcaion for h payoff by way of probabiliis. Considring ha h scond mhod is lss complicad and provids us wih a mor inuiiv way of raching h Black-chols soluion w will us ha. Rcalling our assumpion in quaion of h Gomric Brownian Moion procss undrakn by sock prics and rplacing μ wih r o rflc h risk-fr ra of rurn njoyd by his risk-nural masur w can solv quaion for o g r σ σw 8 5 No ha h nor of h opion conrac and k is h srik pric agrd upon in h conrac. 5

6 wih W ~ N. As such calculaing h xpcd valu of quaion 9 blow will giv us h soluion for a pric of a Europan syl call opion as h xpcd valu of a h payoff of h Opion discound back ovr n as pr h Black-chols rsul. CallOpionPric [ max ] r E k 9 I is imporan o kp a fw poins in mind hr. Firs max k k which mans ha by ingraing k ovr all h k w can ffcivly figur ou h xpcaion of k valus of max. cond by way of h Cnral Limi horm w can safly rplac W wih W wihin h xpcaion and again W and hus w will simplify h xpcaion in rms of W which is h only randomnss involvd and ha way h xpcaion has a sandard normal disribuion. Hnc r r r E [ max k ] r σ / σw E max k r σ / σ W E max k k π r W r σ / σ W / r / W k σ σ ingraing his agains a random variabl wih dnsiy N ~ dw r / W W k r σ σ / σ / k dw σ π r / log W r π r σ / σ W W / r dw k log k/ r / logk/ r σ σ / σ σ Hr w bring som lmns ou of h ingraion and discovr h bounds π W / dw 6

7 log k r / σ / σ / σ π W / σ W dw k log k r / σ / r σ π W / dw Using h propris of normal dnsiis w chang h limis and simplify o g log k r / σ / σ W σ π / dw k log k r / σ / r σ π W / dw sing χ W σ and simplifying furhr w finally hav log k r / σ / σ π χ / dχ k log k r / σ / r σ π W / dw Changing h limis onc again w hav now drivd h Black-chols pric of a Europan Call Opion in h form r d k N N d Wih d log / k r σ / σ I is imporan o no hr ha for A h mony Opions d Δ or h hdg raio of our posiion and d log / k σ r σ / d d σ N rprsns h cumulaiv disribuion funcion of a sandard normal random variabl. 4 Wha w g and wha w do no g from Black-chols h Black-chols modl givs us som vry imporan informaion. Givn h following inpus: bing h sock pric k bing h srik pric r bing h riskfr inrs ra bing h im o xpiraion in yars and σ bing h volailiy of h rlaiv pric chang of h undrlying sock; w can wih nar prfc accuracy driv h pric of an opion conrac. Howvr ha is no all w can also obain a sris of rlaionships implici in h Black-chols modl which hav com o b known as "Grks" Δ Γ Θ ρ and ν vga. Rspcivly hy allow us o masur h Opion 7

8 prics' snsiiviy o movmns in h undrlying sock pric in h cas of Δ and Γ im in h cas of Θ h risk-fr ra in h cas of ρ and volailiy in h cas of ν. L us for h momn discuss wha can arguably b considrd h mos imporan of h grks for h Black-chols framwork namly Δ and Γ and ν. h rasoning hr is ha hy allow us o accuaraly hdg our posiion and mainain a risknural porfolio. Δ is dfind as h ra of chang of h opion pric wih rspc o a chang in h pric of h undrlying ass. In a shor A h Mony AM 6 call Δ will always b ngaiv and i will b approximaly -.5 or -5 in mark rms. h mchanics of Δ hdging ar as follows: if a som im bwn buying h opion and i's xrcis da Δ gos o -.75 on would hav o buy.5 of h undrlying ass in ordr o mainain AM Δ nuraliy. A shor Δ posiion mans a gain if h undrlying pric dcrass and a loss if h undrlying incrass. A long Δ posiion is h invrs. Whil vga ν is an nirly ficional grk lr i plays a somwha imporan rol as i givs us h chang in h pric of h opion if givn a chang in volailiy. ν is h parial drivaiv of h pric of h opion wih rspc o σ. his may sm rivial howvr in h ral world i hlps mark profssionals o assss h impac of hir Γ posiions. W xplaind Δ and ν in ordr o fully xplain Γ. Γ is h ra of chang of Δ wih rspc o changs in h pric of h undrlying ass. ha is o say i is h scond parial drivaiv of h opion pric wih rspc o h undrlying. A high posiiv valu for Γ dnos bing in a long AM posiion whil lowr posiiv Γ valus dno dp in h mony or ou of h mony long posiions and hnc Γ is gnrally viwd as a good indicaor of h nd o adjus h Δ posiion. A high Γ valu says ha w nd o mainain a frqun rgim of Δ adjusmns and a low Γ mans ha Δ changs may occur a a lss rapid pac and hus h frquncy of porfolio adjusmn nd no b high. A long Γ posiion also known as long convxiy or volailiy is whn w buy a call or pu and hdg our Δ wih opposi posions in h undrlying asss. his allows us o bnfi from an incras in volailiy. As on can s Black and chols hav givn us vry imporan rsuls and hy hav all sood h s of im. hr is howvr on vry wak componn in h Black- chols modl namly volailiy. From h brif dscripion of Δ ν and Γ abov w bgin o s a picur mrging in which h mark paricipans nd o coninuously sima σ h volailiy of h undrlying ass. Hnc his paricular opic will xhaus h rmaindr of h coming pags. 5 king igma In ssnc h volailiy σ w ar sking o undrsand is a saisical risk masur which allows us o guag h xn o which h undrlying ass pric may probabalisically mov wihin a givn priod of im δ. For xampl if is h pric of 6 A h Mony rfrs o whn h undrlying pric obsrvd a im and h opion srik pric ar h sam. Whn h srik pric is grar or smallr han h undrlying pric obsrvd a im is rspcivly rfrd o as In h Mony IM or Ou of h Mony OM h posiion 8

9 h undrlying ass a 7 hn or δ σ δ δ σ As such h Black-chols framwork assums σ o b som inrinsic propry of h undrlying ass whrby i dos no chang insofar as any of h ohr modl inpus ar concrnd. Luckily w can driv wha is rfrd o as h implid volailiy srucur from ou of h mony calls and pus if h consan volailiy assumpion wr ru h implid volailiy srucur would display a sraigh lin. Howvr in raliy h volailiy srucur displays ihr a "`smil"' or a "`smirk."' 8 h implid volailiy w can driv from mark quod calls and pus will in a sns giv us an inrpraion as o whhr mark paricipans ar xpcing highr or lowr volailiy. h qusion hn ariss how can w mpirically sima and forcas volailiy wihou mark quod call and pu prics in ordr o pric nw opions. Luckily hr ar wo radiional mhods hrough which w can answr his qusion namly Hisorical olailiy simaion and GARCH. 6 Hisorical olailiy 9 Going back o quaion if by Io's Lmma w can say ha δ implis ha δ ~ N μδ σ δ σ ln ~ N μ σ and ln ~ N σ μ σ ha is o say ha h sock pric follows a lognormal disribuion. hn w can us his propry o xrac informaion rgarding h disribuion of rurns of h sock bwn im and. L ξ b h annualizd coninuously compoundd ra of rurn which 7 Nfci p.7 8 Hull 6 Ch.6. 9 Hull 6 p

10 will hav h disribuion σ ξ i ~ μ σ From quaion w can ascrain ha by xprssing h rurn as coninuously compoundd w ar abl o driv h sock pric volailiy from h sandard dviaion of h rurn. As such if w hav n numbr of obsrvaions h obsrvaions bing closing prics on a paricular sock w can calcula h rurn as ξ ln i i for i 3...n; and i's man bing ξ ξ i n hn h sandard dviaion of ξ can b simad as i υ n i ξ ξ n aking ino considraion from quaion ha w know h sandard dviaion of ξ as σ hn w can sima hisorical volailiy as σ hisorical υ abl : Hisorical olailiy Esimas Horizon olailiy Daily.8 Wkly.4 Monhly.495 Quarrly.85 Yarly.76 7 GARCH Exnding Engl's work in 986 Bollrslv dvlopd h Gnralizd Auorgrssiv Condiional Hroscdasiciy or GARCH modl. In shor GARCH xplains h bhavior of varianc of a im sris by way of wo lags. h firs lag capurs high frquncy ffcs in squard pas rsiduals and h scond shows any

11 longr rm characrisics of h variancs hmslvs. hrough his rlaivly simpl saisical analysis w ar abl o rgisr wo hings. Firs w discovr h xisnc of any clusring bhavior in volailiy and scond w can obsrv any lpokurosis ffcs in h disribuion of rurns. h GARCH framwork is h simpls and mos popular of h modls availabl hrough Bollrslv's work. h ons bwn h parnhsis signify a singl auorgrssiv lag in h quaion and a singl lag in h discovry of a moving avrag. Wihin his framwork w can dfin h condiional man and varianc as μ σ ε wih ω > β γ > and β γ < σ ω βσ γσ ε As on can s abov h condiional varianc will b variabl hroughou h im sris as such if w ak h uncondiional xpcaion of all lmns in h h quaion for h condiional varianc and assuming saionariy w ar abl o solv for h uncondiional varianc as ω σ β γ hus w s h paramric naur of h GARCH framwork. h nx sp naurally would b o sima h paramrs. As h liraur suggss Maximum Liklihood Esimaion is h appropria mhod. Maximizing h following logliklihood funcion σ L logπ log σ μ σ undr an assumpion of normaliy will giv us h mos approria paramr simas. Using hs simas w can hn modl h volailiy srucur of our im sris o obain an sima for volailiy. abl : Garch Esimas Paramr Esima Errors ω.7.9 β γ.89. Σ_annual.865 σ_smi-annual.39 σ_quarrly.93 σ_monhly.538

12 8 Hson's olailiy Dspi h lganc of h Black chols modl w s clarly ha i lavs much o b disrd. h assumpion of consan volailiy snds us ino rriory whr i bcoms ncssary o sima and forcas volailiis in ordr o dal wih h raliy ha volailiy is in fac im varying. In 993 vn Hson dvisd a significan improvmn upon h Black-chols modl. Hson did no bgin dircly from whr Black and chols lf off. h work of co 987 Hull and Whi 987 and Wiggins 987 providd for a concpion of volailiy as im varying hrough a sochasic procs. Howvr his body of work did no produc a gnralizd soluion for h modlling of opion prics. Hson did jus ha. h ramn of volailiy as im varying hrough a sochasic procss is araciv howvr mos imporanly i allows h hory o mach h facs in crain fundmnal rspcs. Firsly h Hson modl provids us wih h bginnings of a scinific undrsanding of h volailiy srucur across mauriis and by xnsion h pu-call pariy. condly man rvrsion which is a paramric componn of h modl allows us o inrpr h phnomnon of volailiy clusring which is a wll obsrvd faur of financial marks. Rcalling quaion Hson modifid h undrlying ass procss o h form d μ 3 d dw Ndlss o say his follows h Black-chols framwork in ha h dpndnc of h chang in h undrlying ass rmains a funcion of h Winr procss xprssd in dw. Howvr h novly ariss as h h variaion ovr im xprssd in h volailiy is of h form d κ θ d σ dw 4 For w s a Cox Ingrsoll and Ross 985 yp man rvring procss wih κ bing h paramr dicaing h spd of man rvrsion θ h long run man volailiy and σ bing h volailiy of volailiy. Hr h rm dw is an addiional Winr procss which corrlas wih dw in quaion 3 hrough dw dw ρd In his fashion Hson has prscribd a probabilisic mark pric for h undrlying ass as wll as is volailiy risk. his modl provids us wih h luxury of fiing h paramrs κ θ σ ρ and h iniial volailiy as wll as h challng of discovring hir valus which calibra our modl o mark raliis as suggsd in scion 7. 9 h Hson aluaion Equaion Rcalling h ncssiy for a a risklss porfolio from quaion and h

13 compoundd ncssiy o valu is movmn w can rformula quaion 3 o incorpora h gnral coningn claim xprssd as paying max k a im. Again h coningn claim is o b rplicad using a slf financing or risklss porfolio. uch a porfolio will ncssarily allow for rading in h undrlying ass h mony mark and anohr drivaiv scuriy which w shall dfin as. Changs in h risklss porfolio can b xprssd as: Hr dπ Δ dπ d Δ d Δ d 5 d Δ Δ γ / γ ρσ σ Δ / γ ρ σ γ σ d 6 sing h firs and scond rms of h abov quaion qual o zro w can now solv for Δ Δ bing h numbr of unis o b invsd in h undrlying a im and Δ bing h numbr of unis o b invsd in h scondary drivaiv scuriy a im. hrough his porfolio allocaion w hav liminad any risk arising from h rms d and d in h porfolio. Givn an appropria slcion of Δ and Δ w can xprss our risk fr porfolio as dπ rπd r Δ Δ d 7 rurning o quaion 6 and combining i wih h rsul in quaion 7w can now collc h lmns of h rms praining o and rspcivly as follows d 3

14 r r / σ σ ρ γ γ r r / γ γ σ σ ρ In ordr for h abov qualiy o hold w mus assum furhr ha boh rms qualiz o som funcion f λ θ κ Hr our arbirary funcion qualizd h PDEs for and o h drif rm lss h produc of h mark pric for volailiy risk λ and. As such and by furhr assigning a valu o γ w ariv a h Hson ochasic olailiy valuaion quaion xprssd as λ θ κ σ σ ρ r r 8 Following Duffi Pan and inglon and Gahral 6 w arriv a a soluion for h valuaion quaion vry similar o h Black-chols soluion ha is o say 9 P K P r whr ar h risk-adjusd probabiliis of h log of h undrlying pric bing highr han h log srik pric a xpiraion and ha of xrcis rspcivly condiional on and. Plugging h soluion xprssd in quaion 9 ino h Hson valuaion PDE and solving using a Fourir ransform chniqu and driving h characrisic funcion as is don in Hakala and Wysup and Gahral 6 w arriv a j P j ln x lnk x x η η η τ η θ τ η π d i x i D C xp R x P j j j I is of imporanc hr o no ha h xisnc of a closd form soluion for h Hson modl rndrs h simaion of his paramr irrlvan for our purpos of solving h modl. 4

15 as h risk-nural probabiliis wih h characrisic funcion bing givn by wih Φ η xp C η τ θ D η τ j j dτ l C j η τ κ hτ log σ l D η τ h l dτ j dτ for τ j and η iη a j ijη b j κ ρσj ρσiη σ c b d d bj 4a jc g σ b d h σ h l g Equaions and 9 ar h Hson ingral and h closd form Hson soluion for h pric of a Europan syl call opion rspcivly. h ingral in quaion 9 is a rlaivly complx on which may only b approximad no calculad prcisly. Nonhlss hr w s h voluion of h Hson ochasic olailiy modl from h concpion of im varying volailiy hrough o a cohrn modl wih a closd form soluion. An alrnaiv o h approximaion rouin dscribd in h foono is o build a Mon-Carlo ngin and probabilisically dfin h valu of h opion undr h Hson Modl. For his i bcoms ncssary o rach an appropria simulaion for h Hson procsss dscribd in quaions 3 and 4. As dscribd in Haasrch and Plssr 8 h xac soluion for h Hson pric dynamic is givn by h procss r d d W and following Gahrall 6 w can us a Milsin discrizaion of h Hson Chny and Kincaid 999 xplain ha using h Basic impson's Rul w could numrically ingra a funcion. ovr wo subinrvals wih hr pariion poins. W us h rcursiv Adapiv impson's Quadraur funcion quadv in MALAB which allows us o mploy his ingraion chniqu ovr as many subinrvals as ncssary o obain a rusd lvl of accuracy. 5

16 volailiy procs givn by W κ θ d σ dw σ d 4 I is imporan o no hr ha w will dfin and h sourcs of randomnss for h pric and volailiy procsss rspcivly as W W and wih W W ~N ρ ρ ζ W ζ ~N aking ino considraion ha Hson's volailiy procss may produc ngaiv valus for w will incorpora a lowr bound in h Mon Carlo ngin whr by h valu of will b Max. Running h runcad and discrizd Hson procsss in quaions and hrough our Mon Carlo ngin an adqua numbr of ims should allow us o produc a purly probabilisic man valu for a plain vanilla Call Opion. abl 3: Opion Prics on DJIA a 993 wih rik 3 Mauriy Mark alu Modl Pric Mon Carlo Pric Jun pmbr Dcmbr Calibraing h Hson Paramrs and Rsuls I is clar ha h Hson modl rsponds adqualy o h shorcomings of h Black chols modl. h Hson framwork howvr sill lavs us wih a furhr ask. W ar lf wih h ncssiy of simaing h modl paramrs so as o obain maningful prics. o rira w nd o sima valus for h man rvrsion paramr κ long run man volailiy θ iniial volailiy h volailiy of volailiy σ and ρ h corrlaion bwn h wo dimnional Winr procsss. Bakshi Cao and Chn 997 show ha mpirical im sris simas of hs paramrs ar vry fragil in h fac of ral world mark prics for coningn claims. In ssnc onc again w fac an invrs problm whr w ar forcd o rly on obsrvd prics o obain sa paramrs. As such h calibraion procss should adhr o h following sps. Firs w 6

17 obain a vcor of liquid opion prics. Using hs prics and h pric of h undrlying scuriy w amp o valua a firs simaion as o wha h iniial paramrs w will us in h calibraion procss ough o b. cond w valua h modl prics using h inial paramrs. hird l χ b a vcor of h modl paramrs w minimiz h wighd pric diffrnc bwn h rurnd modl prics and obsrvd mark prics as follows min N χ i w i i i ModlPric MarkPric 3 Con 5 suggsd using implid volailiis as wighs howvr w follow Moodly 5 in sing w i as. h inuiion hr is ha w allow lssr bidi ask i wighs for rurnd modl prics ha gravia owards h obsrvd mid mark pric. Mor imporanly Moodly's wighing schm allows us o circumvn h fac ha our minimizaion rouin may b rurning a local minimum as oposd o a global on in such a cas w can accp h minimizaion rsuls if h squard sum of h diffrncs in quaion 3 is lss han or qual o h squard sum of h wighd diffrncs bwn h bid and offr nds of h mark. abl 4: Hson Calibraion Rsuls Paramr alu κ θ.7 σ.58 ρ W usd hr daa ss. Firs a im sris of h Dow Jons Indusrial Avrag daily closing prics from Novmbr 998 hrough May 5 8. his daa s was usd o obain an sima of h DJIA hisorical volailiy during ha priod. W also usd a im sris of DJIA monhly closing prics from Jun 3 94 hrough April 3 8 o obain paramr simas and a voliliy sima using h GARCH mhod. cond a spcrum of call opion prics wih varid sriks and hr mauriis Jun 8 pmbr 8 and Dcmbr 8 wrin on h DJIA indx wr usd o obain clos o AM implid volailiis as wll as o calibra h Hson modl paramrs. Blow ar som figurs showing h calibrad opion prics vrsus obsrvd mark prics. Conclusion I is clar from h abov ha h framwork iniiad in Bachlir s work has volvd ino an lgan and uniqu scincific mhodology. W hav prsnd an inuiiv drivaion of h Black-chols and h Hson valuaion quaions and popular Moodly 5 P.- 7

18 soluion mhods using boh modls o valu call syl coningn claims. h inadqua assumpion of consan volailiy in h Black-chols modl was ovrcom in h Hson framwork hrough h us of a man rvring sochasic procss for h volailiy rm. Nonhlss h invrs problm ncssiaing h calibraion of h Black-chols modl o ral world pric rmains inac wih h Hson modl. Albi h Hson modl allows for svral salin faurs of h ral world such as h xisnc of a volailiy smil and volailiy clusring. Going forward w would lik o xr som ffor ino an xposiion of forward volailiy srucurs as wll as h inclusion of jump procsss ino h Hson framwork. Nonhlss w hav offrd an ovrviw and a discussion of wo ky pics of liraur rlad o h pricing of coningn claims wihou which much of oday s financial raliis would rmain unralizd. 8

19 9

20 Rfrncs Bakshi G. Cao C. and Chn Z Empirical Prformanc of Alrnaiv Opion Pricing Modls. Journal of Financ Black F. and chols M h Pricing of Opions and Corpora Liabiliis. Journal of Poliical Economy Bollrslv Gnralizd Auorgrssiv Condiional Hroscdasiciy. Journal of Economrics Brandimar P. 6. Numrical Mhods in Financ and Economics: A Malabbasd inroducion. John Wily ons Inc. Chny W. and Kincaid D Numrical Mahmaics and Compuing. Brooks Col Publishing. Couraul J.M. al.. Louis Bachlir On h Cnnary of hori d la pculaion. Mahmaical Financ Carr P. 7. h alu of olailiy. Bloombrg Marks Fbruary Con R. 5. Rcovring olailiy from Opion Prics by Evoluionary Opimizaion. Cox J.C. Ingrsoll J.E. and Ross.A A hory of h rm rucur of Inrs Ras. Economrica Cuhbrson K. and Nizsch D. 4. Quaniaiv Financial Economics: ocks Bonds Forign Exchang. John Wily ons Inc. Edringon L.H. and Guan W. 5. Forcasing olailiy. Journal of Fuurs Marks Endrs W Applid Economric im ris. John Wily ons Inc. Engl R. F. 98. Auorgrssiv Condiional Hroscdasiciy wih Esimas of h arianc of Unid Kingdom Inflaion. Economrica Figlwski Forcasing olailiy. Financial Marks Insiuions and insrumns Gahral J. 6. h olailiy urfac. A Praciionr's Guid. John Wily ons Inc.

21 Hakala J. and Wysup U.. Forign Exchang Risk Modls Insrumns and ragis. Risk Books Hanslman D. and Lilfild B. 5 Masring Malab 7. Parson Prnic Hall. Haug E. G. 7. h Compl Guid o Opion Pricing Formulas. McGraw-Hill Inc. Hull J. 6. Opions Fuurs and Ohr Drivaivs. Parson Prnic Hall. Hull J. and Whi A h Pricing of Opions on Asss wih ochasic olailiis. Journal of Financ Jorion P alu a Risk: h Nw Bnchmark for Conrolling Mark Risk. McGraw-Hill Inc. Lord R. Kokkok R. and van Dijk D. 6 A Comparison of Biasd imulaion chms for ochasic olailiy Modls. inbrgn Insiu Discussion Papr. Moodly N. 5. h Hson Modl: A Pracical Approach. Univrsiy of Wiwarsrand. Nfci.. Inroducion o h Mahmaics of Financial Drivaivs. Acadmic Prss. Nfci. 4. Principls of Financial Enginring. Elsvir Inc. co L Opion Pricing Whn h arianc Changs Randomly: hory Esimaion and an Applicaion. Journal of Financial and Quaniaiv Analysis aqqu M.. Bachlir and his ims: A Convrsaion wih Brnard Bru. Mahmaical Financ - Bachlir Congrss. Accssd a van Haasrch A. and Plssr A. 8. Efficin Almos Exac imulaion of h Hson ochasic olailiy Modl Wiggins J Opion alus undr ochasic olailiis. Journal of Financial Economics