Numerical Computation of Theta in a Jump-Diffusion

Model by Integration by Parts

Delphine David, Nicolas Privault

To cite this version:

Delphine David, Nicolas Privault. Numerical Computation of Theta in a Jump-Diffusion Modelby Integration by Parts. [Research Report] RR-5829, INRIA. 2006, pp.32. <inria-00070196>

HAL Id: inria-00070196

https://hal.inria.fr/inria-00070196

Submitted on 19 May 2006

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ISS

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249-

6399

ISR

N IN

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29--

FR

+E

NG

ap por t de r ech er ch e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Numerical Computation of Theta in aJump-Diffusion Model by Integration by Parts

Delphine David — Nicolas Privault

N° 5829

February, 8 2006

Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

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Thetat = rtC(x, t, T ) − xrt∂C

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Huht~iL~vwuh#~z|rr~uÇwy~atktrr(Xt)t∈R+

z¤uhwkjLkjuhwkjuas~H³(Wt)t∈R+

[§z¤s0fB xj¦aq:iOk r0vwyk

µ(dy)Huh :huwz¤sk7z¤uaskjuhr0z¤s¬q

λ =

∫ ∞

−∞yµ(dy),

§fwz|%fÇ Huk<ykjwyktrkju-sktr

Xt =

Nt∑

k=1

Uk, t ∈ R+,µ´[½FE¹¸

§fhkjyk(Nt)t∈R+

z|r(#~z|rr~uwy~?tktrr§z¤s0fz¤u-skjuhr0z¤s¾qλHuh

(Uk)k≥1z|rHuzݽ zݽÃϽ#rktavhkjuhtk

~H³8y%Huhw~i ¦Hy0zdHw¥|ktr@§z¤s0fÇwy~3Hwz¤¥¤z¤s¾qÇ[z|r0s0y0z¤wvws0z|~uν(dx) = λ−1µ(dx)

½ ADk"wkjuh~sk-q(Ft)t∈R+

s0fhk :h¥¤s0y%Hs0z|~ukjuhkjy%Hsktaq(Wt, Xt)t∈R+

½

ADk:t~uhr0z|wkjyRy%[z|kju-s·Huh¼[z¤¦kjy0kjuhtk¶~kjy%Hs~yrDHuh

δjs0z¤uwÄ~u s0fhk¯t~uas0z¤u-vh~vhr

t~i~uhkju-s1~H³¿²vwi[«¬[z®­Lvhr0z|~uy%Huhw~i ³Jvwuhjs0z|~u3H¥|r¹½ B kjsD : L2(Ω) → L2(Ω × R+)wkjuh~sks0fhk·µJvwu-~vwuhwkt3¸nH¥¤¥¤zdt¦?z¤uRy%[z|kju-s(~us0fhkA z|kjuhkjyr03tkR[zݽÃkR½

DtF (ωW , ωX) =

n∑

k=1

1[0,tk](t)∂kf(Wt1 , . . . ,Wtn , ωX)

îî ëHJILK±õLM

0% ?, 5/ L 5

³´~yF)y%Huhw~i ¦RHy0zdHw¥|k~H³s0fhk<³´~y0i

F (ωW , ωX) = f(Wt1 , . . . ,Wtn , ωX),

§fhkjykf(·, ωX) ∈ C∞

b (Rn)PX(dωX)

«¾[½Ãr¹½¤Wz|r<vwuwz®³´~y0i¥¤q~vwuhwktÄ~uΩX

½Á<kjuh~sk-q〈·, ·〉L2(R+)

Huh ‖ · ‖ s0fhkr H¥dHywy~?[vhjs@Huh¯uh~y0i z¤uL2(R+)

3Huhwk:huhk

DuF = 〈u,DF 〉, u ∈ L2(Ω × R+).

B kjs

In(fn)(ωW , ωX) = n!

∫ ∞

0· · ·∫ t2

0fn(t1, . . . , tn, ωX)dWt1 · · · dWtn

wkjuh~sk@s0fhk@i+vw¥¤s0z¤w¥|k<r0s~?f3r0s0z|@z¤u-skjRy%H¥W~H³Ws0fhk(r0q?iiOkjs0y0z|³\vwuhjs0z|~ufn ∈ L2(Rn

+ ×ΩX)§z¤s0fÇyktr0Lktjss~·Ê¨y~¹§uwzdHuÇiO~s0z|~u(Wt)t∈R+

½Ìkt H¥¤¥Ïs0f3Hs§©k7f3t¦k

DtIn(fn) = nIn−1(fn(·, t, ωX)), t ∈ R+,

Huhs0fhk<z|r~iOkjs0y0q:³\~y0i"vw¥d

E[In(fn)Im(gm)] = n!1n=mE[〈fn, gm〉L2(Rn+

)].

@u-qF ∈ L2(ΩW × ΩX) ' L2(ΩW ;L2(ΩX))

[iz¤srÇ%f3~Rrwktt~iL~Rrz¤s0z|~u¼~H³s0fhk³´~y0i

F = E[F ] +

∞∑

n=1

In(fn),µ´[½í¸

§fhkjykfn ∈ L2(Rn

+ × ΩX)n ≥ 1

HuhÇs0fhkw~i·Hz¤uDom (D)

~H³Dt~uhr0z|r0srz¤us0fhk+r%kjs

~H³#³\vwuhjs0z|~u3H¥|rF§y0z¤s0skjuÆr+µ´[½í¸Huhr%Hs0z|r²³\qaz¤uw

E

[ ∞∑

n=1

n!n‖fn‖2L2(Rn

+)

]

< ∞.

¾u+s0fhk¨rktavhkj¥-§¨k¨§z¤¥¤¥?kjuhkjy%H¥¤¥¤q+[y~"s0fhk¿z¤uh[z|tktrωW

ωX

½ eBfhk<µJvwu-~vwuhwkt3¸ [z¤¦kjy0kjuhtk~kjy%Hs~y

δ : L2(Ω × R+) → L2(Ω)H²~z¤uas+~H³

D H¥|r%~¶ H¥¤¥|kts0fhk1paË~y~fh~aDz¤u-skjRy%H¥Ý

r`Hs0z|r:3ktrs0fhk7[v3H¥¤z¤s¬q¶ykj¥dHs0z|~u

E[〈DF, u〉] = E[Fδ(u)], F ∈ Dom(D), u ∈ Dom(δ),

Huhs0fhk7[z¤¦kjy0kjuhtk7³´~y0i+vw¥d

δ(uF ) = Fδ(u) − DuF, u ∈ Dom(δ), F ∈ Dom (D),µ´[½í¸

#î

! "#$%&(')+*, -.0/1$2,0 <

§fwz|%fr0fh~¹§rs0f3HsuF ∈ Dom(δ)

wy~¹¦az|wktuF ∈ L2(Ω × R+)

Huh¼s0fhk¶y0z¤Rf-s²«Ýf3Huhrz|wkkj¥|~uwr@s~

L2(Ω)½@kt H¥¤¥#H¥|r~1s0f3Hs

δt~z¤uhjz|wktr@§z¤s0f¾s @wÂÃr@r0s~?f3rs0z|z¤u-skjRy%H¥8~u

r%-v3Hyk±«Ýz¤uaskjRy%Hw¥|k"hHwsktwy~?tktrrktr¹[z¤u¶3Hy0s0z|jvw¥dHy

δ(u) =

∫ ∞

0utdWt

³´~yH¥¤¥ hHwsktÆHuhÇr-v3Hyk±«Ýz¤uaskjRy%Hw¥|kwy~?tktrr(ut)t∈R+

hHuh

δ(u) = I1(u), u ∈ L2(R+).

¾u!s0fhkÆrkt-vhkj¥§¨k§z¤¥¤¥(t~uhr0z|wkjy¶ÄnHy0Ë~¹¦azdHuŲvwi[«¬[z®­Lvhr0z|~u wy0z|tkwy~?tktrr(St)t∈R+Rz¤¦kjuÇ-q

dSt = at(St)dt + bt(St)dWt + ct(St−)dXt,

S0 = x,§fhkjyk

at(·)bt(·)

ct(·)

Hyk C1 BÏz¤hr%fwz¤s0À+³\vwuhjs0z|~uhr vwuwz®³\~y0i¥¤q¶z¤ut ∈ [0, T ]

T > 0

½¾us0fhk rk~H³W<s0z¤iOkfh~iO~kjuhkt~vhr¨kt~iOkjs0y0z|iO~?wkj¥3vwuhwkjyÌs0fhky0z|r0Ëuhkjvws0y%H¥3iOk r0vwykR-s0fhkt~?k4·jz|kju-sr

a(·) b(·) Huh c(·) §z¤¥¤¥WLk<Rz¤¦kjuÇ-q

a(y) = y(r − λη(y)),

b(y) = yσ(y),

c(y) = yη(y),

§fhkjykrHuh

σ(·) η(·) ykjwyktrkjuas:s0fhkz¤uaskjyktr0s¯y%HskR@Huh s0fhkt~uas0z¤u-vh~vhrÇHuh ²vwi¦~¥dHs0z¤¥¤z¤s¾q¯³Jvwuhjs0z|~uhr¹½

¾s @wÂÃrB³´~y0i+vw¥d)³´~y(St)t∈R+

yk wr

φ(St) = φ(Ss) +

∫ t

sφ′(Su)au(Su)du +

∫ t

sφ′(Su)bu(Su)dWu

+1

2

∫ t

sφ′′(Su)b2

u(Su)du +∑

s<u≤t

(φ(Su− + cu(Su−)∆Xu) − φ(Su−)) ,

0 ≤ s ≤ thHuhqaz|kj¥|wr

E[φ(St)] = E[φ(Ss)] + E

[∫ t

sφ′(Su)au(Su)du

]

+1

2E

[∫ t

sφ′′(Su)b2

u(Su)du

]

îî ëHJILK±õLM

0% ?, 5/ L 5

+λE

[∫ t

s

∫ +∞

−∞(φ(Su + zcu(Su)) − φ(Su))ν(dz)du

]

.µ´[½ a¸

3~yk±°wHiw¥|kRhz®³Xt = a1N

1t + · · · + adN

dt , t ∈ R+,

µ´[½í¸§fhkjyk

(Nkt )t∈R+

k = 1, . . . , d

BHykz¤uhwkjLkjuhwkjuas1#~z|rr~uwy~?tktrrktrO§z¤s0f!yktr0ktjs0z¤¦kz¤uaskjuhr0z¤s0z|ktr

λ1, . . . , λd[§¨k+f3t¦k

λ = λ1 + · · · + λdHuh

ν(dx) =λ1

λδa1

(dx) + · · · + λd

λδad

(dx),

Huh

E[φ(St)] = E[φ(Ss)] + E

[∫ t

sφ′(Su)au(Su)du

]

+1

2E

[∫ t

sφ′′(Su)b2

u(Su)du

]

+d∑

k=1

λkE

[∫ t

s(φ(Su + akcu(Su)) − φ(Su))du

]

, 0 ≤ s ≤ t.

¾u¶s0fhk<¥¤z¤uhk Hy@ r%k<§¨k7¥|kjs

a(y) = (r − λη)y,

b(y) = σy,

c(y) = ηy,Huhkjs

dSs = rSsds + σSsdWs + ηSs−(dXs − λds),

S0 = x,

µ´[½í¸

§z¤s0fÆr~¥¤vws0z|~u

St = x exp

((

r − λη − σ2

2

)

t + σWt

)

∏

0<s≤t

(1 + η∆Xs), t ∈ R+.µ´[½=<R¸

¬³(Xt)t∈R+

f3rs0fhk(³´~y0i µ´[½í¸©§¨k+~ws%Hz¤u

St = x exp

((

r − λη − σ2

2

)

t + σWt

)

(1 + ηa1)N1

t · · · (1 + ηad)Nd

t , t ∈ R+.

#î

! "#$%&(')+*, -.0/1$2,0

Cº, )" )82 Theta

¾u:s0fwz|r©r%ktjs0z|~u:§©k(ykj¦az|kj§ s0fhk<t~iwvws%Hs0z|~u~H³Thetat

HuhThetaT

z¤u13Hy0s0z|jvw¥dHy r%ktr vhrz¤uwyktr0Lktjs0z¤¦kj¥¤q¯s0fhkÊ©¥d%Ë-«ªp?%fh~¥|ktr(HuhÁ(vwwz¤yk¿Á(»¿r ½

©~uhrz|wkjy@Hu~ws0z|~uǧz¤s0f3 q~H­³Jvwuhjs0z|~uφHuh¶wy0z|tk

C(x, t, T ) = e−R T

trsds

E

[

φ(ST )∣

∣

∣St = x]

.

paz¤uhtkt 7→ e

R T

trsdsC(St, t, T )

z|r8i·Hy0s0z¤uw-H¥|kRR³Jy~i µ´[½ a¸±C(x, t, T )

r%Hs0z|r:3ktrs0fhk©Ê©¥d%Ë-«p?%fh~¥|ktrÌÁ<»

Thetat =∂C

∂t(x, t, T )

µ´[½FE¹¸

= rtC(x, t, T ) − at(x)∂C

∂x(x, t, T ) − 1

2b2t (x)

∂2C

∂x2(x, t, T )

−λ

∫ +∞

−∞(C(x + zct(x), t, T )) − C(x, t, T ))ν(dz),

§z¤s0f

Delta =∂C

∂x(x, t, T ) = e−

R T

trsds

E

[

YT φ′(ST )∣

∣

∣St = x

]

,µ´[½í¸

§fhkjyk(Yt)t∈R+

= (∂xSt)t∈R+

z|rs0fhk:hyr0s¦Hy0zdHs0z|~uÆwy~atktr%rr~¥¤vws0z|~uÆ~H³

dYt = a′t(St)Ytdt + b′t(St)YtdWt + c′t(St−)Yt−dXt,

Y0 = 1,

Huh

Gamma =∂2C

∂x2(x, t, T ) = e−

R T

trsds

E

[

ZT φ′(ST ) + (YT )2φ′′(ST )∣

∣

∣St = x

]

,µ´[½í¸

§fhkjyk(Zt)t∈R+

= (∂2xSt)t∈R+

z|rs0fhk+rktt~uhǦHy0zdHs0z|~uÆwy~atktrr¹½¨(kjykRa′t(z)

b′t(z)

Huhc′t(z)

wkjuh~sk7s0fhk3Hy0s0zdH¥ wkjy0z¤¦Hs0z¤¦ktr@~H³#s0fhktr%k<³Jvwuhjs0z|~uhr§z¤s0fyktr0ktjss~z½

eBfhk8:hyr0sHuh:r%ktt~uh¶wkjy0z¤¦Hs0z¤¦ktr~uφz¤u:s0fhk7k±°[wyktrr0z|~uhrµ´[½í¸tµ´[½í¸©~H³Á<kj¥¤s%)Huh

ÍHii·¶ HuDkykjiO~ ¦ktD¦?zd¯z¤uaskjRy%Hs0z|~u -q3Hy0sr)HuhÄs0fhknH¥¤¥¤zd ¦az¤u H¥|jvw¥¤vhr r+z¤ukR½ h½);í >¬ ; >¬ws~q?z|kj¥|ÇHuÇk±°[wyktrr0z|~u³\~y

Thetat½Ì(~ §¨kj¦kjy(s0fhkt~iwvws%Hs0z|~u~H³ÌÍHii·

îî ëHJILK±õLM

E 0% ?, 5/ L 5

[z¤yktjs0¥¤qz¤u-¦~¥¤¦ktrs0fhk :hyrs7HuhÄr%ktt~uhƦRHy0zdHs0z|~uÄwy~?tktrrktrYtHuh

Zt½"l(HiOkj¥¤qÆz¤u ;=< >¬

s0fhk7k±°[wyktrr0z|~u µ´[½í¸©z|rykj§y0z¤s0skjur

Delta =∂C

∂x(x, t, T ) =

e−R T

trsds

T − tE

[

φ(ST )

∫ T

t

Ys

bs(Ss)dWs

∣

∣

∣St = x

]

,µ´[½ a¸

vhrz¤uwÆs0fhk:k±°?wyktr%r0z|~uÅ~H³s0fhk:nH¥¤¥¤zdt¦?z¤u wkjy0z¤¦RHs0z¤¦k¯~H³ST ∈ Dom(D)

z¤u skjy0iOr~H³s0fhk:hyrs¦Hy0zdHs0z|~uwy~?tktrrr?D

Ys

bs(Ss)Dsφ(ST ) = YT φ′(ST ), 0 ≤ s ≤ τ, a.s.,

µ´[½í¸

±³0½ ;FE >¬½º¬u¼s0fhk¯uhk±°[sOrktjs0z|~u!§©k¶wyktrkju-s:t~iwvws%Hs0z|~u ~H³ThetaT

-q z¤u-skjRy%Hs0z|~uaqÇ3Hy0sr<§fwz|%fWÏz¤us0fhks0z¤iOk"fh~iO~kjuhkt~vhr" rkRq?z|kj¥|wr<:[z®­Ïkjykju-sykjwyktr%kju-s%Hs0z|~uij´~yThetat = −ThetaT

½#¾s¨w~?ktr¨uh~s©[z¤yktjs0¥¤qOvhr%ks0fhk.:hyrsBHuh1rktt~uh·¦RHy0zdHs0z|~u:wy~atktrr%ktr Huhz¤ua¦~¥¤¦ktr~uw¥¤qÇkj¥|kjiOkju-s%Hy0q1A z|kjuhkjy7z¤u-skjRy%H¥|r7~H³©wkjskjy0iz¤uwz|rs0z|)³Jvwuhjs0z|~uhr<z¤uhr0sk ~H³¾s @r0s~?f3r0s0z|z¤u-skjRy%H¥|r~H³ÌhHwsktwy~?tktrrktr@rz¤u µ´[½í¸j½

Î@us0fhk+~s0fhkjyf3HuhÏ3z¤us0fhk+ r%k~H³»Ìvwy~Lk Hu~ws0z|~uhrz¤uÆOt~u-s0z¤u-vh~vhr(i·Hy0Ëkjs 3zݽÃkR½§z¤s0f

ct(·) = 0at(y) = αty

[§z¤s0fÇ3tq~H­³Jvwuhjs0z|~uφ(x) = (x − K)+

hHuh¶wy0z|tk

C(x, t, T,K) = e−R T

trsds

E[(ST − K)+ | St = x],

Á(vwwz¤ykRÂÃr³\~y0i"vw¥d ;í > yk wr

bT (K) =

√

√

√

√

√

√

2(rT − αT )C +

∂C

∂T+ KrT

∂C

∂K

K2∂2C

∂K2

§fhkjyk ∂C

∂T(x, t, T,K)

t~z¤uhjz|wktr§z¤s0fThetaT

½Ì¾u~s0fhkjyskjy0iOr

ThetaT =∂C

∂T(x, t, T,K)

µ´[½í¸

= (αT − rT )C(x, t, T,K) +K2b2

T (K)

2

∂2C

∂K2(x, t, T,K) − KαT

∂C

∂K(x, t, T,K),

§fhkjyk −∂2C

∂y2(x, t, T, y)

t~z¤uhjz|wktr§z¤s0fs0fhk7wkjuhr0z¤s¾q·³\vwuhjs0z|~udP (ST = y | St = x)/dy

½@kj¥dHs0z|~u µ´[½í¸+ Hu k1wy~ ¦ktÅ-qÅHww¥¤z| Hs0z|~uº~H³¯µ´[½ a¸+~u

[0, T ]8[z®­Ïkjykju-s0zdHs0z|~uº§z¤s0f

#î

! "#$%&(')+*, -.0/1$2,0 EE

yktrLktjsBs~TwHuh1z¤uaskjRy%Hs0z|~u-q13Hy0sr§z¤s0f¯yktr0ktjs©s~

dy~u

R½8eBfhk<t~iwvws%Hs0z|~uÇ~H³

ThetaTwyktrkjuaskt z¤uDs0fhk·uhk±°?s"rktjs0z|~uD³´~¥¤¥|~ §r+s0fhk·r%HiOkOrskjhr ykjw¥djz¤uwÆz¤u-skjRy%Hs0z|~u

aqÅ3Hy0sr·~uR§z¤s0fs0fhk¶[v3H¥¤z¤s¬q ³\~y0i"vw¥dÄ~uºs0fhk A z|kjuhkjyOr03tkR½¼@rr0vh%f!z¤sO Huºk

¦?z|kj§¨kt¼r"kjuhkjy%H¥¤z¤À¹Hs0z|~uº~H³Á(vwwz¤ykRÂÃrHy0RvwiOkju-s)s~ÆHy0wz¤s0y%Hy0q 3tq~H­!³\vwuhjs0z|~uhr+z¤u¼²vwi[z®­vhr0z|~ui·Hy0Ëkjs ½

ThetaT

©~uhrz|wkjy@Hu~ws0z|~uǧz¤s0f3 q~H­³Jvwuhjs0z|~uφHuh¶wy0z|tk

C(x, t, T ) = e−R T

trsds

E

[

φ(ST )∣

∣

∣St = x]

.

ThetaT HukHwwy~t°[z¤i·HsktÇaq:huwz¤sk[z®­Ïkjykjuhtktr@r

ThetaT =C(x, t, (1 + ε)T ) − C(x, t, (1 − ε)T )

2Tε.

µh½FE¹¸

@¥¤skjy0u3Hs0z¤¦kj¥¤qÏs0fhkwkjy0z¤¦RHs0z¤¦k§z¤s0fyktrLktjs(s~T Huk"wvws<z¤uhr0z|wk)s0fhkk±°[Lktjs%Hs0z|~uz®³

φz|r[z®­Ïkjykju-s0zdHw¥|kR½

©~uhr0z|wkjy(Sx

t,s)s∈[t,∞)Rz¤¦kjuaq:s0fhk©²vwi[«¬[z®­vhr0z|~uÆktav3Hs0z|~u

dSxt,s = as(S

xt,s)ds + bs(S

xt,s)dWs + cs(S

xt,s−)dXs,

Sxt,t = x.

µh½í¸

m(r0z¤uwO¾s @wÂÃrB³´~y0i+vw¥dOHuh µ´[½ a¸B§¨k7f3 ¦kD

C(x, t, T ) = e−R T

trsds

E[

φ(Sxt,T )]

= φ(x) − E

[∫ T

trse

−R s

trpdpφ(Sx

t,s)ds

]

+E

[∫ T

te−

R s

trpdpφ′(Sx

t,s)as(Sxt,s)ds

]

+ E

[∫ T

te−

R s

trpdpφ′(Sx

t,s)bs(Sxt,s)dWs

]

+1

2E

[∫ T

te−

R s

trpdpφ′′(Sx

t,s)b2s(S

xt,s)ds

]

+λE

[∫ T

te−

R s

trpdp

∫ +∞

−∞(φ(Sx

t,s + zcs(Sxt,s)) − φ(Sx

t,s))ν(dz)ds

]

îî ëHJILK±õLM

E 0% ?, 5/ L 5

= φ(x) −∫ T

trse

−R s

trpdp

E[

φ(Sxt,s)]

ds +

∫ T

te−

R s

trpdp

E[

φ′(Sxt,s)as(S

xt,s)]

ds

+1

2

∫ T

te−

R s

trpdp

E[

φ′′(Sxt,s)b

2s(S

xt,s)]

ds

+λ

∫ T

te−

R s

trpdp

E

[∫ +∞

−∞(φ(Sx

t,s + zcs(Sxt,s)) − φ(Sx

t,s))ν(dz)

]

ds,

fhkjuhtkThetaT

Huk7k±°?wyktr%rktÇr

ThetaT =∂

∂T

(

e−R T

trsds

E[

φ(Sxt,T )]

) µh½í¸

= −rT e−R T

trpdp

E[

φ(Sxt,T )]

+ e−R T

trpdp

E[

φ′(Sxt,T )aT (Sx

t,T )]

+1

2e−

R T

trpdp

E[

φ′′(Sxt,T )b2

T (Sxt,T )]

+λe−R T

trsds

E

[∫ +∞

−∞(φ(Sx

t,T + zcT (Sxt,T )) − φ(Sx

t,T ))ν(dz)

]

,

§fwz|%fÄ[z®­kjyr@³\y~i>µ´[½FE¹¸jLz¤uhj¥¤vh[z¤uw¯z¤us0fhk"s0z¤iOk±«Ýfh~iO~kjuhkt~vhr+ rkR½<l@~sk"s0f3Hs<vwuw¥¤z¤Ëks0fhk)ÌÁ<» Hwwy~fÄ~H³©p?ktjs0z|~uD[3s0fwz|r@iOkjs0fh~?Æw~?ktruh~s<rktkjis~Ok)Hww¥¤z| Hw¥|k)s~·s0fhkt~iwvws%Hs0z|~u~H³

Thetatz¤uÇ)s0z¤iOk±«Ýz¤uwfh~iO~kjuhkt~vhr(r%kjs0s0z¤uwh½ÌeBfhkHL~¹¦kk±°[wyktrr0z|~u¯³ÝHz¤¥|r

§fhkjuφz|ruh~ss¬§z|tk1[z®­kjykjuas0zdHw¥|kR½:eBfhk1Hz¤i ~H³BÌy~L~Rr0z¤s0z|~u h½FEkj¥|~ §.z|r+s~¶wyktrkju-s

:nH¥¤¥¤zdt¦?z¤uÄs¬q?Lk"³´~y0i+vw¥d1³´~yThetaT

L§fwz|fÅ ¦~z|wr<s0fhkvhrk)~H³ :huwz¤sk[z®­ÏkjykjuhtktrHuhw~?ktr(uh~s7ykt-vwz¤ykOHuaqÆr0iO~?~s0fwuhktrr7~u

φ½+eBfwz|r<§z¤¥¤¥¿H¥¤¥|~ § vhr<z¤u3Hy0s0z|jvw¥dHys~¯t~uhr0z|wkjy

uh~u[«¬riO~a~s0f3tq~H­:³\vwuhjs0z|~uhr r8kR½ h½ z¤us0fhkB rkB~H³[z¤Rz¤s%H¥h~ws0z|~uhr ½eBfhkBwkjy0z¤¦RHs0z¤¦ktr8~uφ§z¤¥¤¥kykjiO~¹¦kt¯-qOz¤uaskjRy%Hs0z|~u¶-q13Hy0sr©~u:s0fhk)A z|kjuhkjyr03tkR?vhr0z¤uw)s0fhk<k±°?wyktrrz|~u

φ′(Sxt,T ) =

Duφ(Sxt,T )

DuSxt,T

, u ∈ L2([t, T ]).µh½ a¸

eBfhk#²vwi·t~i~uhkju-s©~H³Ls0fhkH~ ¦kB³´~y0i+vw¥d7z|r8¥|k±³Js8vwuas~vhfhkt:r0z¤uhtkBz¤s¿w~?ktr8uh~sÌt~u-s%Hz¤uHuaq¶wkjy0z¤¦Hs0z¤¦kR½

Í(z¤¦kjuu, v, w ∈ L2([t, T ])

r0vhfOs0f3HsDuSx

t,T

DvS

xt,T

DwSx

t,T

Hyk[½Ãr ½#uh~u[«ÝÀ kjy~wa¥|kjss0fhk<§©kjz¤Rf-s

Λt,T (u, v, w)k7wk:huhkt¶aq

Λt,T (u, v, w) = −a′T (Sxt,T ) − rT + aT (Sx

t,T )

(

I1(u)

DuSxt,T

+D2

uSxt,T

|DuSxt,T |2

)

#î

! "#$%&(')+*, -.0/1$2,0 E

+1

2

((

b2T (Sx

t,T )

DvSxt,T

I1(v) − 2bT (Sxt,T )b′T (Sx

t,T ) +b2T (Sx

t,T )D2vS

xt,T

|DvSxt,T |2

)(

I1(w)

DwSxt,T

+D2

wSxt,T

|DwSxt,T |2

)

+b2T (Sx

t,T )

DwSxt,T DvSx

t,T

(

I1(v)DwDvS

xt,T

DvSxt,T

− 〈v, w〉 −DwD2

vSxt,T

DvSxt,T

+2DwDvS

xt,T D2

vSxt,T

|DvSxt,T |2

)

−2b′T (Sx

t,T )bT (Sxt,T )

DvSxt,T

(

I1(v) +D2

vSxt,T

DvSxt,T

))

+ b′T (Sxt,T )2 + b′′T (Sx

t,T )bT (Sxt,T ).

@kt H¥¤¥s0f3Hss0fhk A z|kjuhkjy·z¤u-skjRy%H¥|rI1(u)

I1(v)

HuhI1(w)

z¤uºs0fhkÇH~ ¦k:³´~y0i+vw¥dDHyktkjuaskjyktÍ7Hvhr%r0zdHu¶y%Huhw~i ¦RHy0zdHw¥|ktr ½ ÕÌÕ T Õ #

u, v, w ∈ L2([t, T ])/L" %

DuSxt,T

DvS

xt,T

DwSx

t,T

F/J* G / / %

Λt,T (u, v, w) ∈ L2(Ω) +, #

φ : R → R &C / 0+

ThetaT =

e−R T

trsds

E

[

Λt,T (u, v, w)φ(Sxt,T ) + λ

∫ +∞

−∞(φ(Sx

t,T + zcT (Sxt,T )) − φ(Sx

t,T ))ν(dz)

]

.

2G m(r0z¤uw1µ´[½í¸¿Huhǵh½ a¸#§©kkjs8³´~yu ∈ L2([t, T ])

HuhφgT-rv 4·jz|kju-s0¥¤qr0iO~?~s0f D

E[

φ′(Sxt,T )gT (Sx

t,T )]

= E

[

gT (Sxt,T )

DuSxt,T

Duφ(Sxt,T )

]

= E

[⟨

Dφ(Sxt,T ),

gT (Sxt,T )

DuSxt,T

u

⟩]

= E

[

φ(Sxt,T )δ

(

gT (Sxt,T )

DuSxt,T

u

)]

= E

[

φ(Sxt,T )

(

gT (Sxt,T )

DuSxt,T

I1(u) − Du

(

gT (Sxt,T )

DuSxt,T

))]

= E

[

φ(Sxt,T )

(

gT (Sxt,T )

DuSxt,T

I1(u) − g′T (Sxt,T ) +

gT (Sxt,T )D2

uSxt,T

|DuSxt,T |2

)]

.

A z¤s0fgT (·) = aT (·) §¨k~ws%Hz¤u

E[

φ′(Sxt,T )aT (Sx

t,T )]

= E

[

φ(Sxt,T )

(

aT (Sxt,T )

DuSxt,T

I1(u) − a′T (Sxt,T ) +

aT (Sxt,T )D2

uSxt,T

|DuSxt,T |2

)]

,

µh½í¸

îî ëHJILK±õLM

E 0% ?, 5/ L 5

§fwz¤¥|kgT (·) = b2

T (·) q?z|kj¥|wr

E[

φ′′(Sxt,T )b2

T (Sxt,T )]

= E[

φ′(Sxt,T )Γt,T (v)

]

,

§fhkjyk

Γt,T (v) =b2T (Sx

t,T )

DvSxt,T

I1(v) − 2bT (Sxt,T )b′T (Sx

t,T ) +b2T (Sx

t,T )D2vS

xt,T

|DvSxt,T |2

.

Ê©q¶r0z¤iz¤¥dHy@Hy0RvwiOkjuas§¨kkjs

E[

φ′′(Sxt,s)b

2s(S

xt,s)]

= E[

φ′(Sxt,T )Γt,T (v)

]

= E

[

Γt,T (v)Dwφ(Sx

t,T )

Dw(Sxt,T )

]

= E

[

φ(Sxt,T )δ

(

wΓt,T (v)

DwSxt,T

)]

= E

[

φ(Sxt,T )

(

Γt,T (v)

DwSxt,T

I1(w) − Dw

(

Γt,T (v)

DwSxt,T

))]

= E

[

φ(Sxt,T )

(

Γt,T (v)

DwSxt,T

(

I1(w) +D2

wSxt,T

DwSxt,T

)

−2b′T (Sx

t,T )bT (Sxt,T )

DvSxt,T

(

I1(v) +D2

vSxt,T

DvSxt,T

)

+b2T (Sx

t,T )

DwSxt,T DvSx

t,T

(

I1(v)DwDvSxt,T

DvSxt,T

− 〈v, w〉 −DwD2

vSxt,T

DvSxt,T

+2DwDvS

xt,T D2

vSxt,T

|DvSxt,T |2

)

+2b′T (Sxt,T )2 + 2b′′T (Sx

t,T )bT (Sxt,T ))]

.

pavwiiz¤uwĵh½í¸©§z¤s0fs0fhkH~ ¦kykj¥dHs0z|~uHuh¶vhr0z¤uwµh½í¸©§¨k+~ws%Hz¤u

Λt,T (u, v, w) = −a′T (Sxt,T ) − rT + aT (Sx

t,T )

(

I1(u)

DuSxt,T

+D2

uSxt,T

|DuSxt,T |2

)

+1

2

(

Γt,T (v)

(

I1(w)

DwSxt,T

+D2

wSxt,T

|DwSxt,T |2

)

−2b′T (Sx

t,T )bT (Sxt,T )

DvSxt,T

(

I1(v) +D2

vSxt,T

DvSxt,T

)

+b2T (Sx

t,T )

DwSxt,T DvSx

t,T

(

I1(v)DwDvS

xt,T

DvSxt,T

− 〈v, w〉 −DwD2

vSxt,T

DvSxt,T

+2DwDvS

xt,T D2

vSxt,T

|DvSxt,T |2

))

+b′T (Sxt,T )2 + b′′T (Sx

t,T )bT (Sxt,T ).

ADk7t~uhj¥¤vhwk+s0fhk7wy~?~H³#aq¯Hwwy~ °?z¤i·Hs0z|~u~H³φ-q C2

b

³\vwuhjs0z|~uhr ½

#î

! "#$%&(')+*, -.0/1$2,0 E

¾us0fhk7s0z¤iOk+fh~iO~kjuhkt~vhr< rkR3wkjuh~s0z¤uwSx

0,τ

-qSx

τ

HuhΛ0,τ (u, v, w)

-qΛτ (u, v, w)§©k7f3t¦k

C(x, t, T ) = e−(T−t)rE

[

φ(ST )∣

∣

∣St = x]

= e−τrE [φ(Sx

τ )] ,

§z¤s0fτ = T − t

hHuh

ThetaT = −Thetat

= e−τrE

[(

Λτ (u, v, w)φ(Sxτ ) + λ

∫ +∞

−∞(φ(Sx

τ + c(Sxτ )y) − φ(Sx

τ ))ν(dy)

)]

,

§fhkjyku, v, w ∈ L2([0, τ ])

HykOr0vh%fDs0f3HsDuSx

τ

DvS

xτ

DwSx

τ

Hyk·[½Ãr ½uh~u[«ÝÀ kjy~HuhΛτ (u, v, w) ∈ L2(Ω)

½Ìh~yt~uhr0s%HuasrσHuh

ηhzݽÃkR½z¤us0fhk7¥¤z¤uhk Hy rkR[§©k7f3t¦k

a(y) = (r − λη)y,

b(y) = σy,

c(y) = ηy,

Huhŵ´[½=<R¸©qaz|kj¥|wr

DuSxτ = σ

∫ τ

0usdsSx

τ ,

fhkjuhtkD2

vSxτ /|DvS

xτ |2 = 1/Sτ

Huh¯§¨k7kjs

Λτ (u, v, w) = −r +r

σ

I1(u)∫ τ0 usds

− σ

2

I1(w)∫ τ0 wsds

+I2(v w)

2∫ τ0 vsds

∫ τ0 wsds

,

§fhkjykr = r − λη

½ÌeBfhkjs%z|rs0fhkjuÇRz¤¦kju-q

Theta = e−rτE

[

Λτ (u, v, w)φ(Sxτ ) + λ

∫ +∞

−∞(φ(Sx

τ (1 + ηy)) − φ(Sxτ ))ν(dy)

]

,

zݽÃkR½z¤uÇs0fhk7iO~awkj¥W~H³"µ´[½í¸B§¨k7f3 ¦k

Theta = e−rτE

[

Λτ (u, v, w)φ(Sxτ ) +

d∑

k=1

λk(φ(Sxτ (1 + ηak)) − φ(Sx

τ ))

]

,

HuhÆz®³(Xt)t∈R+

z|r(1r0s%HuhhHy#~z|rr~uwy~atktrr(§z¤s0fz¤u-skjuhr0z¤s¾qλHuh·ªvwir0z¤À k

a§¨k

kjsTheta = e−rτ

E [Λτ (u, v, w)φ(Sxτ ) + λ(φ((1 + aη)Sx

τ ) − φ(Sxτ ))] .

îî ëHJILK±õLM

E 0% ?, 5/ L 5

¬³(Xt)t∈R+

f3r@z¤u :huwz¤skj¥¤q¶i·Hu-q²vwihr(~uÇ~vwuhwkts0z¤iOkz¤uaskjy0¦H¥|r¹3zݽÃkR½Bz®³µ(R) = ∞

s0fhkju(St)t∈R+

z|rRz¤¦kjuaq

dSt = at(St)dt + bt(St)dWt + ct(St−)dXt,

S0 = x,

§fhkjyk(Xt)t∈R+

z|r+s0fhk1t~iLkjuhr%Hskt wvwyk+ªvwiÅi·Hy0s0z¤uw-H¥|k¶rr~?jzdHsktDs~(Xt)t∈R+

½¾u¶s0fwz|r rk7¾s @wÂÃrB³´~y0i+vw¥dOyk wr

φ(St) = φ(Ss) +

∫ t

sφ′(Su)au(Su)du +

∫ t

sφ′(Su)bu(Su)dWu +

∫ t

sφ′(Su)bu(Su)dXu

+1

2

∫ t

sφ′′(Su)b2(Su)du +

∑

s<u≤t

(

φ(Su− + cu(Su−)∆Xu) − φ(Su−) − cu(Su−)∆Xuφ′(Su−))

,

0 ≤ s ≤ t[s~Oqaz|kj¥|

E[φ(St)] = E[φ(Ss)] + E

[∫ t

sφ′(Su)au(Su)du

]

+1

2E

[∫ t

sφ′′(Su)b2

u(Su)du

]

+λE

[∫ t

s

∫ +∞

−∞(φ(Su + zcu(Su)) − φ(Su) − zcu(Su)φ′(Su))ν(dz)du

]

.

(~ §©kj¦kjy Ìz¤u¼s0fwz|r rk1s0fhk:¥dr0st~iL~uhkjuasz¤uφ′(Su)

Hu uh~sk:z|r~¥dHsktºHuh wk H¥¤s§z¤s0fÇ-q:z¤u-skjRy%Hs0z|~u-q:3Hy0sr ½

+` A8)

Ê©qÆs0fhkyktr0vw¥¤sr7~H³¨s0fhkwykttkt[z¤uwÇr%ktjs0z|~uWWs0fhk)¦H¥¤vhk~H³BeBfhkjs%¯z¤us0fhkkt~iOkjs0y0z|iO~?wkj¥µ´[½í¸Bz|rRz¤¦kju-q

Theta = e−rτE

[

φ(Sxτ )Λ(u, v, w) + λ

∫ +∞

−∞(φ(Sx

τ + ηSxτ y) − φ(Sx

τ ))ν(dy)

]

.µ´[½FE¹¸

3~y@Hu-qu ∈ L2([0, τ ])

rvhfs0f3Hs ∫ τ0 usds 6= 0

w¥|kjs0s0z¤uw

ut =ut

∫ τ0 usds

, t ∈ [0, τ ],

#î

! "#$%&(')+*, -.0/1$2,0 E?<

s0fhk<§©kjz¤Rf-sΛτ (u, v, w)

z|rk±°[wyktrrktÇr

Λτ (u, v, w) = −r +r

σI1(u) − σ

2I1(w) +

1

2I2(v w).

Î@vwy:~H¥(z|r·uh~¹§s~ :huh³Jvwuhjs0z|~uhru, v, w

§fwz|%f iz¤uwz¤iz¤À kVar[Λτ (u, v, w)]

z¤u s0fwz|rr%kjs0s0z¤uwh½ ÕÌÕ T Õ ! (%

Var[Λτ (u, v, w)] /. G, #+J* ?G J*

/# L+ "/u, v, w

% #Lus = c1

vs = c2

ws = c3

s ∈ [0, τ ]

+, / #

infu,v,w

Var[Λτ (u, v, w)] = Var[Λopt] =1

2τ2+

1

σ2τ

∣

∣

∣

∣

r − σ2

2

∣

∣

∣

∣

2

,

"G r = r − λη %%

Λopt = −r +Wτ

στ

(

r − σ2

2

)

+1

2τ

(

W 2τ

τ− 1

)

.µ´[½í¸

2Gkt H¥¤¥Ïs0f3Hss0fhk BHvhfaq-«ªp?%f-§©Hy0Àz¤uhktav3H¥¤z¤s¬q¯q?z|kj¥|wr

‖u‖2 ≥ 1

τ,

µ´[½í¸

§z¤s0fÆkt-v3H¥¤z¤s¾q:z®³ÌHuh¶~uw¥¤q¯z®³ut = 1/τ

t ∈ [0, τ ]

½

B kjs

F (u, v, w) = Var[Λτ (u, v, w)]

=r2

σ2‖u‖2 − r〈u, w〉 +

σ2

4‖w‖2 +

1

4‖v‖2‖w‖2 +

1

4〈v, w〉2

=1

σ2

∥

∥

∥

∥

ru − σ2

2w

∥

∥

∥

∥

2

+1

4‖v‖2‖w‖2 +

1

4〈v, w〉2.

eBfhk+~ws0z¤i·H¥W¦H¥¤vhk+~H³(u, v, w)

r~¥¤¦ktr

d

dεF (u + εh, v, w)|ε=0 = 0

d

dεF (u, v + εh,w)|ε=0 = 0

d

dεF (u, v, w + εh)|ε=0 = 0,

µ´[½ a¸

îî ëHJILK±õLM

E 0% ?, 5/ L 5

³´~yH¥¤¥h ∈ L2([0, τ ])

zݽÃkR½

2r2

σ2

(

〈h, u〉 − ‖u‖2

∫ τ

0hsds

)

− r

(

〈h, w〉 − 〈u, w〉∫ τ

0hsds

)

= 0,

1

2‖w‖2

(

〈h, v〉 − ‖v‖2

∫ τ

0hsds

)

+1

2

(

〈v, w〉〈h, w〉 − 〈v, w〉2∫ τ

0hsds

)

= 0,

Huh

σ2

2

(

〈h, w〉 − ‖w‖2

∫ τ

0hsds

)

+1

2‖v‖2

(

〈h, w〉 − ‖w‖2

∫ τ

0hsds

)

+1

2

(

〈v, w〉〈h, v〉 − 〈v, w〉2∫ τ

0hsds

)

− r

(

〈h, u〉 − 〈u, w〉∫ τ

0hsds

)

= 0.

¨¥|k Hy0¥¤q³\~y¶H¥¤¥c1, c2, c3 6= 0

s0fhkt~uhr0s%Huas1³\vwuhjs0z|~uhrus = c1

vs = c2

ws = c3

s ∈ [0, τ ]

wHykr%~¥¤vws0z|~uhr¿~H³Ïs0fwz|rÌwy~w¥|kji½ B kjs¿vhr¿rfh~ §s0f3Hs¿s0fwz|r¨r~¥¤vws0z|~u1z|r8vwuwz|avhkR½8h~yH¥¤¥

h ∈ L2([0, τ ])r0vh%fs0f3Hs ∫ τ

0 hsds = 0hktav3Hs0z|~uŵ´[½ a¸©q?z|kj¥|wr

2r2

σ2〈h, u〉 − r〈h, w〉 = 0

‖w‖2〈h, v〉 + 〈v, w〉〈h, w〉 = 0

σ2〈h, w〉 + ‖v‖2〈h, w〉 + 〈v, w〉〈h, v〉 − 2r〈h, u〉 = 0.

¬³Dr~¥¤vws0z|~u(u, v, w)

[z®­kjykjuasO³\y~i(1/τ, 1/τ, 1/τ)

k±°?z|r0sr¹¿s0fhkju ~uhkÇ Hu :huhh ∈

L2([0, τ ])r0vh%f s0f3Hs ∫ τ

0 hsds = 0Huh

(〈h, u〉, 〈h, v〉, 〈h, w〉) 6= (0, 0, 0)7fhkjuhtks0fhk

wkjskjy0iz¤u3Huas‖v‖2‖w‖2 − |〈v, w〉|2 = 0

µ´[½í¸~H³s0fhkH~ ¦k¥¤z¤uhk Hy@r0q?rskji ¦Huwz|r0fhktr¹½8wy~i µ´[½í¸Huh µ´[½í¸B§¨k7kjs

F (u, v, w) =1

σ2

∥

∥

∥

∥

ru − σ2

2w

∥

∥

∥

∥

2

+1

4‖v‖2‖w‖2 +

1

4|〈v, w〉|2

=1

σ2

∥

∥

∥

∥

ru − σ2

2w

∥

∥

∥

∥

2

+1

2‖v‖2‖w‖2

≥ 1

τσ2

∣

∣

∣

∣

∫ τ

0

(

rus −σ2

2ws

)

ds

∣

∣

∣

∣

2

+1

2τ2

#î

! "#$%&(')+*, -.0/1$2,0 E

=1

τσ2

∣

∣

∣

∣

r − σ2

2

∣

∣

∣

∣

2

+1

2τ2,

§fwz|%fÆz|rRyk Hskjy@s0f3Hus0fhk+~ws0z¤i·H¥#¦RH¥¤vhk7³´~vwuh§fhkjuuvwHyk+t~uhr0s%Huas³\vwuhjs0z|~uhr ½

n~ykt~ ¦kjy kt-v3H¥¤z¤s¾q¶~atjvwyr~uw¥¤q:§fhkju ‖v‖2 = 1/τ ‖w‖2 = 1/τ

Huh

∥

∥

∥

∥

ru − σ2

2w

∥

∥

∥

∥

2

=1

τ

∣

∣

∣

∣

r − σ2

2

∣

∣

∣

∣

2

,

zݽÃkR½§fhkjuru − σ2

2 wvwHyk7t~uhrs%Hu-s w§fwz|%fz¤iw¥¤z|ktrs0f3Hs

uz|rH¥|r~t~uhrs%Hu-s hk±°wtkjws

§fhkjur = 0

[z¤u§fwz|%fÇ rk7uh~t~uhrs0y%Hz¤u-sz|rz¤iL~RrktÇ~uu½

ADk)uh~¹§ uhktkts~1wy~¹¦ks0f3Hs7s0fwz|r7r~¥¤vws0z|~uDt~y0yktr0~uhwr<s~¯s0fhk)R¥|~3H¥8iz¤uwz¤i+vwi ~H³F½

paz¤uhtkF (u, v, w) ≥ 0

Ls0fhk+z¤u :hi"vwik±°[z|r0sr<Huhǧ©kwk±uh~sk)z¤s@-ql½(ʨqt~uas0z¤u-vwz¤s¾q~H³

F~uL2([0, τ ])3

s0fhkjykk±°[z|r0srktavhkjuhtk(un, vn, wn)n∈N

r0vh%fs0f3Hs

l = limn→∞

E[Λτ (un, vn, wn)2].

ADkÄ Hu rr0vwiOks0f3Hs(un, vn, wn)

z|r¶~vwuhwkt Dºz®³)uh~s (ykjw¥dtkÄz¤sÇ-q s0fhkÄL~vwuhwktr%kt-vhkjuhtk

(

un

‖un‖,

vn

‖vn‖,

wn

‖wn‖

)

n∈N

,

~u§fwz|%fFs%HËktrs0fhkBr%HiOk©¦RH¥¤vhktrÌr#~u

(un, vn, wn)n∈N

½m@uhwkjys0fwz|rf-q?L~s0fhktr0z|r¹s0fhkjykk±°[z|r0sr©r0vw[«¬rktavhkjuhtk

(unk, vnk

, wnk)k∈N

t~u-¦kjy0Rz¤uw§©k HËa¥¤qs~(u, v, w)

z¤uL2([0, τ ])3

½ADk<f3 ¦k

E[Λτ (u, v, w)Λτ (unk, vnk

, wnk)]

= r2 +r2

σ2〈u, unk

〉 − r

2〈u, wnk

〉 − r

2〈unk

, w〉 +σ2

4〈w, wnk

〉

+1

4〈u, unk

〉〈w, wnk〉 +

1

4〈u, wnk

〉〈w, unk〉,

Huh-q:§¨k H˶t~u-¦kjy0kjuhtk)~H³(unk

, vnk, wnk

)k∈N

s~(u, v, w)

§¨k7kjs

limn→∞

E[Λτ (u, v, w)Λτ (unk, vnk

, wnk)] = E[|Λτ (u, v, w)|2].

n~ykt~ ¦kjy

0 ≥ l − E[|Λτ (u, v, w)|2]

îî ëHJILK±õLM

R 0% ?, 5/ L 5

= limn→∞

E[Λτ (unk, vnk

, wnk)2] − E[|Λτ (u, v, w)|2]

≥ limn→∞

E[|Λτ (u, v, w) − Λτ (unk, vnk

, wnk)|2]

+2E[Λτ (u, v, w)Λτ (unk, vnk

, wnk)] − 2E[|Λτ (u, v, w)|2]

≥ limn→∞

E[(Λτ (u, v, w) − Λτ (unk, vnk

, wnk))2]

+2 limn→∞

E[Λτ (u, v, w)Λτ (unk, vnk

, wnk)]

−2E[|Λτ (u, v, w)|2]≥ lim

n→∞E[(Λτ (u, v, w) − Λτ (unk

, vnk, wnk

))2]

≥ 0,

fhkjuhtklimn→∞ Λτ (unk

, vnk, wnk

) = Λτ (u, v, w)z¤u

L2(Ω)Huh

l = E[|Λτ (u, v, w)|2].eBfavhrs0fhk<R¥|~3H¥iz¤uwz¤i+vwi z|r@Hs0s%Hz¤uhkt¯³\~y

u = v = w = 1/τ½

l(~skDs0f3Hsinfu,v,w∈L2([0,τ ]) Var[Λτ (u, v, w)]

z|rÇiz¤uwz¤i·H¥z¤u skjy0iOr~H³σHuh

r§fhkju

(Sxt )t∈R+

z|rHuk±°?~uhkju-s0zdH¥#Ê©y~ §uwzdHuÇiO~s0z|~uWhzݽÃkR½r = σ2/2

½Ì¬u¶s0fwz|r rk<§¨k+f3t¦k

infu,v,w∈L2([0,τ ])

Var[Λτ (u, v, w)] =1

2τ2.

23 ")2

©~uhrz|wkjys0fhk7wy0z|tk7wy~?tktrr(St)t∈R+

Rz¤¦kjuÇ-q

dSxt = a(Sx

t )dt + b(Sxt )dWt + ηSx

t−dXt,

Sx0 = x,

zݽÃkR½§©k¥|kjsct(z) = ηz

½Ìkt H¥¤¥ s0f3Hs h±³²½.;FEE>¬hs0fhk<wy0z|tk

C(x, y, t, T ) = e−(T−t)rE

[

(

1

T

∫ T

0Sx

udu − K

)+∣

∣

∣Ft

]

Hss0z¤iOkt~H³ÌHuÇ@rzdHu~ws0z|~u§z¤s0fwy0z|tk7wy~?tktrr

(Sxt )t∈R+

HuLk7k±°[wyktrrktr

C(x, y, t, T ) = xe−(T−t)rE

[

(

y +1

T

∫ T

tS1

s−tds

)+]

µ´[½FE¹¸

#î

! "#$%&(')+*, -.0/1$2,0 E

Huh

x = StHuh

y =1

x

(

1

T

∫ t

0Sudu − K

)

.

(rBz¤us0fhkr%kjs0s0z¤uwO~H³»8vwy~k HuÆ~ws0z|~uhr¹w§¨k7i· q¶t~uhr0z|wkjy

Thetat =∂C

∂t(x, y, t, T )

HuhThetaT =

∂C

∂T(x, y, t, T ),

§fwz|%f!fh~ §©kj¦kjy1[z®­Ïkjy³Jy~ik %f~s0fhkjyOz¤uºkjuhkjy%H¥Ý¨z¤uhj¥¤vh[z¤uw §fhkju(St)t∈R+

z|rs0z¤iOkfh~iO~kjuhkt~vhr¹½Ì@kjykR

Thetat Hu1Lkt~iwvwsktO³\y~i.s0fhkÊ©¥d%Ë-«ªp?%fh~¥|ktr©¿Á(»Å³´~y¿(r0zdHu

~ws0z|~uhrz¤uÇ(²vwi[«¬[z®­Lvhr0z|~uÇiO~awkj¥Ýw±³0½Ì8y~~Rr0z¤s0z|~uÆ~H³.;FE#>¬hr

Thetat = rC(x, y, t, T ) +

(

1

T− a(y)

)

∂C

∂y(x, y, t, T ) − 1

2b2(y)

∂2C

∂y2(x, y, t, T )

−(1 + η)λ

∫ +∞

−∞

(

C(

x,yz

1 + η, t, T

)

− C(x, y, t, T ))

ν(dz).

¾u¯s0fhk<rkjs0s0z¤uwO~H³#(r0zdHu¶~ws0z|~uhr w~vwyiOkjs0fh~?Hww¥¤z|ktrL~s0f¯s~Thetat

HuhThetaT

Huhw~?ktruh~syktavwz¤yks0fhk:hyr0s@Huhrktt~uhÆwkjy0z¤¦Hs0z¤¦ktr ∂C

∂y

Huh ∂2C

∂y2

½¿¾sH¥|r~Ow~?ktruh~svhrk¾s @ H¥|jvw¥¤vhr ½ ÕÌÕ T Õ

u ∈ L2([0, τ ])/L" % ∫ τ

0 DuSxs ds 6= 0

5/ s ∈ [0, τ ]

#

Λ(u) = −r − DuSxτ

∫ τ0 DuSx

s ds+

Sxτ −

∫ τ0 Sx

s ds/T∫ τ0 DuSx

s ds

(

I1(u) +

∫ τ0 D2

uSxs ds

∫ τ0 DuSx

s ds

)

,

+, / /L % Λ(u) ∈ L2(Ω)

ThetaT = xe−(T−t)rE

[

Λ(u)

(

y +1

T

∫ τ

0S1

sds

)+]

.

2G @kjw¥djz¤uwx 7→ (x − K)+

z¤u µ´[½FE¹¸-qφ C1

b

³\vwuhjs0z|~uŧ©k1wk:huhk1s0fhk1s0z¤iOkwkjy0z¤¦RHs0z¤¦k

eBfhkjs% φT =

∂C

∂T(x, y, t, T )

= xe−(T−t)rE

[

−rφ

(

y +1

T

∫ T

tS1

s−tds

)

+

(

S1T−t

T− 1

T 2

∫ T

tS1

s−tds

)

φ′(

y +1

T

∫ T

tS1

s−tds

)

]

.

îî ëHJILK±õLM

R 0% ?, 5/ L 5

m(r0z¤uwƵ´[½í¸B§¨k7f3 ¦kD

ThetaφT = −rxe−(T−t)r

E

[

φ

(

y +1

T

∫ τ

0S1

sds

)]

+xe−(T−t)rE

[

(

S1τ − 1

T

∫ τ

0S1

sds

)

Duφ(

y +∫ τ0 S1

sds/T)

Du

∫ τ0 S1

sds

]

= −rxe−(T−t)rE

[

φ

(

y +1

T

∫ τ

0S1

sds

)]

+xe−(T−t)rE

[⟨

Dφ

(

y +1

T

∫ τ

0S1

sds

)

,S1

τ −∫ τ0 S1

sds/T∫ τ0 DuS1

sdsu

⟩]

= −rxe−(T−t)rE

[

φ

(

y +1

T

∫ τ

0S1

sds

)]

+xe−(T−t)rE

[

φ

(

y +1

T

∫ τ

0S1

sds

)

δ

(

S1τ −

∫ τ0 S1

sds/T∫ τ0 DuS1

sdsu

)]

= xe−(T−t)rE

[

φ

(

y +1

T

∫ τ

0S1

sds

)(

−r − DuS1τ

∫ τ0 DuS1

sds

+S1

τ −∫ τ0 S1

sds/T∫ τ0 DuS1

sds

(

I1(u) +

∫ τ0 D2

uS1sds

∫ τ0 DuS1

sds

))]

= xe−(T−t)rE

[

Λ(u)φ

(

y +1

T

∫ τ

0S1

sds

)]

.

¾s(ykji·Hz¤uhr7s~¶Hwwy~t°[z¤i·Hskx 7→ (x − K)+

-qÄ:rktavhkjuhtk(φn)n∈N

~H³ C1b

³Jvwuhjs0z|~uhr ½

Thetat Huk7t~iwvwsktz¤uÇr0z¤iz¤¥dHy§Btq:³\y~i

C(x, y, t, T ) = xe−(T−t)rE

[

(

y +1

T

∫ T−t

0S1

sds

)+]

,

Huh

eBfhkjs% φt = xe−(T−t)r

E

[

rφ

(

y +1

T

∫ T−t

0S1

t−sds

)

+S1

T−t

Tφ′(

y +1

T

∫ T−t

0S1

sds

)

]

,

§fwz|%fq?z|kj¥|wr@Z³\skjyz¤u-skjRy%Hs0z|~u-q:3Hy0sr2D

eBfhkjs%t = xe−(T−t)r

E

[

Λ(u)

(

y +1

T

∫ T−t

0S1

t−sds

)+]

,

#î

! "#$%&(')+*, -.0/1$2,0 R

§z¤s0f

Λ(u) = r − DuSxτ

∫ τ0 DuSx

s ds+

Sxτ

∫ τ0 DuSx

s ds

(

I1(u) +

∫ τ0 D2

uSxs ds

∫ τ0 DuSx

s ds

)

.

¾u¶s0fhk<kt~iOkjs0y0z|iO~?wkj¥¨µ´[½í¸©§©k7f3t¦k

Λ(u) = −r −∫ τ0 usdsSx

τ∫ τ0

∫ s0 updpSx

s ds+

Sxτ −

∫ τ0 Sx

s ds/T

σ∫ τ0

∫ s0 updpSx

s ds

(

I1(u) + σ

∫ τ0

(∫ s0 updp

)2Sx

s ds∫ τ0

∫ s0 updpSx

s ds

)

Huh

Λ(u) = r −∫ τ0 usdsSx

τ∫ τ0

∫ s0 updpSx

s ds+

Sxτ

σ∫ τ0

∫ s0 updpSx

s ds

(

I1(u) + σ

∫ τ0

(∫ s0 updp

)2Sx

s ds∫ τ0

∫ s0 updpSx

s ds

)

.

23% " )2

¾u¶s0fwz|rrktjs0z|~uǧ¨kt~uhr0z|wkjys0fhk7kt~iOkjs0y0z|+Ê©y~ §uwzdHuÆiO~awkj¥

dSs = rSsds + σSsdWs

St = x,

Huh Hww¥¤qºs0fhkÆnH¥¤¥¤zdt¦?z¤u ³\~y0i"vw¥d µ´[½FE¹¸O§z¤s0fus = vs = ws = 1/τ

s ∈ [0, τ ]

s~t~iwvwsk

ThetaT = −Thetat

³´~y»8vwy~k HuÆs¾qak~ws0z|~uhr§z¤s0fÇuh~u[«¬r0iO~?~s0f3tq~H­Ä³\vwuhjs0z|~uhr ½©h~y0i+vw¥dµh½FE¹¸©z|rvhrkt³´~y :huwz¤sk[z®­Ïkjykjuhtktr@Hwwy~ °?z¤i·Hs0z|~uhr ½

¾u rkφ =

[K,+∞)

[s0fhk7¦RH¥¤vhk7~H³eBfhkjs%· Hu¶Lk7t~iwvwsktHu3H¥¤qas0z| H¥¤¥¤qr

Theta = e−rτ

e−a2

2τ

( r

σ− σ

2+

a

2τ√

2πτ

)

− r

∫ ∞

a√τ

e−y2

2

√2π

dy

,

§z¤s0fa = (ln(K/x) − rτ)/σ

h±³0½8kR½ h½(;í >¬[§fwz|%fÇq?z|kj¥|wrs0fhk<³´~¥¤¥|~ §z¤uw·Ry%Hwf D

îî ëHJILK±õLM

0% ?, 5/ L 5

Number of trials=2e+04

Theoretical value

80 90 100 110 120 130 140 150 160Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t-0.5

0

0.5

1

1.5

Theta value

T ER

x E R[

σ [½FE [

r [½íR[

ε = 0.001z¤RvwykE$D8»°hjs¦H¥¤vhk7~H³8eBfhkjs%Or³Jvwuhjs0z|~uÇ~H³tHuh

K

eBfhkuhk±°?s¿Ry%Hwfhr©H¥¤¥|~¹§vhrÌs~"t~i3Hyks0fhk@nÆ~uask BHy0¥|~+rz¤i+vw¥dHs0z|~uhrB~ws%Hz¤uhktOaq :huwz¤sk[z®­ÏkjykjuhtktrHuh1z¤u-skjRy%Hs0z|~uaqO3Hy0sr :hyr0sBz¤u1skjy0iOrB~H³ uavwi+kjy~H³r%Hiw¥|ktrHuh:s0fhkjur³\vwuhjs0z|~uhr~H³

tHuh

K½

#î

! "#$%&(')+*, -.0/1$2,0 R

0.312

0.3125

0.313

0.3135

0.314

0.3145

0.315

5e+08 1e+09 1.5e+08 2e+09

The

ta V

alue

Number of trials

Theoretical valueIntegration by parts

Finite difference method

T ER

t [½í[

x E R[

σ [½í[

r [½FER

K EE [

ε [½í E

z¤Rvwyk" D8»Ìr0s0z¤i·Hs0z|~uÆ~H³8eBfhkjs%¦[ru-vwi"Lkjy(~H³#s0y0zdH¥|rNumber of trials=2e+04

Finite difference method

80 90 100 110 120 130 140 150 160Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t-0.5

0

0.5

1

1.5

Theta value

T ER

x E R[

σ [½FE [

r [½íR[

ε [½íR E

z¤Rvwyk" D8eBfhkjs%Or"³\vwuhjs0z|~uÇ~H³tHuh

K-q:huwz¤sk[z®­Ïkjykjuhtktr

îî ëHJILK±õLM

R 0% ?, 5/ L 5

Number of trials=2e+04

Integration by parts

80 90 100 110 120 130 140 150 160Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t-0.5

0

0.5

1

1.5

Theta value

T ER

x E R[

σ [½FE [

r [½íR[

ε [½íR E

#z¤Rvwyk D8eBfhkjs%Or@)³Jvwuhjs0z|~uÆ~H³tHuh

Kaq¯z¤u-skjRy%Hs0z|~uÇaq¯3Hy0sr

¾u rkφ(x) = (x − K)+

weBfhkjs% HuÇLk7t~iwvwsktr

Theta =xσ

2√

2πτe−c2/2 + Kre−rτN(c − σ

√τ),

§fhkjykc =

(

σ − aτ

)√τHuh

N(x) =∫ x−∞ e−

y2

2dy√2π

@±³²½ kR½ h½ ;í >¬§fwz|f q?z|kj¥|wr¯s0fhk³´~¥¤¥|~ §z¤uw·Ry%HwfW½

#î

! "#$%&(')+*, -.0/1$2,0 <

Number of trials=2e+04

Theoretical value

80 90 100 110 120 130 140 150 160Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t

0

5

10

15

20

Theta value

T ER

x E R[

σ [½FE [

r [½íR[

ε [½íR E

z¤Rvwyk" D8»°hjs¦H¥¤vhk7~H³8eBfhkjs%Or³Jvwuhjs0z|~uÇ~H³tHuh

K

(r#z¤u ; >¬s0fhk©³\~¥¤¥|~¹§z¤uw7Ry%HwfhrÌr0fh~¹§ºs0f3Hs8s0fhk :huwz¤sk[z®­ÏkjykjuhtkHwwy~t°[z¤i·Hs0z|~u·kjy²³´~y0iOrkjs0skjy¨³\~yB»8vwy~k Hu¯ H¥¤¥|r©§z¤s0f

φ(x) = (x−K)+?wvwsBrBrktkju1z¤u:s0fhk@wykj¦?z|~vhr©r%ktjs0z|~uW

s0fhk7nH¥¤¥¤zdt¦?z¤uÆ H¥|jvw¥¤vhr(Hwwy~fÆLkjy²³´~y0iOrkjs0skjy³\~y[z¤Rz¤s%H¥~ws0z|~uhr ½

îî ëHJILK±õLM

R 0% ?, 5/ L 5

Number of trials=2e+04

Finite difference method

80 90 100 110 120 130 140Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t

0

5

10

15

20

Theta value

T ER

x E R[

σ [½FE [

r [½íR[

ε [½íR E

z¤Rvwyk" D8eBfhkjs%Or"³\vwuhjs0z|~uÇ~H³tHuh

K-q:huwz¤sk[z®­Ïkjykjuhtktr

Number of trials=2e+04

Integration by parts

80 90 100 110 120 130 140Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t

0

5

10

15

20

Theta value

T ER

x E R[

σ [½FE [

r [½íR[

ε [½íR E

#z¤Rvwyk < D8eBfhkjs%Or@)³Jvwuhjs0z|~uÆ~H³tHuh

Kaq¯z¤u-skjRy%Hs0z|~uÇaq¯3Hy0sr

eBfhk)¥|~? H¥¤z¤À¹Hs0z|~uwy~atkt[vwykOHww¥¤z|ktz¤us0fhk"uhk±°[s7rktjs0z|~u§z¤¥¤¥8H¥¤¥|~¹§ vhr(s~·z¤iwy~¹¦ks0fhkyktrvw¥¤s~H³#s0fhknH¥¤¥¤zdt¦?z¤uiOkjs0fh~?Ͻ

#î

! "#$%&(')+*, -.0/1$2,0 R

7 %

tq~H­Ä³\vwuhjs0z|~uhr~H³s0fhk<³´~y0i

φ(y) =[K,+∞)(y)

Huhφ(y) = (y − K)+

f3 ¦k¶Çr0z¤uwRvw¥dHy0z¤s¾q Hsy = K

½ÆeBfhk1z|wk Æ~H³¥|~? H¥¤z¤À¹Hs0z|~u¼z|r"s~wktt~i~Rrk·s0fhk:3 q~H­³\vwuhjs0z|~u

φr

φ = gε + hε

z¤uºr0vh%fƧ©tq s0f3Hshε

z|rs¾§z|tk¶[z®­kjykjuas0zdHw¥|kÇHuhgεt~u-s%Hz¤uhrs0fhk¶rz¤uwRvw¥dHy0z¤s¬q¼~H³

φ

r%ktk7kR½ h½;í >W³´~y[z¤Rz¤s%H¥~ws0z|~uhr(Huh;í >W³´~y»8vwy~Lk Hu~ws0z|~uhr ½¿@ww¥¤q?z¤uwOs0fhknH¥¤¥¤zdt¦?z¤uHwwy~%fÇs~

gεHuh¯vhrz¤uwµh½í¸©§©k7kjs

Theta = e−τrE

[(

Λ(u, v, w)gε(Sxτ ) + λ

∫ +∞

−∞(φ(Sx

τ + c(Sxτ )y) − φ(Sx

τ ))ν(dy)

)]

−re−τrE [hε(S

xτ )] + e−τr

E[

h′ε(S

xτ )a(Sx

τ )]

+1

2e−τr

E[

h′′ε(S

xτ )b2(Sx

τ )]

.

ADk(Hww¥¤z|kt·s0fhk@z¤u-skjRy%Hs0z|~u¯aq3Hy0sr¿iOkjs0fh~?·s~+kjs¨y0z|1~H³Ws0fhk.:hyr0sBHuh1r%ktt~uh·wkjy0z¤¦RZ«s0z¤¦ktr~u

gε½8¬u¶s0fhkkt~iOkjs0y0z|iO~?wkj¥Ï§z¤s0fs0fhk~ws0z¤i·H¥W§©kjz¤Rf-s

Λ(u, v, w)s0fwz|rRz¤¦ktr?D

Theta = −re−rτE [φ(Sx

τ )]

+re−rτ

στE [gε(S

xτ )Wτ ] +

e−rτ

2τE

[

gε(Sxτ )

(

W 2τ

τ− σWτ − 1

)]

+re−rτE[

Sxτ h′

ε(Sxτ )]

+σ2

2e−rτ

E[

Sxτ

2h′′ε(S

xτ )]

.

3~y[z¤Rz¤s%H¥ ~ws0z|~uhr§©k7s%HËk

hε(y) =1

2

(

1 +y − K

ε

)2 ]−ε,0](y − K) +

(

1 − 1

2

(

1 − y − k

ε

)2)

]0,ε[(y − K)

+[ε,+∞)(y − K),

HuhÇ~ws%Hz¤us0fhk(³\~¥¤¥|~¹§z¤uw·Ry%Hwfhr-q¶nÆ~uask±« BHy0¥|~·r0z¤i"vw¥dHs0z|~u D

îî ëHJILK±õLM

R 0% ?, 5/ L 5

Number of trials=2e+04

Localized Malliavin formula

80 90 100 110 120 130 140 150 160Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t-0.5

0

0.5

1

1.5

Theta value

T = 1x = 100

σ = 0.15

r = 0.05

ε = 0.001

½#z¤Rvwyk+ D8eBfhkjs%Or"³\vwuhjs0z|~uÇ~H³

tHuh

K-q¯s0fhk<¥|~? H¥¤z¤À ktnH¥¤¥¤zd ¦az¤uÆiOkjs0fh~a

¾u¶s0fhk7 rk~H³8»8vwy~k HuÇ~ws0z|~uhr§©k7s%HËk

hε(y) =1

4ε(y − (K − ε))2

[−ε,ε](y − K) + (y − K)

]ε,+∞)(y − K),

§fwz|%fq?z|kj¥|wrs0fhk(³´~¥¤¥|~ §z¤uw·Ry%HwfÇ-q¶nÆ~uask BHy0¥|~Or0z¤i"vw¥dHs0z|~u D

#î

! "#$%&(')+*, -.0/1$2,0 E

Number of trials=2e+04

Localized Malliavin formula

80 90 100 110 120 130 140 150 160Strike K 0.5

0.6

0.7 0.8

0.9

1

Time t

0

5

10

15

20

Theta value

T ER

x E R[

σ [½FE [

r [½íR[

ε [½íR E

#z¤Rvwyk+ D8eBfhkjs%Or"³\vwuhjs0z|~uÇ~H³tHuh

K-q¯s0fhk<¥|~? H¥¤z¤À ktnH¥¤¥¤zd ¦az¤uÆiOkjs0fh~a

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Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399