A Moments and Strike Matching Binomial Algorithm for Pricing American Put Options

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HAL Id: inria-00070437

https://hal.inria.fr/inria-00070437

Submitted on 19 May 2006

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A Moments and Strike Matching Binomial Algorithm for Pricing American Put Options

Benjamin Jourdain, Antonino Zanette

To cite this version:

Benjamin Jourdain, Antonino Zanette. A Moments and Strike Matching Binomial Algorithm for Pricing American Put Options. [Research Report] RR-5569, INRIA. 2005, pp.15. �inria-00070437�

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ISSN 0249-6399 ISRN INRIA/RR--5569--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

A Moments and Strike Matching Binomial Algorithm for Pricing American Put Options

Benjamin Jourdain and Antonino Zanette

N° 5569

May, 8 2005

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Unité de recherche INRIA Rocquencourt

Domaine 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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y?zSzSm ry{~1\¤y{h `-|Wr{jÄi\^`+y?zSzSm r-µTn_ y?inpr{j²r?ª |TnGFUk^h npr{jÄPgGdWy?m§r»³G~ \y?nj^ha¡Him `-`"_a`Zi\^rE|ShY£CUi\^`"r{i\^`ZmÎr{j^`9nph

i\^`"y?jy?gPinp~´y?zSzSm ry{~1\Gm `Zy?i `-|®i r+i\^`}|Tnph ~Zm `ZinN<»y?inpr{jWr?ª0i\^`}³y?mny?inpr{jy?¶nj^`-oPky?niõg®h1y?inph*«`-| Pgi\^`}r{zSinpr{j

³y?k^`¡ý«^jSni `"|TnGF`Zm `Zj^~-`"r{m«^jSni `´`Zp`Z_a`ZjPi h_a`Zi\^rP|ShY£C±¦[\^`´_"kSinzSnp~y?in³`Snj^r{_ny?T6¦rµP¨õxrh h*¨õxkSSnj^h i `Znj

_arE|S`Z$-*I/\y{h®y?j njAi `Zm `-hi®r{j \Snphr»)jy{h yJy{hnp~Ä|Tnph1~Zm `Zi `C¨Ñin_a`Ä_arE|S`Zª¢r{mi\^`GkSj^|S`ZmgEnjS y{h h `Zi r?ª"y

«^jy?j^~Zny?¦|S`Zmn³{y?in³`0~-r{jA³`ZmnjSWi r i\^`|TnGFUk^h npr{j²zSm rP~-`-h1h ¡*»£kSj^|S`Zm"y?zSzSm r{zSmny?i ` ~-r{j^|Tninpr{j^h»± , j^`r?ª¦ni h

_ y?njWy?iim1y{~Zin³`}ª·`y?ikSm `}nph)i\^`´`y{hnj^`-h1hr?ª4h i1y?j^|^y?m |'r{zSinpr{j'zSmnp~ZnjS Pgy{~1§Py?m |'nj^|Tk^~Zinpr{j¶±

¶`Zi

N |S`Zj^r{i `¬i\^` h i `ZzjPkS_9`Zmr?ªi\^`im `-`¬y?j^|

∆T = _N^T i\^`¬~-r{mm `-hzr{j^|TnjSWin_a`C¨õh i `Zz¶±G[\^`pr{{¨

j^r{m_ y?5|TnGFUk^h npr{j'zSm rP~-`-h1h

(S_n∆T)₀_≤_n_≤_N nphy?zSzSm r-µTn_ y?i `-|GPg i\^`L6¦r-µE¨õxrh h¨õxkSSnj^hi `Znj'Snj^r{_ny?êzSm rP~-`-h h

(s₀Q_n

j=1Y_j)₀_≤_n_≤_N )\^`Zm `i\^`m1y?j^|Sr{_ ³{y?mny?Sp`-h

)ni\Ä³{y?k^`-hnj

{d, u}± ¶`Zik^h|S`Zj^r{i `´Ag

pu =P(Yn=u) = 1−P(Yn=d)±Îk^h\Sj^`Zm èhi\^`-r{m `Z_ -Ü0/0h1y-gTh i\y?i¯i\^`prE~y?~-r{j^hnph i `Zj^~Zg+~-r{j^|Tninpr{j^h»Ai\y?i¯nph°i\^`_ y?i ~1\SnjS+y?i¯i\^`«^m hi¦r{m |S`Zm¯r?ªOi\^`)«^m hi¦y?j^| h `-~-r{j^| _ar?¨

_a`ZjPi h r?ªUi\^`¯pr{Ay?mni\S_np~nj^~Zm `Z_a`ZjAi h r?ªUi\^`y?zSzSm r-µTn_ y?injS9~ \y?nj )ni\i\^rh `r?ªi\^`)~-r{jAinjPk^r{k^h*¨Ñin_a`n_ni

m1y?jPi´i\^`+~-r{jA³`Zm`Zj^~-`ar?ª¦i\^`+`CµEz`-~Zi1y?inpr{j^h´r?ª¦h _arPr{i\Wª¢kSj^~Zinpr{jy?ph±9[\^`-h `"«^m hiÎià¯r®_ar{_a`ZjPi h}_ y?i ~ \SnjS

~-r{j^|Tninpr{j^hm `y{|

(pulogu+ (1−pu) logd= (r− ¹₂σ²)∆T p_u(logu)²+ (1−p_u) (logd)²=σ²∆T.

¡·£

wàji\^`(6¦rµP¨õxrh h*¨õxkSSnj^h i `Znja_arP|S`ZÑAi\^`)zSmnp~-`y?i in_a`

n∈ {0, . . . , N}r?ªi\^`)¸°kSm r{zU`y?j¡Hm `-hz¶±4t_a`Zmnp~y?j£

q°kSir{zSinpr{jGnphn³`ZjGAg

v(n, s₀Qn

j=1Y_j) )\^`Zm `}i\^`Îª¢kSj^~Zinpr{j^h

v(n, x) ~y?jG`}~-r{_zSkSi `-|'Ag i\^`´ª¢r{pr»)njS

y{~1§Py?m |Ä|TgEjy?_np~}zSm r{m1y?__njS¬`-oAky?inpr{j^h







v_N(N, x) = (K−x)⁺ v_N(n, x) = max

ψ(x), 1

upu+d(1−pu) h

p_uv_N(n+ 1, xu) + (1−p_u)v_N(n+ 1, xd)i ,

¡%*£

)\^`Zm `

ψ≡0 ^{¡Hm `-hz¶±} ψ(x) = (K−x)⁺^£C± wõj®i\^`´¸°kSm r{zU`y?jG~y{h1`

vN(n, x) = 1

(up_u+d(1−p_u))^N⁻ⁿ E

K−x

N

Y

j=n+1

Yj

+

. ^¡2P£

[\^`azSm `-h `Zj^~-`r?ª

E[Y₁] =up_u+d(1−p_u) nji\^`ay{~Ziky?nN<»y?inpr{j²ªÑy{~Zi r{m h}nj%¡%*£Îy?j^| ¡2P£`Zj^hkSm `-hÎi\y?i´i\^`

zSmnp~-`-h9~-r{_zSkSi `-|njWi\^`Snj^r{_ny? _arE|S`Z°y?m `ay?mSnim1y?`C¨·ªHm `-`±"wõj¤r{m |S`ZmÎi r r{Si1y?nj²`CµEzSnp~Zni´`CµTzSm `-h hnpr{j^hª·r{m

i\^`Îzy?m1y?_a`Zi `Zm hr?ª4d¤fEd2im `-`-h»T¯`ÎzSm `Cª·`Zm)i r+n_zUrh `Îi\^`«^m h i`-oAky?niàgnjJ¡·£¯m1y?i\^`Zm)i\y?j®i\^`Î_ar{m `}k^h ky?

~-r{j^|Tninpr{j

upu +d(1−pu) = e^r∆t± br{inp~-`ai\y?i´kSj^|S`Zm ¡·£Z¶i\^`ay{hi9`-oPky?niõg¤\^r{p|Sh´kSzi rGy i `Zm_2)ni\

r{m |S`Zm

O((∆T)²)^±

[\^`njSniny? zSmnp~-`®¡Hm `-hz¶±}|S`Zi1y£r?ª i\^`+q kSi}r{zSinpr{jnji\^`+¥¯y{~1§A¨©fE~1\^r{p`-h}_arE|S`Z4~y?j`y?zSzSm rµEn_ y?i `-|²Ag

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(8)

v_N(0, s₀) ¡Hm `-hz¶±'i\^`¬y?_ar{kSjAi vN(1,s0u)−vN(1,s0d) s0(u−d)

r?ªmnph §PgJy{h h1`Zi´nji\^` m `ZzSnp~y?inpr{j/zUr{mi*ª·r{nprÄr{ji\^`a«^m h i

in_a`C¨õh i `ZzJnj²i\^` 6¦rµP¨õxrh h*¨õxkSSnj^hi `Znj²_arE|S`Zý£C±abr{inp~-`ai\y?i9nj²r{m |S`Zm9i r®r{Si1y?nji\^` y?zSzSm r-µTn_ y?i `azSmnp~-`

y?j^|G|S`Zi1yT^r{j^`´r{jSg¬j^`-`-|Shi ra~-r{_zSkSi `

vN(n, s0u^kdⁿ⁻^k), 0≤k ≤n^Ag y{~1§Py?m |'nj^|Tk^~Zinpr{jÄr{j

n^±

[\^`k^h ky?@6¦rµP¨õxrh h*¨õxkSSnj^h i `Znjim `-`¡H\^`Zj^~-`Cª·r{mi\ |S`Zj^r{i `-|K6¦xx£ ~-r{mm `-h zUr{j^|Sh¦i r"i\^`~1\^r{np~-`

u= ¹_d = e^σ^√^∆T ^{y?j^|} d+ (u−d)p_u =e^r∆T±}[\Snphp`y{|Shi r

p_u = ^e^r^∆^T⁻^e^−σ

√∆T

e^σ^√^∆T−e⁻^σ^√^∆T

± 9 \^`Zj

∆T nphÎh_ y?5`Zj^r{kS\¤nÑ±è`±

)\^`Zj

N nph}y?m``Zj^r{kS\¶êi\^`y?Ur»³`³{y?k^` r?ª

p_u U`Zpr{jSh9i r

]0,1[±v^r{m9i\Snph´~1\^r{np~-` r?ª

u d ^{y?j^|} p_u^{êi\^`}

h1`-~-r{j^|'`-oAky?niàg nj¡·£¯nphh1y?inph*«`-| )\^`Zm `y{hi\^`´|TnGFO`Zm `Zj^~-`}U`Zià¯`-`ZjÄUr{i\Gh np|S`-h)r?ª5i\^`Î«^m h i`-oAky?niàg¬nj¡·£

nphr?ªr{m |S`Zm

(∆T)²±²[\Snph+nph+hkT¼ ~Znp`ZjPii rW`Zj^h kSm `®~-r{jA³`Zm`Zj^~-`®i rWi\^` ¥¦y{~ §P¨©fE~ \^r{p`-ha_arE|S`Znj/i\^`n_ni

N → +∞^± ^¶`Zi P_N^CRR ^{y?j^|} P_BS |S`Zj^r{i `+i\^`+njSniny?4zSmnp~-`r?ª¦i\^`¸°kSm r{zU`y?jJq°kSiÎr{zSinpr{j¤)ni\¤_ y?ikSmniàg

T

y?j^|Jhimn§`

K m `-h zU`-~Zin³`ZgWnj¤i\^` 6¦r-µE¨õxrh1h*¨õxkSSnj^hi `Znj¤_arE|S`Z y?j^|¤nj¤i\^`¥¦y{~ §P¨©fE~ \^r{p`-h"_arE|S`ZÑ±chnjS

i\^` 6y?¨õq kSizy?mniàg¬m `Zy?inpr{j^hnj'r{i\®_arP|S`Zphy?j^| m `-hkSi hn³`Zj®ª·r{m)i\^`L6y?¶r{zSinpr{jÄnjJ-2/õ^r{j^`}`Zi h

P_N^CRR=P_BS−Ke⁻^rT N e⁻^d

2 2 2

r2 π

h

κ_N(κ_N −1)σ√

T +D₁i +O

1 N^3/2

,

)\^`Zm `

d₂ = ^log(

s0

K)+(r−^σ2²)T σ√

T

κ_N |S`Zj^r{i `-hÎi\^`ªHm1y{~Zinpr{jy? zy?mi´r?ª log(^K_s0) 2σ

qN

T −^N₂ ^{y?j^|} D₁ ^nph´y ~-r{j^hi1y?jAi±

¥`-~y?k^h `Îi\^`Îh `-oPk^`Zj^~-`

(κN)N

nph)rh ~Zny?injS y?i\^r{kS\®r{kSj^|S`-|OTnj®i\^`}y?zSzSm r-µTn_ y?inpr{j

2P_2N^CRR−P_N^CRR ^r?ª P_BS r{Si1y?nj^`-|ak^h njS9xnp~1\y?m |Sh r{j`CµTim1y?zUr{y?inpr{j¶Pi\^`)i `Zm_ )ni\¬r{m |S`Zm

1/N |SrE`-h°j^r{i ³{y?jSnph\ )\Snp~ \`CµTzSy?nj^h i\^`ÎzrEr{m)jAkS_a`Zmnp~y?¶`Z\y-³Enpr{kSmr?ª0i\Snphy?zSzSm r-µTn_ y?inpr{j¶± wõª0i\^`9fPimn§`

K nph)`-oAky?Oi rr{j^`}r?ª5i\^`«^jy?Oj^rP|S`-h

(s₀e^(2k⁻^N)σ^√^∆T)₀_≤_k_≤_N r?ª i\^`}im `-`i\^`ZjÄr{j^`}\y{h

κ_N = 0±¦[\Snph s*k^hin«`-hr{kSmnjAi `Zm `-hinjGim `-`-hh k^~ \'i\y?i i\^`fPimn§`´~-r{nj^~Znp|S`-h)ni\'r{j^`9r?ª5i\^`Î«^jy?êj^rP|S`-h±

¬¾ JS OI17 V O ? % 7 O""?

K =s₀ ^O N = 2m ^I

= ? )

K ^{%)T ; =} (m+ 1) ^&@D O !#"7 : ? I

P_2m^CRR=P_BS−D₁Ke⁻^rT 2m e⁻^d

2 2 2

r2 π +O

1 m^3/2

.

'L = %$ = " = &I%)

2P_4m^CRR−P_2m^CRR P_BS O

1 m³^/²

'

N = 2m+ 1 ^1(:? ^I κ_N = 1/2 ^. ^{) O} P_2m+1^CRR =P_BS−Ke⁻^rT

2m e⁻^d

2 2 2

r2 π

"

D₁−σ√

∆T 4

# +O

1 m^3/2

.

O $ &I%? 7

2P_4m+1^CRR −P_2m+1^CRR P^BS O

1 m³^/²

¬¾ JS*) ' " ( & ,+=7 - O %+ )

D₁ ^.I.^T &&G0/1.2435/76(

? ) 8:9 1 V O(%+O==

1 2

1 + _σ^r −ε^σ₂q

T N +

ε_2σ^r² −^rσ₆ +ε^σ₂₄³

T N

3/2 + O _N¹²^0&^;

p_u= ^e^r^∆^T⁻^e^−σ

√∆T

e^σ^√^∆^T−e^−σ^√^∆^T

T ε= 1 _(d+(u^up₋^u_d)p

u) = ^e^σ

√∆T−e^−r^∆^T e^σ^√^∆^T−e^−σ^√^∆^T

T ε=−1 ^< ^&N

= A? )) I: @? ? &> OA )?T

1/√

N ⁼ ^&C ^O:^?T 1/N^3/2

; = O 4%1 A@ $7B = &GC<"C

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S7¶16, } U

wàj^hi `y{|+r?ªm `-oPkSnmnjS

u= ¹_d y{h5nj+i\^` 6¦r-µE¨õxrh h*¨õxkSSnj^hi `Znj_arP|S`ZÑ¦`¦zSm r{zUrh `¦nji\^`dfEd_a`Zi\^rE|i rÎ`ZjT¨

h kSm `i\y?i°i\^`fPimn§`

K ^{nph4i\^`} (k+1)¨Ñi\Ä¡H)ni\

k ∈ {1, . . . , N−1}£ê«^jy?Sj^rE|S`)r?ªUi\^`im `-` 8

K=s₀u^kd^N⁻^k

)\Snp~1\Wy?ph r+)mni `-h

1

N log(^K_s₀) =qlogu+ (1−q) logd ^{)\^`Zm `} q = _N^k ^±

6¦r{_zSp`Z_a`ZjPinjS/i\Snpha`-oAky?inpr{j%)ni\8i\^` «^m hiiõ¦r²_ar{_a`ZjPia_ y?i ~ \SnjS~-r{j^|Tninpr{j^hW¡·£C¯¯` y?jAi i r¤«^j^|

(logu,logd, p_u) ^)ni\ logu > logd ^{y?j^|} p_u ∈]0,1[ h r{³EnjSGi\^`aª¢r{pr»)njSWhgThi `Z_ r?ª`-oPky?inpr{j^h9)ni\kSjT¨

§Ej^r)j^h

(x, y, p)







qx+ (1−q)y=α px+ (1−p)y=β px²+ (1−p)y²=γ

¡·£

)\^`Zm `

α= _N¹ log(^K_s₀) β = (r−^σ₂²)∆T ^{y?j^|} γ =σ²∆T±0[\^`)~-rP`C¼ ~Znp`ZjPi h

(q, α, β, γ)`Zpr{jSÎi r

]0,1[×R²× R^∗₊^±

¥`-~y?k^h `r?ª¦i\^`h imnp~Zi9~-r{jA³`CµTniõg¤r?ªi\^`h oAky?m `ª¢kSj^~Zinpr{j¶0i\^`+iõ¯rGy{hi´`-oPky?inpr{j^h}n_zSgi\y?i

γ > β²

nphy¬j^`-~-`-h h1y?mg¤~-r{j^|Tninpr{j²ª·r{m}i\^` `CµTnphi `Zj^~-`ar?ªy®h r{kSinpr{j¶±[\Snph´~-r{j^|Tninpr{jr{P³Pnpr{k^h g¤\^r{p|Sh}nª

N nph}y?m`

`Zj^r{kS\¶±¯wõi)ikSmj^hr{kSii r`}hkT¼ ~Znp`ZjAi8

¾ ² '

γ > β² q ∈]0,1[^* =H ="I , ,$ = =¤¡·£ $ ,+ &

& %=

(p_i, x_i, y_i)_i_∈{_1,2_} ]0,1[×{(x, y) ∈R² :x6=y} ^O) ^&^{O =} ⁼ (x₁−y₁)(x₂−

y₂)<0 ⁼ ^{&I %#} ^L ^1I1 ^O^,+) x₁> y₁ ^? x₂ < y₂

¬¾ JS 9 I7 O 7 O &O

(¯p_i,x¯_i,y¯_i)_i_∈{_1,2_} ^O ^I ⁼ ⁼ ^" ^8&N q

=

¯

q= 1−q ^¡·£ ^.I ⁼ (¯p_i,x¯_i,y¯_i) = (1−p₃₋_i, y₃₋_i, x₃₋_i) i∈ {1,2} ^O (p_i, x_i, y_i)_i_∈{_1,2_}

O7 &O %= 7 O N ?&TI 9) O % &GA O

q = ¹₂ ^O ^M ^&^{O =} ^¡·£

&= =

(p2, x2, y2) = (1−p1, y1, x1)

ÂOÂ 9²`9y{h h kS_a`}i\y?i

γ > β² ^{y?j^|} q ∈]0,1[^± 9²`9y?m `Î«^m h ir{njSai ra|S`y?¶)ni\Gi\^`´~y{h `

α6=β ^{`Cª¢r{m `}

im `y?injSi\^`´~y{h `

α=β^±

9S ¾

α 6= β ^{[\^`Zj} i\^`Ä«^m hi iõ¦r8`-oAky?inpr{j^h'y?m ``-oPkSn³{y?p`ZjPi®i r

p 6= q x = α + (1 −q)^β_p⁻₋^α_q ^{y?j^|}

y=α−q^β_p⁻₋^α_q± x`ZzSy{~ZnjSi\^`-h1``CµTzSm `-h hnpr{j^h)r?ª

x ^{`Zj^|} y nj¬i\^`y{hi`-oPky?inpr{j¶T¯`}|S`-|Tk^~-`i\y?ii\^`}hgEh i `Z_

¡·£nph`-oAkSn³y?p`ZjAii r





 p6=q

x=α+ (1−q)^β_p⁻₋^α_q, y=α−q^β_p⁻₋^α_q ap²+bp+c= 0

)\^`Zm `







a= (α−β)²+ (γ−β²) b=−(α−β)²−2q(γ−β²) c=q²(γ−β²)

.

[\^`"|Tnph ~Zmn_njy?jAi

∆ = (α−β)²

(α−β)²+ 4q(1−q)(γ−β²) r?ª4i\^`´y{hi`-oPky?inpr{jGnphzUrhnin³`"y?j^|'¯`

|S`-|Tk^~-`9i\y?ii\Snph`-oPky?inpr{jÄy{|T_ni hià¯r m rPr{i h 8

p₁ = −b+ (β−α)p

(α−β)²+ 4q(1−q)(γ−β²) 2a

y?j^|

p₂ = −b+ (α−β)p

(α−β)²+ 4q(1−q)(γ−β²)

2a .

fPnj^~-`´i\^`ÎzUr{gEj^r{_ny?

P(z) =az²+bz+c y?ph r+)mni `-h

P(z) = (α−β)²(z²−z) + (γ−β²)(z−q)²^{Sr{j^`}

`y{h ng'~ \^`-~ §Thi\y?i

P(z) >0 ^ª·r{m z∈R\]0,1[ y?j^|Gi\y?i

P(q)<0±[\^`Zm `Cª·r{m `9r{j^`"r?ª i\^`´m rEr{i h`Zpr{jShi r

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(10)

]0, q[ y?j^|®i\^`"r{i\^`Zmr{j^`}i r

]q,1[±dWr{m `}zSm `-~Znph `Zg

sign(p₁−q) = sign(β−α) = sign(q−p₂)±¯u`Zj^~-`´i\^`

h gEhi `Z_ ¡·£\y{h`CµSy{~Zig ià¯rah r{kSinpr{j^h

(p₁, x₁, y₁) ^{y?j^|} (p₂, x₂, y₂) ^nj ]0,1[×{(x, y) ∈R² :x6=y} ^)ni\

xi=α+ (1−q)β−α p_i−q

y?j^|

yi =α−qβ−α p_i−q

ª·r{m

i∈ {1,2}.

br{inp~-`}i\y?i

xi=yi+_p^β⁻^α

i−q

T)\Snp~ \Ä`Zj^h kSm `-h)i\y?i

x1> y1

y?j^|

x2 < y2

±

9S

¾ α=β fPnj^~-`}¯`y?m `ÎprPr{§EnjSª·r{mh r{kSinpr{j^h))ni\

x6=yTi\^`Î«^m h i)iõ¯r`-oAky?inpr{j^hnj®i\^`´hgThi `Z_ ¡·£

y?m `+`-oPkSn³{y?p`ZjPii r

p=q ^{y?j^|} x = (β+ (q−1)y)/q±Îx`ZzSy{~ZnjS¬i\^`-h `"`CµTzSm `-h hnpr{j^hÎr?ª

p ^{y?j^|} x njWi\^`9y{h i

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 p=q

x= (β+ (q−1)y)/q y²−2βy+^β²₁⁻₋^qγ_q = 0.

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∆ = ^4q(γ₁₋⁻_q^β²⁾ nph)zrhnin³`9y?j^|'i\Snph`-oAky?inpr{jG\y{h)iõ¦r h r{kSinpr{j^h

y₁ =β− s

q(γ−β²) 1−q

y?j^|

y₂ =β+ s

q(γ−β²) 1−q .

[\^`9~-r{mm `-hzUr{j^|TnjS³{y?k^`-h)ª¢r{m

x x₁ =β+

s

(1−q)(γ−β²) q

y?j^|

x₂ =β− s

(1−q)(γ−β²)

q ,

r{P³Pnpr{k^h gGh1y?inph*ª¢g

x₁ > y₁ ^{y?j^|} x₂ < y₂±u`Zj^~-`9i\^`9hgThi `Z_ ¡·£)\y{h`Cµ^y{~Zig®iõ¦rh r{kSinpr{j^h

(q, x₁, y₁) ^{y?j^|}

(q, x₂, y₂) ^nj ]0,1[×{(x, y)∈R²:x6=y}^±

ê`Zi®k^hikSmjy{~1§%i ri\^`~-r{j^himk^~Zinpr{j r?ª"d¤fEd im `-`-h»± th1hkS_njS/i\y?i

γ > β²^ª¢r{m q = k/N ^)ni\

k ∈ {1, . . . , N −1}Si\^`´h r{kSinpr{j^h

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p_u = p₁ logu = x₁ ^{y?j^|} logd = y₁ nphahk^~1\8i\y?iai\^`

(k + 1)¨Ñi\8«^jy?j^rE|S`

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K

i\^`h `-~-r{j^|r{j^`)ni\

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(N−k)¨Ñi\«^jy?Uj^rE|S`

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K^±

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q= (N−k)/N p`y{|Sh¯i ri\^`iõ¦rhYy?_a`im `-`-h»±

tj^|®)\^`Zj

N nph`Z³`ZjÄy?j^|

k=N/2Sr{i\'im `-`-hy?m `9`-oAky?Ñ±

Â Â ýS

¿

'

N ≥ ^r_σ −^σ₂2

T ^O ^& k ∈ {1, . . . , N−1} ^O ^$ ^=K

T

N ⁴⁼ ⁷ (p_u,logu,logd) ^8T p_u ∈]0,1[ ^? logu >logd^"$ . " O

A"7 :" K ? %='¡·£"? =. # =L

K ^$,$ ^=& ⁼ (k+ 1) ^& ¹

, OL

K=s₀u^kd^N⁻^k

= = 7 $ $ .I =

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

p_u = ^(α⁻^β)

2+2q(γ−β²)−(α−β)√

(α−β)²+4q(1−q)(γ−β²) 2((α−β)²+(γ−β²))

logu=α+ (1−q)_p^β⁻^α

u−q

logd=α−q_p^β⁻^α

u−q

T K &I

β−α pu−q =q

γ−β² q(1−q)

O

α =β

O

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



 q = _N^k α = _N¹ log

K s0

β =

r−^σ₂²

T N

γ =σ²_N^T

.

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