Numerical Computation of Theta in a Jump-Diffusion Model by Integration by Parts

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Submitted on 19 May 2006

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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 Model by Integration by Parts. [Research Report] RR-5829, INRIA. 2006, pp.32. �inria-00070196�

ISRN INRIA/RR--5829--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Numerical Computation of Theta in a

Jump-Diffusion Model by Integration by Parts

Delphine David — Nicolas Privault

N° 5829

February, 8 2006

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Theta_t = ∂C

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∂T(x, t, T).

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³\y~ˆi s0fhk¨~ˆuhk¨šRz¤¦”kju)³\~ˆy

Thetat =−ThetaT

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∂x

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∂x²

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C(x, t, T, K) =e^−RtTrsdsE[(S_T −K)⁺|S_t=x],

~ˆvwyiOkjs0fh~?‰k±°[skjuh‰wrs0fhk+†Hy0šRvwiOkjuas@~H³.;í— >W§fwz|ŠfÆqaz|kj¥|‰wrs0fhk+Á@vw‡wz¤yk+}ÌÁ<»

Theta_T =−Kr_T ∂C

∂K(x, t, T, K) + 1

2K²σ_T²(K)∂²C

∂K²(x, t, T, K).

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p?ktŠjs0z|~ˆuº˜Ç§¨k1ykj¦az|kj§Ès0fhk1Št~ˆi•‡wvws%†Hs0z|~ˆu!~H³

Theta_t ¦?zd†s0fhk:Ê©¥d†ˆŠËa«ªp?Šfh~ˆ¥|ktr•}ÌÁ<» z¤uÅs0fhk

²vwi•‡[«¬‰[z®­vhr0z|~ˆu!Š †ˆrk¯vhr0z¤uwšÄs0fhkÇÍ(yktkjË[rOÁ<kj¥¤s%†Ä†Huh‰Í7†Hi•i·†[“B†Huh‰¼s0fhk¶Št~ˆi•‡wvws%†Hs0z|~ˆu ~H³

Theta_T ¦azd†"s0fhk7Á(vw‡wz¤yk}ÌÁ<»¼³´~ˆy»8vwy~ˆ‡k †HuÇ~ˆ‡ws0z|~ˆuhrz¤u†Št~ˆu-s0z¤uavh~ˆvhri·†Hy0˔kjs ½ ADk<uh~ˆsk s0f3†Hs"s0fh~Rrk•ykj‡wyktrkjuas%†Hs0z|~ˆuhr)³\~ˆy0i"vw¥d†ˆr"‰[z®­Ïkjyz¤uŚ”kjuhkjy%†H¥Ì³\y~ˆi s0fhk·~ˆuhk·~ˆ’ws%†Hz¤uhkt‰ z¤uÅs0fwz|r

‡3†H‡kjy ½„¬u¶p?ktŠjs0z|~ˆuh“avhr0z¤uwš"s0fhkn†H¥¤¥¤zd†t¦?z¤u:Š †H¥|Šjvw¥¤vhr “a§¨k(~ˆ’ws%†Hz¤u¶†Hu1k±°[‡wyktrr0z|~ˆu1~H³

Theta_T

z¤udž<ªvwi•‡[«¬‰[z®­vhr0z|~ˆuiO~?‰wkj¥Ï§fwz|ŠfW“3z¤us0fhk<s0z¤iOk7fh~ˆiO~ˆš”kjuhkt~ˆvhr(Š †ˆrkR“wyk †ˆ‰wr

Theta =e^{−τ r}E

Λ_τ(u, v, w)φ(S_τ) + Z +∞

−∞

(φ(S_τ +c(S_τ)y)−φ(S_τ))µ(dy)

,

§z¤s0f¯†Hy0’wz¤s0y%†Hy0qO‡3†tq”~H­¶³\vwuhŠjs0z|~ˆu

φ(·)^†Huh‰ S₀ =x^½‚@kjykR“ Λ_τ(u, v, w) z|r©†7y%†Huh‰w~ˆiȧ¨kjz¤šRfas

§fwz|Š%f+z|rWk±°[‡w¥¤z|Šjz¤s0¥¤q<Št~ˆi•‡wvwskt‰Ï“

µ(dx)^†Huh‰ c(·)†Hyk8yktr0‡ktŠjs0z¤¦”kj¥¤q@s0fhk@µ:huwz¤sk¹¸ BWxj¦?q@iOk †ˆrvwyk

†Huh‰"s0fhk©¦”~ˆ¥d†Hs0z¤¥¤z¤s¬q)Št~ak4·Šjz|kjuasÌ~H³3s0fhk¨Št~ˆi•‡~ˆvwuh‰}~ˆz|r%r~ˆu‡wy~aŠtktrr‰[y0z¤¦?z¤uwš

(S_t)_t∈^R+

½€(r#z¤u

;>¬“ ;íŒ >¬“[‰[z®­Ïkjykju-s0zd†Hs0z|~ˆuz|r©‡kjy²³\~ˆy0iOkt‰¯~ˆuw¥¤qO§z¤s0f¯yktr0‡ktŠjsBs~+s0fhk<Ê©y~ §uwzd†HuŠt~ˆi•‡~ˆuhkju-s~H³

s0fhk@‡wy0z|Štk(‡wy~aŠtktr%r ½8eBfhk7†H’L~¹¦”k(k±°[‡wyktrr0z|~ˆu:z|r©z¤uh‰wkj‡Lkjuh‰wkjuas~H³ s0fhk@³JvwuhŠjs0z|~ˆu3†H¥‡3†Hy%†HiOkjskjyr

u, v, w“̆Huh‰Dz¤uºp?ktŠjs0z|~ˆu¼Œ§¨k:‰wkjskjy0i•z¤uhk·s0fhk·‡3†Hy%†HiOkjskjyr)qaz|kj¥|‰[z¤uwšs0fhkO’ktr0s+uavwiOkjy0z|Š †H¥

‡kjy²³´~ˆy0i·†HuhŠtk’-q•i•z¤uwz¤i•z¤À¹†Hs0z|~ˆu¶~H³Ïs0fhk¦R†Hy0zd†HuhŠtk(~H³s0fhk§¨kjz¤šRfas

Λ_τ(u, v, w)^“h†Huh‰ :huh‰·s0f3†Hs s0fwz|rOi•z¤uwz¤i+vwi z|r·†Hs0s%†Hz¤uhkt‰!§fhkju

u, v, w †HykNJt~ˆuhr0s%†Hu-s·³JvwuhŠjs0z|~ˆuhr ½ €@rzd†Hu!~ˆ‡ws0z|~ˆuhr1†Hyk

Št~ˆuhrz|‰wkjykt‰ºz¤u p?ktŠjs0z|~ˆu —[½ paz¤i"vw¥d†Hs0z|~ˆuhrO³´~ˆyO‰[z¤šRz¤s%†H¥@†Huh‰!»Ìvwy~ˆ‡Lk †Hu~ˆ‡ws0z|~ˆuhr¹“¨vhrz¤uwšÄs0fhk

n~ˆu-skB†Hy0¥|~•iOkjs0fh~a‰Ï“†Hyk7‡wyktrkju-skt‰z¤up?ktŠjs0z|~ˆu1<"s~OŠt~ˆi•‡3†Hyk7s0fhk‡Lkjy²³´~ˆy0i·†HuhŠtk~H³s0fhk

:huwz¤sk·‰[z®­kjykjuhŠtk•iOkjs0fh~?‰Äs~¯s0f3†Hs+~H³Bs0fhkOn†H¥¤¥¤zd†t¦?z¤u Š †H¥|Šjvw¥¤vhr"†H‡w‡wy~”†ˆŠ%fW½·€¥|~aŠ †H¥¤z¤À¹†Hs0z|~ˆu

†H‡w‡wy~”†ˆŠ%fÄz|r‡wyktr%kju-skt‰z¤uÄp?ktŠjs0z|~ˆuD·s~·ykt‰[vhŠtk)s0fhk)r0z¤uwšRvw¥d†Hy0z¤s¬qƇL~ˆz¤uasr<‡wfhkjuh~ˆiOkjuh~ˆuÄ~ˆu

s0fhk7rz¤i+vw¥d†Hs0z|~ˆuǚRy%†H‡wfhr¹½

B

„¾uDs0fwz|r)rktŠjs0z|~ˆu §©k·yktŠ †H¥¤¥©r~ˆiOk•³Ý†ˆŠjsr†Huh‰Duh~ˆs%†Hs0z|~ˆu¼~ˆu s0fhk·n†H¥¤¥¤zd†t¦?z¤u¼Š †H¥|Šjvw¥¤vhr"~ˆuÅs0fhk

A z|kjuhkjyBr0‡3†ˆŠtkR“aŠ±³#kR½šh½ ;FE –>¬“?†Huh‰:~ˆu:rs~aŠ%f3†ˆr0s0z|Š@Š †H¥|Šjvw¥¤vhr©§z¤s0f)²vwi•‡hr “ar%ktkkR½šh½Ì†Huh‰1;í˜ >³´~ˆy

†)yktŠtkjuasz¤uas0y~a‰[vhŠjs0z|~ˆuƧz¤s0fyk±³\kjykjuhŠtktr¹½CADk7§¨~ˆy0˶~ˆudž‡wy~?‰[vhŠjs

(Ω, P) = (ΩW ×ΩX, PW ⊗PX)

~H³#‡wy~ˆ’3†H’wz¤¥¤z¤s¬q¶r0‡3†ˆŠtktr~ˆu¶§fwz|Š%fdžHyk<yktr0‡LktŠjs0z¤¦”kj¥¤q¯‰wk:huhkt‰†)r0s%†Huh‰h†Hy‰Ê©y~ §uwzd†HuÇiO~ˆs0z|~ˆu

(W_t)_t∈^R+

†Huh‰†Št~ˆi•‡L~ˆvwuh‰}#~ˆz|rr~ˆuLJwy~aŠtktrr

(X_t)_t∈^R+

z¤uh‰wkj‡Lkjuh‰wkjuas~H³

(W_t)_t∈^R+

“[§z¤s0f

B xj¦aq:iOk †ˆr0vwyk

µ(dy) ^†Huh‰ :huwz¤sk7z¤uaskjuhr0z¤s¬q

λ= Z _∞

−∞

yµ(dy),

§fwz|Š%fNJ †Hu’k<ykj‡wyktrkju-skt‰†ˆr

X_t =

Nt

X

k=1

U_k, t∈R+, ^{µ´Ž[½FE¹¸}

§fhkjyk

(N_t)_t∈R₊

z|r†(}#~ˆz|rr~ˆu‡wy~?Štktrr§z¤s0fz¤u-skjuhr0z¤s¾q

λ^†Huh‰ (U_k)_k≥1 ^z|r†HuzݽzݽÉϽ#rkt{avhkjuhŠtk

~H³8y%†Huh‰w~ˆi ¦ˆ†Hy0zd†H’w¥|ktr@§z¤s0fLJwy~ˆ’3†H’wz¤¥¤z¤s¾qlj[z|r0s0y0z¤’wvws0z|~ˆu

ν(dx) =λ⁻¹µ(dx)^½ ADk"‰wkjuh~ˆsk’-q

(Ft)_t∈R₊

s0fhk :h¥¤s0y%†Hs0z|~ˆuš”kjuhkjy%†Hskt‰’aq

(W_t, X_t)_t∈R₊

½

ADk:Št~ˆuhr0z|‰wkjyšRy%†ˆ‰[z|kju-s·†Huh‰¼‰[z¤¦”kjy0š”kjuhŠtk¶~ˆ‡kjy%†Hs~ˆyr

D ^†Huh‰ δ †ˆŠjs0z¤uwšÄ~ˆu s0fhk¯Št~ˆuas0z¤u-vh~ˆvhr

Št~ˆi•‡~ˆuhkju-s1~H³¿²vwi•‡[«¬‰[z®­Lvhr0z|~ˆuy%†Huh‰w~ˆi ³JvwuhŠjs0z|~ˆu3†H¥|r¹½ B kjs

D : L²(Ω) → L²(Ω×R+)

‰wkjuh~ˆsks0fhk·µJvwu-’~ˆvwuh‰wkt‰3¸n†H¥¤¥¤zd†t¦?z¤ušRy%†ˆ‰[z|kju-s(~ˆus0fhkA z|kjuhkjyr0‡3†ˆŠtkR“[zݽÃkR½

D_tF(ω_W, ω_X) =

n

X

k=1

1_[0,tk](t)∂_kf(W_t1, . . . , W_tn, ω_X)

³´~ˆy

F †)y%†Huh‰w~ˆi ¦R†Hy0zd†H’w¥|k~H³s0fhk<³´~ˆy0i

F(ωW, ωX) =f(Wt1, . . . , Wtn, ωX),

§fhkjyk

f(·, ωX) ∈ Cb^∞(Rⁿ)^“ PX(dωX)«¾†[½Ãr¹½¤“Wz|r<vwuwz®³´~ˆy0i•¥¤q’~ˆvwuh‰wkt‰Ä~ˆu

ΩX

½Á<kjuh~ˆsk’-q

h·,·iL²(R₊)

†Huh‰

k · k s0fhkrŠ †H¥d†Hy‡wy~?‰[vhŠjs@†Huh‰¯uh~ˆy0i z¤u

L²(R+)“3†Huh‰‰wk:huhk

D_uF =hu, DFi, u∈L²(Ω×R+).

B kjs

I_n(f_n)(ω_W, ω_X) =n!

Z _∞

0 · · · Z _t2

0

f_n(t₁, . . . , t_n, ω_X)dW_t1· · ·dW_tn

‰wkjuh~ˆsk@s0fhk@i+vw¥¤s0z¤‡w¥|k<r0s~?Šf3†ˆr0s0z|Š@z¤u-skjšRy%†H¥W~H³Ws0fhk(r0q?i•iOkjs0y0z|Š³\vwuhŠjs0z|~ˆu

f_n∈L²(Rⁿ₊×Ω_X)

§z¤s0fÇyktr0‡LktŠjss~·Ê¨y~¹§uwzd†HuÇiO~ˆs0z|~ˆu

(W_t)_t∈R₊

½Ì…ktŠ †H¥¤¥Ïs0f3†Hs§©k7f3†t¦”k

D_tI_n(f_n) =nI_n−1(f_n(·, t, ω_X)), t∈R+,

†Huh‰s0fhk<z|r~ˆiOkjs0y0q:³\~ˆy0i"vw¥d†

E[I_n(f_n)I_m(g_m)] =n!1_{n=m}E[hf_n, g_miL²(Rⁿ

+)].

€@u-q

F ∈ L²(Ω_W ×Ω_X) ' L²(Ω_W;L²(Ω_X)) †ˆ‰[i•z¤sr•†ÇŠ%f3†ˆ~Rr•‰wktŠt~ˆi•‡L~Rrz¤s0z|~ˆu¼~H³s0fhk

³´~ˆy0i

F =E[F] + X∞

n=1

In(fn), ^{µ´Ž[½íŽ”¸}

§fhkjyk

fn∈L²(Rⁿ+×ΩX)^“ n≥1“†Huh‰Çs0fhk‰w~ˆi·†Hz¤u

Dom (D) ^~H³ D Št~ˆuhr0z|r0srz¤us0fhk+r%kjs

~H³#³\vwuhŠjs0z|~ˆu3†H¥|r

F §y0z¤s0skjuƆˆr+µ´Ž[½íŽ”¸†Huh‰r%†Hs0z|r²³\qaz¤uwš

E

"_∞ X

n=1

n!nkf_nk²L²(Rⁿ

+)

#

<∞.

„¾u+s0fhk¨rkt{avhkj¥-§¨k¨§z¤¥¤¥?š”kjuhkjy%†H¥¤¥¤q+‰[y~ˆ‡"s0fhk¿z¤uh‰[z|Štktr

ω_W^“ ω_X^½ eBfhk<µJvwu-’~ˆvwuh‰wkt‰3¸ ‰[z¤¦”kjy0š”kjuhŠtk

~ˆ‡kjy%†Hs~ˆy

δ : L²(Ω×R+) → L²(Ω) †ˆ‰H²~ˆz¤uas+~H³

D^“ †H¥|r%~¶Š †H¥¤¥|kt‰s0fhk1pa˔~ˆy~ˆfh~a‰Dz¤u-skjšRy%†H¥Ý“

r`†Hs0z|r:3ktrs0fhk7‰[v3†H¥¤z¤s¬q¶ykj¥d†Hs0z|~ˆu

E[hDF, ui] =E[F δ(u)], F ∈Dom (D), u∈Dom (δ),

†Huh‰s0fhk7‰[z¤¦”kjy0š”kjuhŠtk7³´~ˆy0i+vw¥d†

δ(uF) =F δ(u)−D_uF, u∈Dom (δ), F ∈Dom (D), ^{µ´Ž[½í˜”¸}

§fwz|Š%fr0fh~¹§r•s0f3†Hs

uF ∈ Dom (δ) ‡wy~¹¦az|‰wkt‰

uF ∈ L²(Ω×R+) †Huh‰¼s0fhk¶y0z¤šRf-s²«Ýf3†Huh‰

rz|‰wk’kj¥|~ˆuwš”r@s~

L²(Ω)½…@ktŠ †H¥¤¥#†H¥|r~1s0f3†Hs

δ Št~ˆz¤uhŠjz|‰wktr@§z¤s0f„¾s@wÂÃr@r0s~?Šf3†ˆrs0z|Šz¤u-skjšRy%†H¥8~ˆu r%{-v3†Hyk±«Ýz¤uaskjšRy%†H’w¥|k"†ˆ‰h†H‡wskt‰‡wy~?Štktrrktr¹“[z¤u¶‡3†Hy0s0z|Šjvw¥d†Hy

δ(u) = Z _∞

0

u_tdW_t

³´~ˆy†H¥¤¥ †ˆ‰h†H‡wskt‰Æ†Huh‰Çr{-v3†Hyk±«Ýz¤uaskjšRy%†H’w¥|k‡wy~?Štktrr

(u_t)_t∈R₊

“h†Huh‰

δ(u) =I₁(u), u∈L²(R+).

„¾u!s0fhkÆrkt{-vhkj¥§¨k§z¤¥¤¥(Št~ˆuhr0z|‰wkjy¶†Än†Hy0˔~¹¦azd†HuŲvwi•‡[«¬‰[z®­Lvhr0z|~ˆu ‡wy0z|Štk‡wy~?Štktrr

(S_t)_t∈R₊

šRz¤¦”kjuǒ-q







dSt=at(St)dt+bt(St)dWt+ct(S_t−)dXt, S₀ =x,

§fhkjyk

at(·)^“ bt(·)^“ ct(·) ^†Hyk C¹ BÏz¤‡hrŠ%fwz¤s0À+³\vwuhŠjs0z|~ˆuhr “vwuwz®³\~ˆy0i•¥¤q¶z¤u

t∈[0, T]^“ T >0^½„¾u

s0fhkŠ †ˆrk~H³W†<s0z¤iOkfh~ˆiO~ˆš”kjuhkt~ˆvhr¨š”kt~ˆiOkjs0y0z|ŠiO~?‰wkj¥3vwuh‰wkjyÌs0fhky0z|r0Ëuhkjvws0y%†H¥3iOk †ˆr0vwykR“-s0fhk

Št~?k4·Šjz|kju-sr

a(·)^“ b(·) ^†Huh‰ c(·) §z¤¥¤¥W’Lk<šRz¤¦”kjuǒ-q















a(y) =y(r−λη(y)), b(y) =yσ(y), c(y) =yη(y),

§fhkjyk

r ^†Huh‰ σ(·)^“ η(·) ykj‡wyktrkjuas:s0fhkz¤uaskjyktr0s¯y%†HskR“@†Huh‰ s0fhkŠt~ˆuas0z¤u-vh~ˆvhrdžHuh‰ ²vwi•‡

¦”~ˆ¥d†Hs0z¤¥¤z¤s¾q¯³JvwuhŠjs0z|~ˆuhr¹½

„¾s@wÂÃrB³´~ˆy0i+vw¥d†)³´~ˆy

(S_t)_t∈R₊

yk †ˆ‰wr

φ(S_t) = φ(S_s) + Z _t

s

φ⁰(S_u)a_u(S_u)du+ Z _t

s

φ⁰(S_u)b_u(S_u)dW_u +1

2 Z _t

s

φ⁰⁰(S_u)b²_u(S_u)du+ X

s<u≤t

(φ(S_u−+c_u(S_u−)∆X_u)−φ(S_u−)),

0≤s≤t“h†Huh‰qaz|kj¥|‰wr

E[φ(S_t)] = E[φ(S_s)] +E Z t

s

φ⁰(S_u)a_u(S_u)du

+1 2E

Z t s

φ⁰⁰(S_u)b²_u(S_u)du

+λE Z t

s

Z _+∞

−∞

(φ(S_u+zc_u(S_u))−φ(S_u))ν(dz)du

. ^µ´Ž[½a¸

ƒ3~ˆyk±°w†Hi•‡w¥|kR“hz®³

X_t =a₁N_t¹+· · ·+a_dN_t^d, t∈R+, ^{µ´Ž[½íŒ”¸}

§fhkjyk

(N_t^k)_t∈R₊

“ k = 1, . . . , d“B†Hykz¤uh‰wkj‡Lkjuh‰wkjuas1}#~ˆz|rr~ˆu‡wy~?ŠtktrrktrO§z¤s0f!yktr0‡ktŠjs0z¤¦”k z¤uaskjuhr0z¤s0z|ktr

λ₁, . . . , λ_d“[§¨k+f3†t¦”k

λ=λ₁+· · ·+λ_d ^†Huh‰

ν(dx) = λ₁

λδ_a1(dx) +· · ·+λ_d

λδ_ad(dx),

†Huh‰

E[φ(St)] = E[φ(Ss)] +E Z _t

s

φ⁰(Su)au(Su)du

+1 2E

Z _t

s

φ⁰⁰(Su)b²_u(Su)du

+

d

X

k=1

λ_kE Z t

s

(φ(S_u+a_kc_u(S_u))−φ(S_u))du

, 0≤s≤t.

„¾u¶s0fhk<¥¤z¤uhk †Hy@Š †ˆr%k<§¨k7¥|kjs















a(y) = (r−λη)y, b(y) =σy, c(y) =ηy,

†Huh‰š”kjs







dS_s=rS_sds+σS_sdW_s+ηS_s−(dX_s−λds), S₀ =x,

µ´Ž[½í—”¸

§z¤s0fÆr~ˆ¥¤vws0z|~ˆu

S_t =xexp

r−λη−σ² 2

t+σW_t

Y

0<s≤t

(1 +η∆X_s), t∈R+. µ´Ž[½=<R¸

„¬³

(Xt)t∈R₊

f3†ˆrs0fhk(³´~ˆy0i µ´Ž[½íŒ”¸©§¨k+~ˆ’ws%†Hz¤u

S_t =xexp

r−λη−σ² 2

t+σW_t

(1 +ηa₁)^Nt1· · ·(1 +ηa_d)^Ntd, t∈R+.

Cº, )" )82

Theta

„¾u:s0fwz|r©r%ktŠjs0z|~ˆu:§©k(ykj¦az|kj§ s0fhk<Št~ˆi•‡wvws%†Hs0z|~ˆu~H³

Theta_t ^†Huh‰ Theta_T z¤u1‡3†Hy0s0z|Šjvw¥d†HyŠ †ˆr%ktr “ vhrz¤uwš•yktr0‡LktŠjs0z¤¦”kj¥¤q¯s0fhkÊ©¥d†ˆŠ%Ë-«ªp?Š%fh~ˆ¥|ktr(†Huh‰Á(vw‡wz¤yk}¿Á(»¿r ½

©~ˆuhrz|‰wkjy@†Hu~ˆ‡ws0z|~ˆuǧz¤s0f‡3† q”~H­³JvwuhŠjs0z|~ˆu

φ†Huh‰¶‡wy0z|Štk

C(x, t, T) =e^−RtTrsdsEh φ(S_T)

S_t =xi .

paz¤uhŠtk

t7→e^RtTrsdsC(S_t, t, T)z|r8†i·†Hy0s0z¤uwš-†H¥|kR“R³Jy~ˆi µ´Ž[½a¸±“

C(x, t, T)r%†Hs0z|r:3ktrs0fhk©Ê©¥d†ˆŠ%Ë-«

p?Š%fh~ˆ¥|ktr}ÌÁ<»

Theta_t = ∂C

∂t (x, t, T) ^{µ´˜[½FE¹¸}

= rtC(x, t, T)−at(x)∂C

∂x(x, t, T)−1

2b²_t(x)∂²C

∂x²(x, t, T)

−λ Z +∞

−∞

(C(x+zct(x), t, T))−C(x, t, T))ν(dz),

§z¤s0f

Delta = ∂C

∂x(x, t, T) =e^−RtTrsdsE h

Y_Tφ⁰(S_T)

S_t=xi

, ^{µ´˜[½íŽ”¸}

§fhkjyk

(Y_t)_t∈R₊ = (∂_xS_t)_t∈R₊

z|rs0fhk:hyr0s¦ˆ†Hy0zd†Hs0z|~ˆuƇwy~aŠtktr%rr~ˆ¥¤vws0z|~ˆuÆ~H³







dY_t =a⁰_t(S_t)Y_tdt+b⁰_t(S_t)Y_tdW_t+c⁰_t(S_t−)Y_t−dX_t, Y₀ = 1,

†Huh‰

Gamma = ∂²C

∂x²(x, t, T) =e^−RtTrsdsEh

Z_Tφ⁰(S_T) + (Y_T)²φ⁰⁰(S_T)

S_t =xi

, ^{µ´˜[½í˜”¸}

§fhkjyk

(Z_t)_t∈R₊ = (∂_x²S_t)_t∈R₊

z|rs0fhk+rktŠt~ˆuh‰Ç¦ˆ†Hy0zd†Hs0z|~ˆuƇwy~aŠtktrr¹½¨‚(kjykR“

a⁰_t(z)^“ b⁰_t(z) ^†Huh‰

c⁰_t(z) ‰wkjuh~ˆsk7s0fhk‡3†Hy0s0zd†H¥ ‰wkjy0z¤¦ˆ†Hs0z¤¦”ktr@~H³#s0fhktr%k<³JvwuhŠjs0z|~ˆuhr§z¤s0fyktr0‡ktŠjss~

z^½

eBfhk8:hyr0s†Huh‰:r%ktŠt~ˆuh‰¶‰wkjy0z¤¦ˆ†Hs0z¤¦”ktr~ˆu

φz¤u:s0fhk7k±°[‡wyktrr0z|~ˆuhrµ´˜[½íŽ”¸t“µ´˜[½í˜”¸©~H³Á<kj¥¤s%†)†Huh‰

͆Hi•i·†¶Š †HuD’kykjiO~ ¦”kt‰D¦?zd†¯z¤uaskjšRy%†Hs0z|~ˆu ’-q‡3†Hy0sr)†Huh‰Äs0fhk•n†H¥¤¥¤zd† ¦az¤u Š †H¥|Šjvw¥¤vhr “†ˆr+z¤u

kR½šh½);íŒ >¬“ ;>¬“ws~•q?z|kj¥|‰Ç†HuÇk±°[‡wyktrr0z|~ˆu³\~ˆy

Theta_t½Ì‚(~ §¨kj¦”kjy(s0fhkŠt~ˆi•‡wvws%†Hs0z|~ˆu~H³Ì͆Hi•i·†

‰[z¤yktŠjs0¥¤qz¤u-¦”~ˆ¥¤¦”ktrs0fhk :hyrs7†Huh‰Är%ktŠt~ˆuh‰Æ¦R†Hy0zd†Hs0z|~ˆućwy~?Štktrrktr

Y_t ^†Huh‰ Z_t½"l(†HiOkj¥¤qÆz¤u ;=< >¬“

s0fhk7k±°[‡wyktrr0z|~ˆu µ´˜[½íŽ”¸©z|rykj§y0z¤s0skju†ˆr

Delta = ∂C

∂x(x, t, T) = e^−RtTrsds T−t E

φ(S_T)

Z _T

t

Y_s b_s(S_s)dW_s

S_t =x

, ^µ´˜[½a¸

vhrz¤uwšÆs0fhk:k±°?‡wyktr%r0z|~ˆuÅ~H³s0fhk:n†H¥¤¥¤zd†t¦?z¤u ‰wkjy0z¤¦R†Hs0z¤¦”k¯~H³

S_T ∈ Dom (D) ^z¤u skjy0iOr~H³s0fhk :hyrs¦ˆ†Hy0zd†Hs0z|~ˆu‡wy~?Štktrr†ˆr?D

Y_s

bs(Ss)D_sφ(S_T) =Y_Tφ⁰(S_T), 0≤s≤τ, a.s., ^{µ´˜[½íŒ”¸}

Š±³0½ ;FE – >¬½º„¬u¼s0fhk¯uhk±°[sOrktŠjs0z|~ˆu!§©k¶‡wyktrkju-s:†Št~ˆi•‡wvws%†Hs0z|~ˆu ~H³

Theta_T ^’-q z¤u-skjšRy%†Hs0z|~ˆu

’aqLJ3†Hy0sr<§fwz|Š%fW“Ïz¤us0fhks0z¤iOk"fh~ˆiO~ˆš”kjuhkt~ˆvhr"Š †ˆrkR“q?z|kj¥|‰wr<†:‰[z®­Ïkjykju-sykj‡wyktr%kju-s%†Hs0z|~ˆuij´~ˆy

Theta_t=−Theta_T½#„¾s¨‰w~?ktr¨uh~ˆs©‰[z¤yktŠjs0¥¤qOvhr%ks0fhk.:hyrsB†Huh‰1rktŠt~ˆuh‰·¦R†Hy0zd†Hs0z|~ˆu:‡wy~aŠtktrr%ktr “

†Huh‰z¤ua¦”~ˆ¥¤¦”ktr~ˆuw¥¤qÇkj¥|kjiOkju-s%†Hy0q1A z|kjuhkjy7z¤u-skjšRy%†H¥|r7~H³©‰wkjskjy0i•z¤uwz|rs0z|Š)³JvwuhŠjs0z|~ˆuhr<z¤uhr0sk †ˆ‰~H³

„¾s@•r0s~?Šf3†ˆr0s0z|Šz¤u-skjšRy%†H¥|r~H³Ì†ˆ‰h†H‡wskt‰‡wy~?Štktrrktr@†ˆrz¤u µ´˜[½íŒ”¸j½

Î@us0fhk+~ˆs0fhkjyf3†Huh‰Ï“3z¤us0fhk+Š †ˆr%k~H³»Ìvwy~ˆ‡Lk †Hu~ˆ‡ws0z|~ˆuhrz¤uƆOŠt~ˆu-s0z¤u-vh~ˆvhr(i·†Hy0˔kjs “3zݽÃkR½

§z¤s0f

ct(·) = 0^“ at(y) =αty“[§z¤s0fLJ3†tq”~H­³JvwuhŠjs0z|~ˆu

φ(x) = (x−K)⁺“h†Huh‰¶‡wy0z|Štk

C(x, t, T, K) =e^−RtTrsdsE[(S_T −K)⁺|S_t=x],

Á(vw‡wz¤ykRÂÃr³\~ˆy0i"vw¥d† ;í— > yk †ˆ‰wr

b_T(K) = v u u u u t 2

(rT −αT)C+∂C

∂T +KrT ∂C

∂K K²∂²C

∂K²

§fhkjyk ∂C

∂T(x, t, T, K) Št~ˆz¤uhŠjz|‰wktr§z¤s0f

Theta_T½Ì„¾u~ˆs0fhkjyskjy0iOr “

Theta_T = ∂C

∂T(x, t, T, K) ^{µ´˜[½í—”¸}

= (α_T −r_T)C(x, t, T, K) +K²b²_T(K) 2

∂²C

∂K²(x, t, T, K)−Kα_T∂C

∂K(x, t, T, K),

§fhkjyk

−∂²C

∂y²(x, t, T, y) Št~ˆz¤uhŠjz|‰wktr§z¤s0fs0fhk7‰wkjuhr0z¤s¾q·³\vwuhŠjs0z|~ˆu

dP(S_T =y|S_t =x)/dy^½

…@kj¥d†Hs0z|~ˆu µ´˜[½í—”¸+Š †Hu ’k1‡wy~ ¦”kt‰Å’-qņH‡w‡w¥¤z|Š †Hs0z|~ˆuº~H³¯µ´Ž[½a¸+~ˆu

[0, T]“8‰[z®­Ïkjykju-s0zd†Hs0z|~ˆuº§z¤s0f

yktr‡LktŠjsBs~

T“w†Huh‰1z¤uaskjšRy%†Hs0z|~ˆu’-q1‡3†Hy0sr§z¤s0f¯yktr0‡ktŠjs©s~

dy ^~ˆu R½8eBfhk<Št~ˆi•‡wvws%†Hs0z|~ˆuÇ~H³

Theta_T ‡wyktrkjuaskt‰ z¤uDs0fhk·uhk±°?s"rktŠjs0z|~ˆuD³´~ˆ¥¤¥|~ §r+s0fhk·r%†HiOkOrskj‡hr “ ykj‡w¥d†ˆŠjz¤uwšÆz¤u-skjšRy%†Hs0z|~ˆu

’aqŇ3†Hy0sr·~ˆu

R §z¤s0fs0fhk¶‰[v3†H¥¤z¤s¬q ³\~ˆy0i"vw¥d†Ä~ˆuºs0fhk A z|kjuhkjyOr0‡3†ˆŠtkR½¼€@r•r0vhŠ%f!z¤sOŠ †Huº’k

¦?z|kj§¨kt‰¼†ˆr"†š”kjuhkjy%†H¥¤z¤À¹†Hs0z|~ˆuº~H³Á(vw‡wz¤ykRÂÃr•†Hy0šRvwiOkju-s)s~ƆHy0’wz¤s0y%†Hy0q ‡3†tq”~H­!³\vwuhŠjs0z|~ˆuhr+z¤u¼†

²vwi•‡‰[z®­vhr0z|~ˆui·†Hy0˔kjs ½

Theta_T

©~ˆuhrz|‰wkjy@†Hu~ˆ‡ws0z|~ˆuǧz¤s0f‡3† q”~H­³JvwuhŠjs0z|~ˆu

φ†Huh‰¶‡wy0z|Štk

C(x, t, T) =e^−RtTrsdsE h

φ(S_T)

S_t =xi . Theta_T Š †Hu’k†H‡w‡wy~t°[z¤i·†Hskt‰Ç’aq:huwz¤sk‰[z®­ÏkjykjuhŠtktr@†ˆr

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

2T ε . ^{µ h½FE¹¸}

€@¥¤skjy0u3†Hs0z¤¦”kj¥¤q”“Ïs0fhk‰wkjy0z¤¦R†Hs0z¤¦”k§z¤s0fyktr‡LktŠjs(s~

T Š †Hu’k"‡wvws<z¤uhr0z|‰wk)s0fhkk±°[‡LktŠjs%†Hs0z|~ˆuz®³

φz|r‰[z®­Ïkjykju-s0zd†H’w¥|kR½

©~ˆuhr0z|‰wkjy

(S_t,s^x )_s∈[t,∞) šRz¤¦”kju’aq:s0fhk©²vwi•‡[«¬‰[z®­vhr0z|~ˆuÆkt{av3†Hs0z|~ˆu







dS^x_t,s=a_s(S_t,s^x )ds+b_s(S_t,s^x )dW_s+c_s(S_t,s^x ₋)dX_s, S_t,t^x =x.

µ h½íŽ”¸

m(r0z¤uwšO„¾s@wÂÃrB³´~ˆy0i+vw¥d†O†Huh‰ µ´Ž[½a¸B§¨k7f3† ¦”kD

C(x, t, T) =e^−RtTrsdsE

φ(S_t,T^x )

= φ(x)−E Z T

t

r_se^−Rtsrpdpφ(S_t,s^x )ds

+E Z T

t

e^−Rtsrpdpφ⁰(S_t,s^x )as(S_t,s^x )ds

+E Z T

t

e^−Rtsrpdpφ⁰(S_t,s^x )bs(S_t,s^x )dWs

+1 2E

Z _T

t

e^−Rtsrpdpφ⁰⁰(S_t,s^x )b²_s(S_t,s^x )ds

+λE Z _T

t

e^−Rtsrpdp Z _+∞

−∞

(φ(S_t,s^x +zc_s(S_t,s^x ))−φ(S^x_t,s))ν(dz)ds