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Munich Personal RePEc Archive Multidimensional Black-Scholes options Esposito, Francesco Paolo 10 December 2010 Online at https://mpra.ub.uni-muenchen.de/42821/ MPRA Paper No. 42821, posted 24 Nov 2012 17:49 UTC

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Page 1: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

Munich Personal RePEc Archive

Multidimensional Black-Scholes options

Esposito, Francesco Paolo

10 December 2010

Online at https://mpra.ub.uni-muenchen.de/42821/

MPRA Paper No. 42821, posted 24 Nov 2012 17:49 UTC

Page 2: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

t♠♥s♦♥

♦s ♣t♦♥s

r♥s♦ P s♣♦st♦

r♦♣ rt s ♥♠♥t rs ♥∗

strt

♥ ts rt ♣r♦♣♦s ♥ ①t♥s♦♥ ♦ t ss −♦s♦♣t♦♥ ♥ ♠t♠♥s♦♥ st♣ ♥r②♥ ♥♥ sst s st ♦ qt② st♦s ♦♥ ♥r r♦♣♥ t②♣ ♦♣t♦♥ ♣②−♦s ♦♥sr ❯s♥ t strt♦♥ ♦rr tr♥s♦r♠ r ♥r ♦r♠ s♦t♦♥ ♥ ♣r♦ s♥t ♦♥t♦♥ t♦ ♦♥strt t♦r♠r ①♣t② ♥ r② r st ♦ ♥t♦♥s ♥② ♦♣ t♦rt ♦♣t♦♥s r ♥ ♥ ♦s−♦r♠ t rst ♦♣t♦♥ ♥ ①♣rss s ♥r ♦♠♥t♦♥ ♦ t ss ♣t ♦♣t♦♥s t s♦♥ ♦♥ s ♥ ♦♣t♦♥ t ♠t♠♥s♦♥ ♥r②♥ ♥♠② χ

2−♦♣t♦♥

②♦rs ♦s ♠♦ ♣r♥ qt♦♥ ♥r ♦♥st♥t ♦♥ts P str

t♦♥ ♦rr tr♥s♦r♠ ♣♥ ♥ ♦♣t♦♥ χ2−♦♣t♦♥

∗ ♦♣♥♦♥s ①♣rss r r ♠♥ ♥ ♦ ♥♦t ♥ssr② rt t s ♥ ♦♣♥♦♥s♦ ♥② ♠st ♦♥t♥ ♥ ts ♦r r♠♥s ♦ ♦rs ♠♥

Page 3: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♦♥t♥ts

♥tr♦t♦♥

t♠♥s♦♥ ♦s ♠♦

♦ t♣ ♣t♥ ♣♦rt♦♦

r♥ t ♥r s♦t♦♥

♦♥ t P rt Pr♥

♣♣t♦♥s

ss st ♦♣t♦♥ χ2−♦♣t♦♥

♦♥s♦♥s

Page 4: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♥tr♦t♦♥

♥ ts rt ♦♣t t rtr r♠♥t ♥tr♦ ♥ t s♠♥ ♣♣r❬❪ t t ♠ t♦ ♣r♦ ♥r s♦t♦♥ t♦ t ♣r♥ qt♦♥ ♦ ♥②r♦♣♥ ♦♣t♦♥ rtt♥ ♦♥ st ♦ st♦s ♦s ②♥♠s s sr ② ♠t♠♥s♦♥ rs♦♥ ♦ t ♦r♥ −♦s ♠♦ s♦t♦♥s ♦t♥ tr♦ t ①♣♦♥♥tt♦♥ ♦ t ♦♣rt♦r ♥♥ t ♣rt r♥t qt♦♥ P ♥♦ ♥ ♣r♥ ②♥ t ♦♥♦t♦♥ ♦♣rt♦rG ♣r♦s t ♦♣t♦♥ ♣r ♥ ♣♣ t♦ t ♣②−♦ ♥t♦♥ ❲♥t② r② r ♠② ♦ ♥t♦♥s ♦r rr strt♦♥s tt stsst st ♦ ♠ss ♣②−♦s r♥t ♥♠r ♦r ♦s ♦r♠ s♦t♦♥st♦ t ♣r♦♠ ♦ rt ♣r♥ ♥ t s♥ ♦ t r♣t♥ ♣♦rt♦♦❲ ♥♦t tt ts ♠② ♦ st② ♥r ♦♥sr♥ s♥r strt♦♥s t ♦ ♥♦t rtr sss ts ♣♦♥t ♥ t ♣rs♥t ♣♣r♣♥♥ ♦♥ t ♦r♠ ♦ t ♣②♦t t ♦rrt♦♥ strtr ♦ t r♥♦♠s♦rs ♥trs ♥t♦ t ♣r♥ ♦r♠ t♥ t ❲ ♥② ♣r♦t♦ ♦s ♦r♠ ①♠♣s r ♥ t rst ♦♥ t ♠♥s♦♥t② ♦ t♣r♦♠ ♠t srr ♥ t s♦t♦♥s tr♥ ♦t t♦ t ♥r ♦♠♥t♦♥ ♦ ♥ ♣♥ ♥ ♦♣t♦♥s ♥ t s♦♥ ①♠♣ ♦♣ ♥♦ r♦♣♥ ♦♣t♦♥ t χ2−♦♣t♦♥ ♦♥ st ♦ qt② st♦s ♦r s ♦r♥③ s ♦♦s ♥ st♦♥ r t ♠t♠♥s♦♥−♦s ♠♦ ♥ st t ♣r♥ qt♦♥ ♦r ♥② rt rtt♥ ♦♥t qt② st s ♦♥sq♥ ♦ t rtr r♠♥t ♥ st♦♥ r t ♥r s♦t♦♥ ♦r t ♣r♥ qt♦♥ ♥ st ♥t♦♥ s♣♥ st♦♥ sss t♦ ♣♣t♦♥s tr♦ ♦s ♦r♠ s♦t♦♥sr ♦♥ ♥ st♦♥ r t ♦♥s♦♥s

t♠♥s♦♥ ♦s ♠♦

♦ t♣

♥ ts ♦r ♦♦ ❬❪ ♥ ♠♦♥ t st♦ ♣rs ♠rt ♣r ②♥♠s ♦ s♥ qt② st♦ s ♠♦ t t ♦♦♥ st♦str♥t qt♦♥

dSt = St(µ dt+ σ dBt ),

r µ ♥ σ r r ♦♥st♥ts r♣rs♥t♥ t tr♥ ♥ t ♦tt② ♦ tst♦ ♣r S st♦st ♣r♦ss Bτ , τ ∈ [0, T ], s t ♥♦♥ r♦♥♥♠♦t♦♥ ♦sr r♦♠ t♠ 0 t♦ t♠ T qt♦♥ rrr s t−♦s ♠♦ s t ♦♠tr r♦♥♥ ♠♦t♦♥ ♥ ♥ ♥♥ tr♠s

r♦♠ sttst ♣rs♣t t st♠t♦♥ ♦ t ♦♥t µ ♦r ♥−str♣♣st♦ s r② t ②♥ ♥♦♥ s♥♥t② r♥t r♦♠ ③r♦ ♣r♠trs ♦♥ rq♥② t s♠♣ ♦ r st tr r♠rs ♦t ♦♥♥ t♦ s♦rttr♠♦r③♦♥s ♦♥sst ♥ t ♦♠♥t ♦tt② str♥ ♣♥♦♠♥♦♥ ♥ st s②♠♠tr② ♥ t ♠♣r ♣r strt♦♥s r ❬❪ ❬❪ r ♥♦t r♣r♦ ② t♦♥t♦♥ ♣r♦t② strt♦♥s ♥rt ② t

Page 5: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

t stts tt t ♦♥t♥♦s t♠ ♦♠♣♦♥♥ s st t♦ ss♥r♥♦♠ ♥♦s ts t r♦t rt❲♥ ♦♥sr♥ ♥ qt② st♦ st s t rt rr♥ ♥r②♥t qt♦♥ s t♦ t ♥t♦ ♦♥t t ♥trt♦♥ ♠♦♥ t ♠♥ts ♦t st ♦ ♥st♠♥t ♦♣♣♦rt♥ts r♦r ♦♦s t♦ ♠♦ t ♠rt♣r ②♥♠s ♦ t st♦s t t ♠t♠♥s♦♥ −♦s ♠♦

tt s ♥ sdSj

t = Sjt (µj dt+ σj dW

jt )

j=1,...,n,

r Wj , j = 1, . . . , n r t ♦rrt r♦♥♥ ♠♦t♦♥s strt♥ t t =

0 t Bj0 = 0 s ∀j ♦rrt r♥♦♠ s♦rs r ss♠ t♦

♥rt ② t qt♦♥ Wt = Q · Bt t Q sqr r♥ ♥ s tt

Q =[√

aii]−1 ·U ♥ A = U ·U ′ A > 0 ② ♦♥strt♦♥ Q·Q′ s t ♦rrt♦♥

♠tr① ♦ t r♦t rt ♦ t st ♦♠♣♦♥♥ts ♥ d[W i, W j

]= ρijdt

r♥ ss♠♣t♦♥ ♦ r♦♣♣ t♦t ♠♦r ♠♦t♦♥s

♣t♥ ♣♦rt♦♦

♥ ♦rr t♦ ♣r t r♦♣♥ ♦♣t♦♥ ♦♥ t ♦♥t ♦ t st ♦♠♣♦♥♥ts sr ♥ ♠trs t t♠ T ∈ R ♥ 1R ∼ ②r ①♣♦tt rtr r♠♥t s t s ♥ ♣rs♥t ♥ ❬❪ ♥ tr ♦r♠③ ♥❬❪ ❬❪ s ♣♣r♦ ♦s t♦ rt ①♣t② t ♣rt r♥t qt♦♥♦s s♦t♦♥ ②s t ♣r ♦ t s♦t ♥♥ rt s ♣r♠tr③ ♥t♦♥ ♦ t qt② ♣rs

❲ ♥ s t♦ ♣r♦r♠ t ♥ ♦ r Xjt = log

(Sjt

Sj

0

), ♥ t♦ ①♣rss

t rts ♣r ♥ tr♠s ♦ t r♦t t♦rs ♦ t st ♦♠♣♦♥♥tsf(X1, . . . , Xn) rtr t♥ tr ♣r ♥t② t ♣r♥ ♦r♠ ♥ ♦♥rt ♥t♦ t S ♣♥♥t ♥t♦♥ h = f Xt Π t ♣♦rt♦♦ ♠ ♦ s♦rt ♣♦st♦♥ ♥ t rt f ♥ ①♣♦st♦ t n st♦s

Sj

j=1,...,n

Π =

n∑

j=1

∆jSj − f ⇒ dΠ =∑

j∆jdSj − df,

r ∆jSj s t s ♠♦♥t ♥st ♥ t qt② st♦ j t t♠ t r

s② ♣♣②♥ t♦s r ♥ stt♥ ∆j =∂jfSj

r t♦ ♥rt ♥ sst

tt s r r♦♠ st♦st rt♦♥s ♥ rt♦♥ss ♦r ts rt sst♦ r♦ t t rs−r rt r ≥ 0 r♦r st dΠ = rΠdt ♥ t ∂j t ♣rt rt t rs♣t t♦ xj ♥ ∂t t ♣rt rt trs♣t t♦ t♠ ♦t♥ t ♦t♦♥r② ♣rt r♥t qt♦♥ P

∂tf = rf −∑

j (r − σ2j/2) ∂jf − 1

2

∑j

∑kρjkσjσk∂

2jkf,

s t ♣r♥ qt♦♥♥② rrs t t♠ ①s ♥ st t = T −t t② tr♥♥ t tr♠♥

Page 6: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♣②−♦ ♦♥t♦♥ ♥t♦ t ② ♣r♦♠ ❲ rt t ♣r♥ qt♦♥ ♥t ♦♦♥ ♦♠♣t ♦r♠

∂tf = −D (∂X) · f

f |t=0 = g.

r ♥ t ♣♦②♥♦♠ ♦♣rt♦rD (∂X) = r−∑j αj∂j−1/2

∑j

∑k ρjkσjσk∂

2jk

t αj = (r − σ2j/2) ♥t♦♥ g : X ∈ R

n 7→ g(X) ∈ R r♣rs♥ts t ♣rsr ♣②−♦ ♣r♦

r♥ t ♥r s♦t♦♥

♦t ♦ ts st♦♥ s t♦ ♦♥strt ♥r s♦t♦♥ ♦ tt ♦s s t♦ ♣r ♥ ♥② st rt ♦ t r♦♣♥ t②♣ ♦s♥r②♥ ②♥♠s s r♣rs♥t ② s♦t♦♥ ♦ t ♣r♥ qt♦♥♣♥s ♦♥ t s♣ ♥t ♣r♦♠ ♥ ♥♥ tr♠s s r♣rs♥t ② t ♥ ♥ t rrs t♠ ♣②−♦ ♦ t st ♦♣t♦♥ s rst t ♦♣t♦♥ ♦r♠ s t ♠ ♦ t tr♠♥ ♣②−♦ tr♦ t♦♥ ♦♣rt♦r ♦♥t♦♥s ♣♦♥ t st ♦ ♠ss tr♠♥ ♥t♦♥s rs ♥tr② r♥ t ♣r♦ss ♦ s♦♥ t ♠♥ qt♦♥ ts♦r t s ♦ ♦♣♥ ♣♣t♦♥s t t st ♦ ♠ss ♥t♦♥t♦♥s s r s ♣♦ss s ♣r♦s② r♠r t s ♣♦ss t♦ ♥rts st ♥tr♦♥ t ss ♦ ♣s♦−♥t♦♥s

❲ st ♣ ♦r st② ♥ t ♦♥t①t ♦ t t♦r② ♦ strt♦♥s ①♣♦t♥t ♣r♦♣rts ♦ ts ♠t♠t ♦ts t♦ ♥t② ♥ rtr③ ts♣ ♦ s♦t♦♥s ♥ ♥ ♥♦t ①♣t② ♥♦t t ♥t♦♥s ♥ rsts♠st t♥ ♥ t strt♦♥ s♥s ♥ ts ♦r t ♦♦♥ ♥♦tt♦♥ ss r ❬❪ ❲ rr t♦ t strt♦♥ ♦rr tr♥s♦r♠ (u) = u♥ t ♥t♦♥ s♣s D∗ t s♣ ♦ strt♦♥s S∗ t s♣ ♦ t♠♣rstrt♦♥s E∗ t s♣ ♦ strt♦♥s t ♦♠♣t s♣♣♦rt ♥ Z∗ := (D∗) ❲ s♦ rr t♦ t s♣ ♦ ♥t♦♥s ♦ r♣ s♥t S ts♣ ♦ ♥♥t② s♠♦♦t ♥t♦♥s E := C∞ ♥ t s♣ ♦ ♦② s♠♠♥ s♦t② s♠♠ ♥t♦♥s rs♣t② 1

loc♥ 1 ♥ ♥① t s

♦♥♥t♦♥② ♣♣♥ t♦ t s②♠♦ ♦ t s♣ t♦ ♥t tt t s♣♣♦rt♦ ts ♠♥ts ♦♥s t♦ R

n×T r T := [0,∞) ❲♥ t f ∈ ♦s ♥♦t♣♥ ♦♥ t♠ t ♥① t s r♦♣♣ ♥ supp f ⊆ R

n

♦♥ t P

❲ sr ♦r s♦t♦♥ ♥ t s♣ ♦ strt♦♥s D∗t t ♦

Dt s♦t♦♥ strt② ♦♥ssts ♥ ♦♥strt♥ t−♦ ♦ t ♣②−♦♥t♦♥ ♥ t♥ ①t♥♥ t s♦t♦♥ ♥ t ♠t ❲ ♦♥ s♦ ♠st♦♥sr t ♣rt② ♦ t ♣r♦♠ t rs♣t t♦ t t♠ ♣r♠tr♣♥♥② r♦r st♥s t t♦ ss r t ∈ (0, T ]

Page 7: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♥ t = 0 stt♦♥ ♥ T → +∞ s ♥♦t ♣rtr② ♥trst♥ ♥t ♥♥ ♣rs♣t s ②s ① ♥t stt♠♥t t ♦rt ♦♣t♦♥ ❲ ♥♦t tt ♥ ♦rr ♦r t qt♦♥ t♦ s♦t♦♥ trts ♣r♦ t ♠trt② s ♥t♦♥ g : Rn → R ♠st ♥ st

ss♣ ♦ D∗ t =e−‖x+α‖2

α∈R

♣r♠tr ♠② ♦ ♥t♦♥s

♦ r♣ s♥t tt s ⊂ S❲ ss♠ tt t ♥t ♥t♦♥ g s ♥ 1

loc♥ s s tt

gh ∈ 1, ∀h ∈ .

♦♥t♦♥ ♦r t st stt♠♥t t♦ r s tt g s ♥ ♣♣r♦♣rt♦r t ♥♥t② t

=g ∈ 1

loc: g ∈ O

(‖x‖−αe‖x‖

2), α > n

.

t s ♥rst♦♦ tt g ∈ ⇒ gh ∈ 1, ∀h ∈ ② ss ♦♠♥t ♦♥r♥ r♦r t♥ t sr st♣ r ♥ t ♣♦st♦♥t♦ s♦ t ♣r♦t♦t②♣ qt♦♥

Pr♦♣♦st♦♥ t g ∈ ♥ P (ξ) = δ + α · ξ + 1/2 ξ ·Ω · ξ t ξ ∈ Cn

♣♦②♥♦♠ s tt Ω > 0 ♥ V =√Ω s♦t♦♥ f ∈ D∗

t ♦ t qt♦♥

∂tf = P (∂X) · f

f |t=0 = g

s t f s tt

f = (G ⋆ g) eδt

r G s t ss♥ ♥t♦♥

G = |V |−1

√2πt

n e−1

2t‖X‖2

♥ X = (V ′)−1 · [X + tα]

Pr♦♦ t ♣♣♥①

❲ rtr ♥♦t tt qt♦♥ ♠ts st② stt ♦r ♥δ ≤ 0 ♠♣②♥ f t♦ ♦♥r t♦ t ③r♦ strt♦♥ ♥ t t → +∞

rt Pr♥

♥② r ♥ t ♣♦st♦♥ t♦ s♦ t ♣r♥ qt♦♥ ♦r t rth(T − t, S) ♠♥ st♣ ♦ t ♣r♦r ♦♥ssts ♥ ♦♥rt♥ t ♥t♣②−♦ ♥r ①♣rss ♥ ♣r s ♥t♦ t ♣r♦r♠♥ ♣♥♥t♥t♦♥s g (X) ② ♠♥s ♦ t tr♥s♦r♠t♦♥ S = S0e

X ♥ ♦rr t♦ ♣♣② ♥ ♦t♥ rt ♣r s ♦♠♣tt♦♥② ♠♦r ♥② ♥rtr t♦ rrs t t♠ rt♦♥ ♥ st t = T − t ❲ ♥ s② ♦r♠tt s♦t♦♥ t♦ t ♣r♥ ♣r♦♠ ♦♥sr♥ t s②♠♦ tr♥s♦r♠t♦♥f = h t S r♦r t g ∈

Page 8: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♦r♦r② ♣r ♦ t rt f (t, X) rtt♥ ♦♥ t ♣r♦r♠♥

♦ t st Sjj=1,...,n ♣②s ♦t g (X) t ♠trt② tt s t = 0 st f s tt

f = (G ⋆ g) e−rt

Pr♦♦ s s strt♦rr ♣♣t♦♥ ♦ t ♣r♦♣♦st♦♥

♦r♦r② ♦s t♦ ♦♥strt ♥♠r ♥♦r ♦s−♦r♠ s♦t♦♥s♦r ♥② rt rtt♥ ♦♥ t rr♥ st Pr♦ tt t sttst♠sr s ♥ ② t t ♣r ♦ ♥② rt ♦♥ t ♣r♦r♠♥♦ t st ♦♠♣♦♥♥ts ♥ s♥ s ♥t♦♥ ♦ t tr♠♥ ♣②−♦g (XT ) ② t ♦r♠ ♥ssr② t ♣r♥ qt♦♥ ♠t ♦♥rt t♦ t h = f t X

Page 9: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♣♣t♦♥s

♥ t ♦♦♥ st♦♥s ♦♣ t♦ ♣♣t♦♥s ♣r♦ ♦s−♦r♠s♦t♦♥s ♥ t rst ♦♥ t s ♥t ♦ t ♣②−♦ ♥t♦♥ ♠t ♠♣tt ♦r♠ ♥ tr♠s ♦ t ♦♥t ♣r♦r♠♥ ♦ t st ♦♠♣♦♥♥ts ♥t② r♥ ♦ t ♦rrt♦♥ strtr ♦ t ♥r②♥ t♠s s♦♥♣♣t♦♥ s ♥♦ ♠t♠♥s♦♥ ♦♣t♦♥

ss st ♦♣t♦♥

♥ t rst ♣♣t♦♥ t s♣ ♦r♠ ♦ p ♦ t♦ ♣rtt♦♥ t s♦t♦♥♦ t qt♦♥ ♥t♦ ♣rtr② s♠♣ ♥trs tt ♥ s♦ tt ♠t♦ ♦ st♦♥s r ❬❪ ♥ ts s t ♠trt −♦s♦r♠ tr♥s ♦t t♦ ♥r ♦♠♥t♦♥ ♦ ss qt② ♦♣t♦♥s

t s ss♠ tt t ♦♣t♦♥ ♣②s ♦t ♥r ♦♠♥t♦♥ ♦ t ①ssrtr♥ ♦♥ ♥tr② ♣♥♥t ♦♥ ♦♥t a tt s

p =∑

j

φj

Sj − Sj0e

a, Sj > Sj0e

a

0, Sj0e

−a < Sj ≤ Sj0e

a

Sj0e

−a − Sj , Sj ≤ Sj0e

−a

,

r φj ∈ R r ♠t♣rs qt♦♥ ♠♣s tt ♦♥ s♥tr♠♥ ♦♥t♦♥ t ♦♣t♦♥ ♣②s ♦t t s♦t ①ss rt♦♥ t♥r②♥ ♣r ①s ±a ♦−rtr♥ ♦♥ stt ♥t ♥ r♦♠♥♦tr ♣♦♥t ♦ ♦♥ ♥r②♥ t ♦♣t♦♥ ♣②s−♦t t s♦trt♦♥ t rs♣t t♦ t ♦♥r② strs K±

j = Sj0e

±a♣♣②♥ t ♦♣rt♦r t♦ t ♥t ♦♥t♦♥ g = p S ♦r♠② ♦t♥t ♦♣t♦♥ ♣r ♥ ♦♥♦t♦♥ ♥ ♦♠♣♦s ♥t♦ ♥trs ♦ts ♦rt♦♦♥ t♦ jth ①s tt s

f = e−rt∑

j φjSj0

∫Ω+

j

(exj − ea)Gt +

+∫Ω0

j

Gt +∫Ω−

j

(e−a − exj )Gt

r Ω+j Ω

0j Ω

−j r rs♣t② t st♦rs ♦ t ♥ s♣ ej ·

X > a −a < ej ·X ≤ a ej ·X ≤ −a ♥ ej s t jth ♥♦♥ t♦rtr s♦♠ r t ♦♦♥ ♦s ♦r♠ s♦t♦♥ s ♦t♥

f =∑

j φjSj0

exjF (α1j)− e−rt+aF (ω1j) +

+ e−rt−aF (ω2j)− exjF (α2j)

t ♥t♦♥s F (·) ♥ t ♠t strt♦♥ ♥t♦♥ ♦ st♥r♥♦r♠ r ♣r♠trs α ♥ ω r rs♣t②

Page 10: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

α1j =−a+Xj+(r+σ2

j/2)t

σj

√t

ω1j =a+Xj+(r+σ2

j/2)t

σj

√t

α2j =a−Xj−(r−σ2

j/2)t

σj

√t

ω2j =−a−Xj−(r−σ2

j/2)t

σj

√t

♥ ♥ t s −♦s ♥♦tt♦♥

Xj = logSj

Sj

0

Sj0e

±a = K±j

♦r♠ s ♦♠♥t♦♥ ♦ ♥ ♣t s♥st♦ ♦♣t♦♥s ♦ s ♥ rtt♥ ♦♥ t jth st♦ ♥ ♣②♥ ♦t ♥tr② ①ss rtr♥st rs♣t t♦ t strs Sj

0ea ♥ Sj

0e−a

t ♥ ♥♦t tt t qt♦♥ s ♦ t ♣rtr ♦r♠ ♦g s ♥♣♥♥t ♦ t ♦rrt♦♥ strtr ♦ t st♦st ♣r♦sss ttsr t ②♥♠s ♦ t ♥r♥s ♥ ♦r♠ rsts ♥ ♥r♦♠♥t♦♥ ♦ str− ♦♣t♦♥s ♦ s ♣②♥ ♥tr② trs♣t t♦ t ♥−t−♠♦♥② ♥tr tt s tr♠♥ ② t a ♦♥t♥s t ②♣r tt ♥♦ss t ♦r♥ ♥ tX s♣ t ♦♣t♦♥ ♣②−♦ts rs ♦ts ts ♥♦r♦♦ t ♦♣t♦♥ ♣②s ♦t ♦♠♥t♦♥ ♦♥r② ♥rs♥ rtr♥s ♥ ♥② rt♦♥♥ r ♣♦t t s♣ ♦ t ♥t♦♥ ♥ t ❳ s♣ t st♦sa = 0.03 σ = 0.2 r = 0.02 ♥ T − t = 1.5 1 0.5 ♥ 0.0001

χ2−♦♣t♦♥

❲t t s♦♥ ♣♣t♦♥ ♦♣ ♥ ♦♣t♦♥ ♦♥ ♥ qt② st ♦s②♥♠s s sr ② t ♠trt −♦s ♠♦ ①♣♦t♥ rst s ♦♥ t ♦♠tr ♣r♦♣rts ♦ t ♥r②♥ ♣r♦t② strt♦♥s ♥ t ♣r s♥ ♦ t ♣②−♦ ♥t♦♥ r t♦ ♣r♦ ♦s−♦r♠ s♦t♦♥ ♦r t ♣r♥ qt♦♥ ♦ s♠♣② t ♣r♦r ①t rt ♣r t rs♣t t♦ t ♣r♦r♠♥ ♦ t st ♦♠♣♦♥♥ts

t A = PΛP ′ t t♠ ♥tr② ♦r♥ ♠tr① ♦ t ♣r♦ss Wtr P ♥ Λ ♦♥t♥ rs♣t② t ♥t♦rs ♥ ♥s ♦ ts②♠♠tr ♣♦st ♥t ♠tr① A ♥ t ♣t ♥♦r♦♦ ♦ t♣♦♥t x∗ s

=

x ∈ R

n : ‖z‖2 ≤ 1, z =[√

Λτ]−1

P ′(x− x∗)

.

♥ r s♦ t st t n = 2 τ = 0.5 t ♣♦♥t x∗ = (+10%, +10%)♥ t ♦r♥ ♠tr①

A =

[0.04 0.020.02 0.04

]

Page 11: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♦rrs♣♦♥s t♦ t ♦rrt♦♥ ♦♥t ρ = 0.5 ♥ ②r② ♦ttsσj = 20%, j = 1, 2♦ sr ♦r t ♣r ♥ t ♦ t rtP∗ ♣②♥ ♦t t g(X) t t ①♣rt♦♥ t♠ T t st♦st r♦t t♦r Xt = log (St/S0) s ♥ ♣②−♦ ♣r♦ ♥t♦♥ g (X) s ♥ s tt

g(x) =

K[eφ(1−‖z‖2) − 1

], x ∈

0, x ∈ Rn \

r K s t ♦♣t♦♥ ♥♦t♦♥ ♥ t ♦♥t φ > 0 ♥ s t ♣②−♦ ♦♣t♦♥ t φ = 0.4 ♥ K = 1 s♦t ♦♣t♦♥rs ♣r♦rss② ♥rs♥ ♣②♠♥t s ♠ s t st ♦♠♣♦♥♥ts♣r♦r♠♥ s ♦sr t♦ x∗ t ♠trt② st ♦ ♣♦♥ts r ts ♣②♠♥t s♥♦t ♥ ♦♥s t t ♣t ♥♦r♦♦ ♦ x∗ ♦r♥t ② t ♥s ♦t ♥t♦rs r♠ ♥ s ② t ♥s ♣r♠trs t ♠trt②♦ts t st t ♦♣t♦♥ s ♦rt ♥♦t♥

♣r f (t, X) ♦ t rt P∗ ♦♦s r♦♠ t ♣♣t♦♥ ♦ t♦♣rt♦r

(Gt e

−rt)(⋆) t♦ t ♣②−♦ ♣r♦ ♥t♦♥ g ♥ ♥ ♥②

stt

Pr♦♣♦st♦♥ ♣r ♦ t rt P∗ rtt♥ ♦♥ t ♣r♦r♠♥

♦ t st Sjj=1,...,n ♣②s ♦t g (X) t ♠trt② s t f s tt

f =[α

n2 eφ(1−ακ0)Fχ2

2

(θ2)− Fχ2

1

(θ1)]Ke−r(T−t)

t t ♣r♠trs α =(1 + 2φ

θ1

)−1

θ1 = τT−t θ2 = θ1

α κ0 = ‖η‖2 ♥ t

♥sts χ21= χ2 (n, κ1) χ

22= χ2 (n, κ2) t κ1 = θκ0 ♥ κ2 = ακ1

t♦r η s ♥t♦♥ ♦ t rt ♣♦rt♦♦ ♣r♦r♠♥

η =[√

Λτ]−1

P ′ [X − x∗ + (T − t)(rι− 1

2 diagA)]

Pr♦♦ t ♣♣♥①

r F s t ♠t strt♦♥ ♥t♦♥ ♦ t ♥♦♥−♥tr χ2 (n, κ)t n rs ♦ r♦♠ ♥ ♥♦♥ ♥trt② ♣r♠tr κ ♥ ♣♦tt s♣ ♦ t ♥t♦♥ ♥ t ❳ s♣ t st♦s a = 0.03 σ = 0.2r = 0.02 ♥ T − t = 0.5 0.3 0.1 ♥ 0.001 ❲ r♠r tt t ♦♣t♦♥ rtr♥t ♠trt② ♣♥s ♦♥ t ♣♦sst② tt t ♣r♦r♠♥ t♦r ♣♦♥t ♦ t♥ t ♣t ♥♦r♦♦ ♦ x∗ s ♥② ♦tr ♦♣t♦♥ t ♣r ♦t χ2−♦♣t♦♥ s rt t♦ t ♣r♦t② tt t rr♥ ♥t ♣♣♥s ♥T − t ②rs s ♣r♦t② ♥ s② t ♥ t t ♣r♦t②♦ Xt ♦♥♥ t♦ t −♥♦r♦♦ ♦ x∗ s

Page 12: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

P Xt ∈ |X0 =

‖u−√θζ‖2≤θ

due−

12‖u‖2

√2π

n = Fχ2 (θ) ,

t χ2(n, θ‖ζ‖2

) θ = τ/t ♥ t ♣♦♥t ζ =

[√Λτ

]−1

P ′ (x∗ + 12 t diagA

)

rst♥ r♦♠ t ♥ ♦ r ♥ st ♦r ♥st♥ t = 1 t♥F = 0.179 s t ♣r♦t② ♦ t ♦♣t♦♥P∗ ♥ ♥−t−♠♦♥② ♥ ♦♥ ②r♥t② t ♦♣t♦♥ ♥t♦♥ tr♥s♦r♠ ♥t♦ ♠♦♥tr② t ♣r♦t②♦♥t♥t ♦ t ♥ s♣ ♠♣♣♥ t ♣r♦r♠♥ ♦ t rr② ♦ tst ♦♠♣♦♥♥ts

♦♥s♦♥s

♥ ts ♣♣r sr ♥ ♣♣r♦ t♦ ♦♥strt t s♦t♦♥ t♦ t♣r♦♠ ♦ ♣r♥ ♥ r♦♣♥ ♦♣t♦♥ ♥ t ♠t♠♥s♦♥ −♦s♠♦ ♦♣t♦♥ ♣r s ♥♦t ♦♥② ♦ s♦t♦♥ t t s ①t♥ t♦ t♦♠♣t ♣r ♣r♦r♠♥ s♣ R

n s♦t♦♥ ts t ♦r♠ ♦ ♥t♦♥ ♦♥ r♥ ♦ ♥t♦♥s t ♥ ♣♣r♦♣rt ♦r t t ♥♥t② ss♣ ♦ ♥t♦♥ s t st ♦ ♠ss ♣②−♦s ♦ t ♦♣t♦♥ t t ♠trt②t♠ ❲ ♣r♦ s♥t ♦♥t♦♥ t♦ ♥t② ts st ❲ r ttts s♣ ♥ ♥r ♦♥sr♥ t ② ♣r♥♣ s ♦♠♣t♦♥ s ♥ sr ♥ ts ♦r ♦s ♥t♥t♦♥ s ♥ t ♦♣♠♥t♦ ♠t♦ ♦ s♦t♦♥ ♦r s♣ ♥ ♦ ♥♥ ♣♣t♦♥s s ♠t♦♣r♦s s♦t♦♥ sts t ①♣♦tt♦♥ ♦ ♥♠r t♥qst♦ ♦♥strt ♥② ♣r ♥t♦♥ tt ♥♥♦t ♦t♥ ♥ ♦s ♦r♠♥ t ♥ st♦♥ ♦♥strt t♦ ♣♣t♦♥s r ♦s−♦r♠ ♦t♣t ♥ t rst ①rs r ♣r♦♠♣t t♦ s rt ♦rrt♦♥ strtr ♥♦t ♥tr t ♣r♥ ♥t♦♥ r t ♥ st ♦♣t♦♥ ♦r♠ s t rst ♦ t ♥r ♦♠♥t♦♥ ♦ ♣♥ ♥ ♣t ♦♣t♦♥s ♦♥ s♥ ♥r②♥ s rst s ♦♥sq♥t ♣♦♥ t s♣ ♦r♠t♦♥ ♦ t ♣②−♦t ♥t♦♥ ♥ t s♦♥ ①♠♣ t t②♣stt♦♥ rss r t rtr♥ ♣r♦ ♦ t st ♦♣t♦♥ t t ①♣rt♦♥t♠ rqrs t trt♠♥t ♦ t ♦♣t♦♥ ♣r♥ s ♦♥t ♣r♦♠ ♦♥ t st ♦t ♥r②♥ ❲ ♦rt♥ t ①rs r t♦ ♦t♥ t ♣r♥ ♦r♠ s♥ rst ♦♥r♥♥ t ♣r♦t② ♦♥t♥t ♦ r♦♥s ♦ Rn

♥r ♥♦r♠ strt♦♥ ♣r♥ ♥t♦♥ s ♥t s t χ2−♦♣t♦♥s t ♥♦s t ♥♦♥−♥tr χ2 ♠t strt♦♥ ♥t♦♥ ♥ ttr♠♥t♦♥ ♦ t ♦♣t♦♥ ❯♥r ts ♣r♠ sr ♦tr ♦♣t♦♥stt ♥ ♣r♦t② ♦♥t♥t ♦ r♦♥s ♦ t ♥ s♣ ♠t s②♦t♥

Page 13: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

r♥s

❬❪ ♦s Pr♥ ♦ ♣t♦♥s ♥ ♦r♣♦rt ts

♦r♥ ♦ P♦t ♦♥♦♠② ❱♦ ♦

❬❪ ♦rs ♥r③ t♦rrss ♦♥t♦♥ tr♦sstt②

♦r♥ ♦ ♦♥♦♠trs ❱♦

❬❪ ♥ t♦rrss ♦♥t♦♥ tr♦sstt② t st♠ts

♦ ❱r♥ ♦ ❯♥t ♥♦♠ ♥t♦♥ ♦♥♦♠tr

❬❪ r ♥s tr r

❬❪ rrs♦♥ r♣s rt♥ ♥ rtr ♥ t♣r♦ rt②

rts ♦r♥ ♦ ♦♥♦♠ ♦r②

❬❪ rrs♦♥ Ps rt♥ ♥ t♦st ♥trs ♥ t

♦r② ♦ ♦♥t♥♦s r♥ t♦st Pr♦sss ♥ r ♣♣t♦♥s

❬❪ rt♦♥ ♦r② ♦ t♦♥ ♣t♦♥ Pr♥ ♦r♥ ♦♦♥♦♠s

❬❪ s t♦s rt♥ t♦s ♥ ♥♥ ♦♥

t♦st ♦♥ ♥ ♣♣ Pr♦t② ❱♦

❬❪ ♥ Pr♦t② ♦♥t♥t ♦ r♦♥s ♥r s♣r ♥♦r♠ str

t♦♥s ♥♥ t tt ❱♦

❬❪ ❩♠♥♥ strt♦♥ t♦r② ♥ tr♥s♦r♠ ♥②ss ♦rs♦♥ t♦♥

Page 14: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

♣♣♥①

Pr♦♦ ♦ Pr♦♣♦st♦♥ t f ∈ D∗t X (∂tf) = ∂tf = P (iξ) f

♣♦ss② ft ∈ Z∗ ♠t♠♥s♦♥ ♣♦②♥♦♠ P ∈ E ♥ t ♣rt−tr♥s♦r♠ ♦ t s ♣r♠tr③ ♦r♥r② r♥t qt♦♥ ♦ss♦t♦♥ s t ♣r♦t ♦ t ♥t t♠s t ①♣♦♥♥t F = eP (iξ)t❲ ♦♥sr t f s t ♠t ♦ t sq♥ ♦ ♦ s♦t♦♥s ♦t♥ssttt♥ t ♥t ♦♥t♦♥ g ∈ t ts t−♦ gK = χ

Kg r χ

K

s t ♥t♦r ♥t♦♥ ♦ t ♦♣♥ B0(s) =: K ⊂ Rn r② gK ∈ 1

loc

♥ gK → g s s → ∞ ② t ②−r③ ♥qt② t s♦ ∀K∣∣⟨gK , a

⟩∣∣ ≤ ‖ g ‖1,K‖ a ‖∞,K < +∞, ∀a ∈e−iξ·x

ξ∈Rn

♠♣s tt t ♥t♦♥ E ∋ gK : Rn → C s ♦ ♦♥ rt♦♥ gK ∈ E ∩S∗ ♥ ♣♥ t −tr♥s♦r♠ ♦ t ♥t ♦♥t♦♥ ♥t♦ ts♦t♦♥ ♦ t ♣r♠tr③ ODE t

fK = F gK

♦t tt F ∈ Et ♥ Ft ∈ E ∩ S∗ ∀t ∈ T ♦r s♣② s♣t t♥tr T ♥t♦ t♦ ssts s tt s Ω > 0 t♥ F ∈ S, ∀t ∈ (0, T ] ♥t♥ t limt→0+ F = 1 ∈ E ∩ S∗s gK s ♥t♦♥ ♦ ♦♥ rt♦♥ ♥ F |t s ♥r② ♥ E ∩S∗∀t ∈ T s tt tr ♣r♦t F gK = fK ∈ E ⊂ D∗ ♥ t ♥rstr♥s♦r♠ ♦ ♦t ss t fK = −1

X

(F gK

)=

(G ⋆ gK

)eδt ttr

♦♥♦t♦♥ s ♥ s gK ∈ 1loc

⊂ E∗ ♥ G ∈ S∗t s t

s → ∞ ♥ gK → g ♦t♥ fK → f ♥ t ♦♥r♥ s ♥ 1 ∀t ∈ (0, T ]

s G ∈ O(e−‖x‖2

)♥ g ∈ ❲rs t → 0+ G ♦♥r♥

t♦ δX t ♥tt② ♠♥t t rs♣t t♦ t ♦♥♦t♦♥ ♦♣rt♦♥ s♦♠♣ts t ♣r♦♦

Pr♦♦ ♦ Pr♦♣♦st♦♥ s♦t♦♥ s ♦t♥ ♣♣②♥ t ♦♣rt♦r t♦t ♥ ♣②−♦ ♥ t♥ ♠♥♣t♥ t qt♦♥s ♥ ♦rr t♦ ♥trs s ♥ s t②♣ ♦ ♥trs ♥ s♦ t ♦♠trr♠♥t ♦r tr♦ t ♣♣t♦♥ ♦ t ♠t♦ ♦ st♦♥s r ❬❪

Page 15: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

r D ss ♦s ♦♣t♦♥ ♣r t t♠ t♦ ♠trt② t = 1.51 0.5 ♥ 0.0001

r τ−♥♦r♦♦ ♦ x∗ t τ = 0.5 ♥ ♦r♥ ♠tr① A ♥n = 2

Page 16: Multidimensional Black-Scholes options · 2 Multidimensional Black-Scholes model 2.1 Model Setup In this work we follow [1] in modeling the stock prices. The market price dy-namics

r D χ2−♦♣t♦♥ tr♠♥ ♣②−♦

r χ2−♦♣t♦♥ ♣rs t r♥t t♠ t♦ ♠trt②