Option Pricing in a Fractional Brownian Motion Environment

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Gwen Payne
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1 Opo Pcg a acoal owa Moo vom Cpa Ncula Acamy o coomc u ucha, omaa mal: cpc@yahoo.com h a: buay, Abac h pupo o h pap o oba a acoal lack-chol omula o h pc o a opo o vy [, ], a acoal lack-chol quao a a k-ual valuao hom h ulyg v by a acoal owa moo, < <. o h pupo w wll pov om ul gag h qua-cooal xpcao, pcally h bhavo o a Gaov aom. W wll alo compa ou ul wh h clacal ul ba o h aa owa moo a w coclu ha h ca o h acoal owa moo h pc o h opo o log p oly o.. Iouco I < < h acoal owa moo m wh u paam h couou Gaua poc {,, wh ma [ ] a who covaac gv by: C [ ] {, I h coc wh h aa owa moo. h acoal owa moo a l-mla poc mag ha o ay α α ha h am law a α.

2 h coa m h g o h covaac o h uu a pa cm. h covaac pov wh, zo wh a gav wh <. Aoh popy o h acoal owa moo ha o ha log ag pc h ha w pu Cov, h h l-mlay a log-ag pc pop mak h acoal owa moo a uabl ool applcao lk mahmacal ac. c o h acoal owa moo h a Makov poc, o a mmagal, w ca o u h uual ochac calculu o aalyz. Wo ll a a pahw gao hoy o acoal owa moo wa vlop L 995, Dcuo a Uul 999 wa pov ha h mak coul hav abag og 997. h acoal owa moo wa o log co o mahmacal molg ac. owv a h vlopm o a w k o gal ba o h Wck pouc Duca, u a Pak-Duca, u a Okal call acoal Io gal, wa pov u a Okal ha h copog Io yp acoal lack-chool mak ha o abag. I h am mahmacal mol v by pap u a Okal a omula o h pc o a uopa opo a v. h pupo o h acl o x h omula o vy [, ]. W oba k-ual valuao omula a a acoal lack-chol quao. W wll alo aalyz h vy cao. h pap ogaz a ollow : co, w m om ul o acoal Io gal, co 3 w pov om ul gag h quacooal xpcao, pcally h bhavo o a Gaov aom, co 4 w apply h ul h uy o h uopa opo.. ackgou I h co w wll p om ul w wll o h o h pap. o mo apc o h ma you may coul h uamal pap cocg acoal Io gal Duca, u a Pak-Duca, u a Okal. o a x, < < : L φ, : mauabl. h L φ φ :, φ <

3 Dg h pouc w hav ha g, g : φ, φ φ,, φ L a lb pac. I lm L φ : wh a [, : a a Lmma. Io omy I h I ε L φ L φ : xp φ φ { Lmma. h la pa o ε L µ h pobably law o L. b h m polyomal. h, φ x x x I w pu ha ω, ω L φ I α α,...,α I a ohogoal ba µ a w o: α x,. L wh µ, h o all mul-c o ogav g, L Lmma 3. u a Okal ω : h ω, h ω,... h, α α α ω Lmma.3 acoal W-Io chao xpao hom L L µ h h x coa a wh α : α! α!... α! c a α I uch ha: α! X ω c ω µ X L α I α α α c! µ α α I L X. 3

4 D h o all omal xpao G ω c ω uch ha G, q α I α! c α α I α α αq N < o om q N I ω a ω α α a G ω b α α ω α I Wck pouc o a G by G Lmma.4 L g L ε α I ω a b ω, φ α β α β α, β I. W hav ha, g φ ε g ε g ε ε g h acoal wh o W a m by: W W v φ, v v W hav ha W a w h. I Y : a uco uch ha Y W gabl acoal Io gal by: Y : Y W h Lmma.5 Gomc acoal owa moo Co h acoal al quao: X µ X X, X x W hav ha: X x xp µ o h o o h pac φ D X, D X coul u a Okal., L φ a o h Mallav vav Lmma.6 acoal Io omula Co h acoal al quao: I C, µ ω, ω, µ, L X h w hav:, φ 4

5 x, X, X, X, X µ x φ, X x, X D X Lmma.7 acoal Gaov omula L a γ a couo upp, upp, uch ha uco wh γ [ ] a a uco wh [ ], γ,, upp [, ] by: γ φ L D a pobably mau µ o h - algba, h Do L µ µ xp ω, γ a acoal owa moo u µ. φ h pac o uco ha a ymmc wh pc o vaabl a <. L W h a gal: I : :!,...,... <... < Lmma.8 acoal W-Io chao xpao hom m o L µ L uch ha: a gal L a X. h h x X ω µ I φ L X L L. µ! W ay ha a omal xpao blog o h pac G G q, G q g q N!, g Lφ g L q < L U G. W hav ha L µ G G q N q 5

6 L G g G. W h qua-cooal xpcao o G wh pc o, Lmma.9 a L G b L by: [ G] : [ G ]: g. W hav ha [ ] G, G G. W hav ha [ G] [ ] [ G] c L L µ. [ ] - mauabl W ay ha a - aap ochac poc,ω M G, a [ M ] M,. Lmma. a qua-magal a b L L φ a ε : xp [ ], φ ε a qua-magal c L, L φ M a qua-magal a M :, ω. W hav ha M a quamagal Lmma. acoal Clak-Oco hom a L G a b L L µ - mauabl. h [ D ] G [ ] [ ] ω a D W a, - mauabl. h [ D ] [ ] [ ] ω D. W hav ha L φ a 6

7 3. om ul gag h qua-cooal xpcao L Ω,,µ a pobably l uch ha,ω moo wh pc o µ. a acoal owa hom 3. o vy < < a λ C w hav [ ] λ λ λ Poo: Co h acoal al quao: λ X, X X Ug Lmma.5 w hav ha: c X xp λ λ 3. X ug Lmma. c ollow ha λ X o [ X ] X [ ] λ λ λ q hom 3. L b a uco uch ha [ ] <. h o vy [ ] x xp xx 3.3 Poo: L b h ou aom o : x x x 7

8 h h v ou aom o : W hav ha: x x I ollow ha: [ ] [ ] h 3.4 wh h h v ou aom o h pouc bw a. u h uco h ou aom o x xp, x 3.5 Ug h ac ha h ou aom o a covoluo h pouc o h ou aom o h wo uco ollow ha y y h, y q.. Coollay 3.3 A. h L [ ] x A x xp 3.6 A 8

9 9 L. Co h poc, 3.7 Lmma.7 au u ha h a mau µ uch ha a acoal owa moo u µ. W wll o [] h qua-cooal xpcao wh pc o µ. Co [ ], xp ε 3.8 hom 3.4 L b a uco uch ha [ ] <. h o vy [ ] [ ] 3.9 Poo: Aga w wll o by h ou aom o. W hav [ ] [ ] 3.

10 O h oh ha [ ] 3. h ul ollow om 3. a 3. q...

11 4. Applcao o Mahmacal ac Co a acoal lack-hol mak ha ha wo vm pobl:. a moy mak accou: M M M, wh p h coa kl a.. a ock who pc a h quao:, 4.,, δ 4. wh δ, a coa. u a Okal hav how ha h mak o o hav abag a compl. U h k-ual mau µ w hav ha:,, 4.3 W wll o by [ ] k-ual mau. h qua-cooal xpcao wh pc o h gv by hom 4. acoal k-ual valuao h pc a vy [, ] Poo: o a bou [ ] - mauabl clam L µ 4.4 c h mak compl h a plcag poolo o h clam m, who valu : a m M W hav ha m M

12 y mulplyg wh a gag ollow ha, 4.5 y h acoal Clak-Oco hom Lmma. w hav ha [ ] [ ] D 4.6 om h compl o h mak w g [ ] D, 4.7 o w hav ha [ ] I ollow ha [ ] [ ] 4.8 Ug Lmma. w g ha [ ] [ ] 4.9 om 4.5 a 4.9 w hav ha [ ] q.. hom 4. acoal lack-hol omula h pc a vy [ ], o a uopa call opo wh k pc a mauy gv by, N N C 4. wh l a l a N h cumulav pobably o h aa omal buo.

13 3 Poo: W hav ha [ ], max, C X { { X { { u w o by l w g x { { xp x x z z xp z z xp N 4. Co h poc, 4. Lmma.7 au u ha h a mau µ uch ha a acoal owa moo u µ. W wll o xp 4.3

14 4 Ug hom 3.4 w hav ha x { { x { x { { u l l l 4.4 I w o l w g x { { xp x x z z xp z z xp N 4.5

15 o { N N 4.6 om 4. a 4.6 w g h pc o h opo. q.. hom 4.3 acoal lack-hol quao h pc o a vav o h ock pc wh a bou payo gv by D,, wh D, h oluo o h PD: D D D D D, 4.7 Poo: om hom 4. a hom 3. ollow ha h pc o h vav a a mom a uco o a. A h clacal lack-hol mol w co a poolo ha coa a vav a ock. h valu o h poolo D Π, 4.8 U h mak mau µ ug acoal Io omula Lmma.6 a h ac ha D D u δ D u a w g ha φ D [ ] u, φ, u u D Π, D D δ D δ D 5

16 W wa h poolo o b kl. o D a Π I ollow ha valuao quao gv by: mak D D Π 4.9 D D q.. A h clacal mol h quao o o p o δ h pc o h uopa call gv by hom hom 4. h oluo o h quao 4.7 wh h bouay coo: D, max, hom 4.5 h Gk h Gk a gv by: C N C ϑ C ρ C Θ N C C Γ N N 4. wh z z Poo: W wll v a gal omula. L y b o o h luc aco. W hav C N N N N 6

17 u N xp xp xp xp xp xp xp l xp xp I ollow ha: C N N 4. ubug 4. w g h Gk. q.. mak h acoal lack-hol pc o a uopa call opo o log p oly o. A ao may b h ac ha h acoal owa moo ha log mmoy. h pc o a opo a a mom [, ] wll p o h ock pc, bu p h clacal lack-hol mol, wll ak o coao h voluo o h ock pc h po [,]. h luc lc h acoal lack-hol omula by h u paam. Co h mom a wo opo wh mauy o o hm w o a h oh o o. I h clacal lack-hol mol h pc o h wo opo a h mom w qual. I h acoal lack-hol mol h pc o h wo opo a h mom a o log qual. Du o h log mmoy popy, h pc o h. opo alo luc by h voluo o h ock pc h po [ ], 7

18 c Dcuo, L a A.. Uul 999. ochac aaly o h acoal owa moo. Poal Aaly,, 77-4 Duca,.., Y. u a. Pak-Duca. ochac calculu o acoal owa moo. I. hoy. IAM J. Cool Opm. 38, u, Y. a. Okal. acoal wh o calculu a applcao o ac. Pp, Uvy o Olo u, Y.,. Okal, a A. ulm. Opmal coumpo a poolo a lack-chol mak v by acoal owa moo. Pp, Uvy o Olo 3/. L,.J ochac aaly o acoal owa moo, acoal o a applcao. IAM vw,, og, L.C.G 997. Abag wh acoal owa moo. Mahmacal ac, 7,

By Joonghoe Dho. The irradiance at P is given by

By Joonghoe Dho. The irradiance at P is given by CH. 9 c CH. 9 c By Joogo Do 9 Gal Coao 9. Gal Coao L co wo po ouc, S & S, mg moocomac wav o am qucy. L paao a b muc ga a. Loca am qucy. L paao a b muc ga a. Loca po obvao P a oug away om ouc o a a P wavo