Impmnaion an Caibraion of h Exn Affin son o for Bask Opions an Voaiiy Drivaivs * viana Bykina an Ax Lvin ixh Wor Congrss of h Bachir Financ ociy orono. Jn * Prsnaion a h ixh Wor Congrss of h Bachir Financ ociy orono Jn. h viws xprss in his papr ar hos of h ahors ony an no ncssariy of h Bank of onra an Roya Bank of Canaa. Bank of onra; sviana.bykina@bmo.com Roya Bank of Canaa; ax.vin@rbccm.com
Oin Inrocion rfrncs Gnra affin iffsion mos an pricing of Eropan opions i-facor Affin Exn son mo wih ispac sochasic varianc an sochasic Gassian inrs ras an ivin yis Pric of a Eropan opion in h Exn son o wih Dispac V Pric of a varianc swap in h Exn son o wih Dispac V Gomric Avrag Bask Opion Pric in h Exn Qasi-Eipica son o Paramr Caibraion Bask Pricing an Caibraion Rss. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss.
Inrocion As obsrv in h mark h mpirica isribions of qiy og-rrns ar skw an havy-ai. In aiion qiy prics xhibi mps sochasic voaiiy csring an aocorraion in h sqar rrns. A hs propris of h sock ynamics consir in h risk nra masr rs in h smis an smirks of h corrsponing impi voaiiy srfacs. Prsn papr aboras a spcia cas of h i-facor Affin Exn son mo wih ispac sochasic voaiiy an sochasic inrs ras corra wih h nryings vop in Lvin 8 9. his iffsion mo bongs o a broa affin mp-iffsion cass of mos wihin a gnra framwork of Dffi Pan an ingon. A sysm of DE s consir in h prsnaion has on common sochasic varianc scrib by h CIR procss. ip socks hav iffrn avrag voaiiis an corraions wih his sochasic varianc proviing iffrn vs an smirks of h inivia impi voaiiy srfacs. h Qasi-Eipica son mo is xn in h affin way by iffrn Gassian ispacmns in h sock sochasic varianc. hy aow for iffrn vs of smis in h impi voaiiis an for corraions bwn sock og-rrns an sochasic -Whi inrs ras an qiy coninos ivin yis. imiar qasi-ipica consrcion for mi-facor mos hav bn consir in Lvin an chrnisr 3 Loni an chons 8 for mp-sochasic voaiiy an in many arics on sochasic im chang mos.g. Carr an W 4. A im-pnn man rvrsion v for h son sochasic varianc is consir for br fi ino h rm srcr of h A impi voaiiis an varianc swap prics. im-pnn paramrs in son mo wr consir for xamp in ikhaiov an og 3 an Zh an Zhang 7 for VIX. his papr assms ony man rvrsion v is im-pnn pic-wis consan an ohr paramrs ar consan in orr o prsrv anayica racabiiy for h Eropan opion prics an mivaria characrisic fncion.. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 3
Rfrncs Anrsn L. 8. imp an fficin simaion of h son sochasic voaiiy mo. Jorna of Compaiona Financ 3-4. Brgomi L. 8 mi Dynamics III. Risk Ocobr 9-96. Carr P. an W L. 4. im-chang Lévy procsss an opion pricing. Jorna of Financia Economics 7 3-4. Carr P. an W L. 6. A a of wo inics. Jorna of Drivaivs 3 3 3 9. Dai Q. an ingon K.. pcificaion Anaysis of Affin rm rcr mos. h Jorna of Financ LV5 943-978. Dffi D. Pan J. an ingon K.. ransform anaysis an ass pricing for affin mpiffsions. Economrica 68 6 343-376. Gahra J. 6. h voaiiy srfac. John Wiy & ons. Gassrman P. 4. on Caro hos in Financia Enginring: Appicaions of ahmaics ochasic oing an Appi Probabiiy. pringr. son. 993. A cos-form soion for opions wih sochasic voaiiy wih appicaions o bon an crrncy opions. h Rviw of Financia is 6 37-343. Loni P. an chons W. 8. ivaria smiing. Wimo agazin arch. Lvin A. an chrnisr A. 3. ifacor sochasic varianc mos in risk managmn: maximm nropy approach an Lévy procsss. anbook of avy ai Disribions in Financ E. by.. Rachv Esvir cinc B.V. 443-48.. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 4
Lvin A. 8. Affin mi-facor xnsions of h son mo for mip asss an sochasic inrs ras. Prsnaion a h Fifh Wor Congrss of h Bachir Financ ociy Lonon. Lvin A. 9. Affin xnsions of h son mo wih sochasic inrs ras. Prsnaion a h Fis Insi minar on Acaria ci. an ah. Financ orono Fb. 9. Lor R. Kokkok R. an van Dirk D. 6. A comparison of bias simaions schms for sochasic voaiiy mos. Working Papr inbrgn Insi. ikhaiov. an og U. 3. son s sochasic voaiiy mo impmnaion caibraion an som xnsions. Wimo agazin. Jy 3. Zh Y. an Zhang J. 7. Varianc rm srcr an VIX frs pricing. Inrnaiona Jorna of horica an Appi Financ 7.. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 5
Gnra Affin Diffsion mos A iffsion mo consir in his prsnaion bongs o a broa affin mp-iffsion cass of mos wihin a gnra framwork of Dffi Pan an ingon. ppos h risk nra ynamics of h sa variabs X nr h qivan maringa masr Q is fin by h foowing arkovian procss X µ X σ X W r h rif an covarianc marix ar affin in sa variabs: µ x K K x σ x σ x x K R R x x K R ; xx R Vcor W R is a sanar Q-Brownian moion wih inpnn componns. Cofficin K is im-pnn incing qaions for h sochasic variancs o provi consisncy wih h inrs ra ynamics an aow for h xac fi ino iniia qiy forwar pric crvs an varianc swap pric rm srcrs. Cofficins K an ar consan o nsr anayica racabiiy of h mo. Accoring o Dai an ingon i is sfficin for h affiniy of h iffsions wih affin rifs ha h voaiiy marix σ X is of h foowing canonica form: σ X Σ v X v X L L O L v X. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 6
r Σ is a consan marix in v X χ λ X χ R x R an X λ R. v ar affin fncions wih consan cofficins Chriio Fiipovic an Kimm 7 an Coin-Dfrsn Gosin an Jons 8 sggs mor gnra canonica form wih h nmbr of Winr procsss possiby grar han h nmbr of sa variabs consan marix Σ R x an k Gassian an k sqar roo componns: σ X Σ L L L W s h ar canonica form wih sqar roo procsss ony for h sochasic variancs. Examp of non-affin mi-facor xnsion of h son mo wih -Whi inrs ra: r r X k L L L O L r κ θ r σ W V κ θ V η V V r r V W V X X r.5v V W. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 7
ain rss for Affin Diffsion mos h xn ransform from h Dffi Pan an ingon papr can b prsn in a mor nara for h son mo form of a iscon characrisic fncion i X ϕ X E xp δ r X s F ha combins oghr a finiion of h iscon characrisic fncion an rgar mivaria characrisic fncion sing a fag δ an δ corrsponingy. Unr h sam chnica rgariy coniions as in Dffi Pan an ingon : ϕ X A B r an for a fix C h vcor-fncion B B A A saisfy h foowing compx-va ODEs: B& δρ K B B B B i A& δρ K B B B A X an h fncion Whr ρ an ρ scrib h affin fncion of h omsic shor inrs ra r ρ ρ X. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 8
Pricing of Eropan opions L G a b y no h pric of a scriy ha pays n ra nmbr y an any a an b in R. a X a im in h vn ha b X y for any a y has h foowing rprsnaion via h iscon characrisic fncion ϕ X : G b G a b y; x ϕ ia x π R ivy ϕ ia vb iv x v bx hn a pain-vania Eropan ca opion K K has a pric a im fin by h foowing forma: wih xpiraion im an srik K C Gb b n K; X K G b n K; X bx As h ca opion is in h mony whn bx n K an in ha cas pays X K vcor wih -h mn qa o on an a ohr mns qa o zro. whr b is a. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 9
h choic of a paricar mo in or papr is bas on h foowing rqirmns w wan o saisfy:. h mo sho b affin i.. wih mipicaion of Σ in h iffsion par by V i by comns. An affin mo rss in cos-form Eropan opion prics an ffciv paramr caibraion.. W rsric h mo o on-facor sochasic varianc for ach sock for simp an sab caibraion ohrwis h pairwis corraions bwn iffrn sochasic variancs an sochasic variancs an sock og-rrns n o b caibra as w. 3. W rqir sochasic inrs ras an ivin yis corra wih h qiy prics. 4. W s -Whi mo for h inrs ras an coninos ivin yis. ochasic ivin yis nsr mor raisic ynamics for h qiy forwar pric crvs. 5. W n o capr iffrn smirks an smis of h impi voaiiy srfacs. 6. h mo sho aow for accra fi ino h A impi voaiiy an varianc swap pric rm srcrs. hr ar wo ways o saisfy coniions -: - On can ak iffrn inpnn son sochasic variancs for iffrn socks. hn o hav on varianc in ach row s poin abov h corraions bwn sock prics sho b zro which is nraisic Brgomi 8 consir a wo-facor sochasic varianc wih many mor paramrs for caibraion. - On can sc on common sochasic varianc corrsponing o gnra mark aciviy an prsrv h corraions bwn sock prics. W consir h ar approach ca qasi-ipica mo. Finay w iiz Gassian ispacmns in h V o corra sock prics wih Gassian inrs ras an ivin yis.. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss.
i-facor Affin Exn son o wih Dispac ochasic Varianc an ochasic Inrs Ras an Divin Yis A gobay affin sysm of sochasic iffrnia qaions for on common normaiz son sochasic varianc V sock og-prics X n... inrs ra r ivin yis q... an ingra sochasic varianc IV V s s for varianc swaps is fin as foows: r r r r r κ θ σ q q a r W q q q κ θ σ I V V rq a q W rq... V κ θ V η V W X r q σ θ σ θ a W rq ω a θ ~ ω θ ~ ω V a W G V V ρ W ω V a W. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss.
r V V X n... whr corraions bwn r q an X corraions bwn V an X. r a q a an a fin corraions bwn X i an a fin hisorica V X an ρ ar o smmariz marix A is from h composiion of h consan corraion marix R AA wih h pair-wis hisorica corraions ρ... for h bask consins inrs ra an ivin yis as w as h caibra risk nra corraions ρ... bwn h sochasic varianc V an qiy prics. h sochasic varianc V is normaiz o on avrag an rprsns a common sochasic aciviy of h mark. Fncion θ θ θ... θm m m θm m is impnn pic-wis consan man rvrsion v for h sochasic varianc aso s in h Gassian ispacmns for consisncy wih h imiing Back-chos cas σ >... ar h sock avrag oa voaiiis κ an η ar consan man rvrsion sp an voaiiy for h sochasic varianc r an q ar sochasic risk fr ra an qiy ivin yis W rq... W G... an W... ar inpnn sanar Winr procsss for h Gassian an son componns ω ar wighs for h ispacmns an ~ ω ρ a w.. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss.
i-facor Affin Exn son o wih Dispac ochasic Varianc h firs spcia cas of h gnra mo is h cas wih rminisic inrs ras an ivin yis. W wi ca his cas h Exn son o wih Dispac V. A gobay affin sysm of DEs for on common normaiz son sochasic varianc V an sock og-prics X n... is as foows: I v V V κ θ V η V W X r q σ ω σ θ a W ~ ω θ ~ ω V W G V V ρ V V X n... ω V a W. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 3
i-facor Affin Exn Qasi-Eipica son o h scon spcia cas anayz in his papr is a so-ca Qasi-Eipica i-facor mo wih Dispac ochasic Varianc for zro Gassian ispacmns. W wi ca his cas h Exn Qasi-Eipica son o. h nam coms from h fac ha a mivaria isribion of sock rrns is qasi-ipica for zro corraions ρ bwn V an X. V X κ θ V η V W σ V r q V σ V ρ W a W V V X n.... Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 4
. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 5 Pric of a Eropan opion in h Exn son o wih Dispac V A cos-form pricing forma for Eropan ca opion on sock wih srik K mariy an payoff K is as foows: P r K P q Ca r V K P P n n n ar wo Forir ransforms for n b n : [ ] { } x x i V ib x B ib x A q r K x i P n n G n / n xp R σ π A A A G G ~ G i A θ ω σ n G G A ς θ η κ γ ς G B G ς ς 4 ξ γ ς i ησ ρ κ ς ~ i i ω ξ σ η γ...
. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 6 Pric of a varianc swap in h Exn son o wih Dispac V h varianc swap is a forwar conrac on h raiz annaiz varianc: var og og i i i K A a wap Varianc r is h noiona amon of h swap A is h annaizaion facor an var K is h srik pric. h rif rm in h abov payoff may or may no appar. h pric of h varianc swap in coninos im is fin as: s s V E K var h corrsponing varianc swap pric forma var K for h inivia sock xns h sanar son mo pric forma s for xamp Gahra 6: κ κ κ θ κ κ ω θ ω σ var ~ ~ V K
. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 7 Gomric Avrag Bask Opion Pric in h Exn Qasi-Eipica son o h affin mo aows for a cos-form pricing forma for a Gomric Avrag Bask GAB Eropan opion wih h payoff of h form: xp K X K β β h soion can b prsn in h foowing form: xp P r K P Q Ca GAB β β r V K P P n n n ar wo Forir ransforms for n b n : x x i V ib x β B ib x β A Q β r /K i x P n n β n n xp R π n G G A ς θ η κ γ ς G B G ς ς 4 ξ γ ς i ρ σ η κ ς σ η γ k k k k ρ σ σ σ ξ...
Paramr Caibraion For h mo caibraion w consir on s of paramrs V κ η θ for h normaiz common sochasic varianc an iffrn paramrs σ ρ ~ ω for ach bask componn. h caibraion is achiv by soving an opimizaion probm of h wigh as sqars fi ino h mark impi voaiiis an whr avaiab h varianc swap prics.g. VIX rm rcr: ivo k ivo k Varwap z Varwap min F x min wiv k wv z x x k z ivo k Varwap z ikhonov rgarizaion was impmn o improv sabiiy of caibraion. s Fig.. Join fi ino h &P 5 impi voaiiis an VIX rm rcr significany improvs h varianc swap rm srcr approximaion wiho affcing h qaiy of h impi voaiiy approximaion. h &P 5 &VIX oin caibraion wih im pnn θ cras RE for VIX rm rcr by 5% ovr consanθ. h caibra son paramrs an RE ar prsn bow θ is on Fig. : V.44 κ 3.786 η.448 ρ.7 σ.34 RE.5 Raiv RE.49. s Fig. a-cab compars h caibraion of oifi Qasi-Eipica son mo an Affin Exn son o wih Dispac ochasic Varianc for a bask of socks D O& F. h s of ispacmns ~ ω cras h obciv fnciona by 7% an improv h sabiiy of θ. z. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 8
6% 5% 4% PX Impi Voaiiy Caibraion. o ariy ark Imp.Vo. ony Imp.Vo.VIX rm rc. 6% 5% 4% PX Impi Voaiiy Caibraion. 3 o ariy ark Imp.Vo. ony Imp.Vo.VIX rm rc. 3% 3% % % % % %.4.5.6.7.8.9....3.4.5 %.4.5.6.7.8.9....3.4.5 6% 5% 4% 3% % % % 3% 8% 6% 4% % PX Impi Voaiiy Caibraion. 7 o ariy ark Imp.Vo. ony Imp.Vo.VIX rm rc..4.5.6.7.8.9....3.4.5 VIX rm rcr Approximaion ark Imp.Vo. ony ha Imp.Vo.VIX rm rc. Imp.Vo.VIX rm rc cons ha..5..5..5 6% 5% 4% 3% % %.4...8.6.4.. % PX Impi Voaiiy Caibraion. 5 o ariy ark Imp.Vo. ony Imp.Vo.VIX rm rc..4.5.6.7.8.9....3.4.5 rm rcr of an Rvrsion Lv for V Imp.Vo. ony ha Imp.Vo.VIX rm rc. Imp.Vo.VIX rm rc. cons ha..5..5..5 3. Fig.. Caibraion o &P 5 Impi Voaiiis wih an wiho Fiing ino VIX rm rcr. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 9
6% 5% 4% D Impi Voaiiy Caibraion. o ariy ark ing sock son Bask son Bask Affin G 6% 5% 4% D Impi Voaiiy. o ariy ark Inivia caibraion Bask caibraion Bask Affin G 3% 3% % % % % %.4.5.6.7.8.9....3.4.5 %.4.5.6.7.8.9....3.4.5 6% 5% 4% D Impi Voaiiy Caibraion. 5 o ariy ark ing sock son Bask son Bask Affin G- 6% 5% 4% D Impi Voaiiy Caibraion. o ariy ark ing sock son Bask son Bask Affin G- 3% 3% % % % % %.4.5.6.7.8.9....3.4.5 %.4.5.6.7.8.9....3.4.5 Figr a. Bask Caibraion Rss for Affin Exn son o for om Dpo. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss.
6% O Impi Voaiiy. o ariy 6% O Impi Voaiiy. 4 o ariy 5% 5% 4% 4% 3% 3% % % % ark Inivia caibraion Bask caibraion Bask Affin G.4.5.6.7.8.9....3.4.5 % % % ark Inivia caibraion Bask caibraion Bask Affin G.4.5.6.7.8.9....3.4.5 6% O Impi Voaiiy. 7 o ariy 6% O Impi Voaiiy. o ariy 5% 5% 4% 4% 3% 3% % % % ark Inivia caibraion Bask caibraion Bask Affin G.4.5.6.7.8.9....3.4.5 % % % ark Inivia caibraion Bask caibraion Bask Affin G.4.5.6.7.8.9....3.4.5 6% O Impi Voaiiy. o ariy 5% 4% 3% % % % ark Inivia caibraion Bask caibraion Bask Affin G.4.5.6.7.8.9....3.4.5 Figr b. Bask Caibraion Rss for Affin Exn son o for onsano. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss.
F Impi Voaiiy. o ariy F Impi Voaiiy. 4 o ariy 6% 5% 4% ark Inivia caibraion Bask caibraion Bask Affin G 6% 5% 4% ark Inivia caibraion Bask caibraion Bask Affin G 3% 3% % % % % %.4.5.6.7.8.9....3.4.5 %.4.5.6.7.8.9....3.4.5 F Impi Voaiiy. 7 o ariy F Impi Voaiiy. o ariy 6% 5% ark Inivia caibraion Bask caibraion Bask Affin G 6% 5% ark Inivia caibraion Bask caibraion Bask affin G 4% 4% 3% 3% % % % % %.4.5.6.7.8.9....3.4.5 %.4.5.6.7.8.9....3.4.5 F Impi Voaiiy. o ariy 6% 5% ark Inivia caibraion Bask caibraion Bask Affin G 4% 3% % % %.4.5.6.7.8.9....3.4.5 Figr c. Bask Caibraion Rss for Affin Exn son o for icrosof. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss.
Caibra paramrs Caibraion Rss for h UD Bask of D O an F oifi Qasi-Eipica son mo wih im pnn ha Affin Exn son mo wih im pnn ha an ispacmns V.966.335 Kappa.63 7.639 Ea 9.989 9.975 Rho D -.573 -.6 Rho O -.3 -.5 Rho F -.494 -.534 igma D.45.53 igma O.35.3 igma F.7.76 Dispacmn for D..4 Dispacmn for O..7 Dispacmn for F..4 ha.34.533 ha.45.74 ha 4.78. ha 5.45.7 ha 3 Y.. Fina min fnciona.88.678 RE.5.4 Raiv RE.8.76 ab. Comparison of caibraion rss for UD bask wih an wiho Gassian ispacmns. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 3
Bask Pricing an Caibraion Rss s 3 monsras h mi-facor mo caibraion an pricing rss for an arihmic bask opion β X K of EF s XF XEG XA an XI rprsning for maor sb-inics 98.5% of h orono ock Exchang Inx. h inx isf is rprsn by h EF wih h ickr XIU. h qaiy of caibraion was s by comparison of h mark prics for h XIU Eropan ca opions for varios mariis consir as opions on h bask wih h sima bask opion prics sing fix hisorica qiy corraions an mo caibra paramrs. h on Caro simaion was bas on h mhos from Anrsn 8. On avrag h abso iffrnc in h horica an mark bask opion prics was 5.%. hn h hisorica qiy corraions wr as o br fi ino h inx opion prics. h obain impi qiy corraions wr highr han h hisorica. Bask Componn am ark pric Bask wigh XIU 5.4.% XF 8.7 33.7% XEG 6.4 3.7% XA 5.97.3% XI 5.77.3% ab. XIU bask composiion an wighs. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 4
isorica corraions XF XEG XA XI XF..64.6.59 XEG.64..693.46 XA.6.693..4 XI.59.46.4. Impi corraions XF XEG XA XI XF..756.397.6747 XEG.756..7943.5656 XA.397.794..3679 XI.675.566.368. ab 3. Impi vrss hisorica qiy corraions for h bask of EF s isorica corraions isorica corraions wih Impi corraions Impi corraions wih consan ha rm srcr of ha wih consan ha rm srcr of ha 5.%.%.65%.3% ab 4. Abs. avrag rror for h XIU Inx Eropan opion pric vs. Bask opion pric. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 5
s 4 compars cos form pric wih on Caro sima pric for Gomric Avrag Bask opion. h s focs is o vrify h anayica xprssion for h Gomric Avrag Bask opion in h Exn Qasi-Eipica son an s h obain anayica soion as a conro varia in pricing of h Arihmic Avrag Bask opion. h approach is o s homognos bask firs wih h sam wighs an iniia sock prics b iffrn corraions. Afr ha h obain anayica pric for h Gomric Avrag Bask opion is s as a conro varia for h homognos Arihmic Avrag Bask opion. h opima cofficin simaion givn anohr op 4 * b ha minimizs h varianc of h X... X n wih h known xpcaion ] σ Y b* ρ σ X XY Cov[ X Y ] Var[ X ] Y...Yn ops from n rpicaions of a E [X is as foows Gassrman h raio of h varianc of h opimay conro simaor o ha of h nconro simaor is Var[ Y b * X E[ X ]] ρ Var[ Y ] Whr X an Y ar samp mans. h s rss monsra significan improvmn in h accracy achiving h avrag varianc raio of.75 for b *. 34. For h non-homognos bask of EF s XF XEG XA an XI h rss ar no as goo as for h cas of homognos bask b si saisfacory rsing in h avrag varianc raio of.89 for b *. 5. XY. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 6
Gomric Avrag Bask Opion Anayica pric for homognos bask Opion ariy y. I A O.45..5683.64.378.47.88.3555.567.6.38.5859 Opion ariy y. K simaions simaions I A O I A O Gomric Bask Avrag Opion C QE Pric.45.4.5686.69.3.5687.69.378.446.84.3578.4.83.3566.567.64.398.587.69.39.587 Arihmic Bask Avrag Opion C QE Pric.45.487.585.686.473.585.696.378.4735.844.386.477.839.3795.567.764.6.635.744.53.6354 Arihmic Bask Avrag Opion wih Gomric Bask Avrag as conro varia.45.463.5845.69.46.5844.69.378.468.8373.378.4678.837.378.567.6999.6.636.6995.6.637. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 7
Gomric Avrag Bask Opion Anayica pric for nonhomognos bask Opion ariy y. I A O.45.769.45.473.378.48.3585.95.567.88.65.459 Opion ariy y. K simaions simaions I A O I A O Gomric Bask Avrag Opion C QE Pric.45.7634.45.473.765.4.478.378.43.366.969.4.3597.96.567.847.64.469.837.67.47 Arihmic Bask Avrag Opion C QE Pric.45.637.6365.36.6.6357.369.378.8995.974.6758.8965.974.675.567.5.774.9837.498.766.9838 Arihmic Bask Avrag Opion wih Gomric Bask Avrag as conro varia.45.6.6364.36.6.635.364.378.896.969.6737.894.969.6739.567.47.75.986.457.738.985. Bykina an A. Lvin Impmnaion an Caibraion of Exn Affin son o for Bask Opions an Voaiiy Drivaivs. 6 h BF Congrss. 8