運用圖形處理器增進計算巴黎選擇權價格的效能

୯ҥᆵ᡼εᏢႝᐒၗૻᏢଣၗૻπำᏢس ᅺγፕЎ

Department of Computer Science and Information Engineering College of Electrical Engineering and Computer Science

National Taiwan University Master Thesis

ၮҔკ׎ೀ౛Ꮤቚ຾ीᆉЃᎿᒧ᏷៾ሽ਱ޑਏૈ

Using GPU To Accelerate the Pricing of Parisian Options

஭ඵṓ

Chang Chih-Hsuan

ࡰᏤ௲௤Ǻֈػၰ റγ Advisor: Lyuu Yuh -Dauh, Ph.D.

ύ๮҇୯ 99 ԃ 7 Д July, 2010

୯ҥᆵ᡼εᏢᅺγᏢՏፕЎ

α၂ہ঩཮ቩۓਜ

ၮҔკ׎ೀ౛Ꮤቚ຾ीᆉЃᎿᒧ᏷៾ሽ਱ޑਏૈ

Using GPU To Accelerate the Pricing of Parisian Options

ҁፕЎ߯஭ඵṓ։ȐR96922087ȑӧ୯ҥᆵ᡼εᏢၗૻ

πำᏢ܌ֹԋϐᅺγᏢՏፕЎǴܭ҇୯ 99 ԃ 7 Д 5 В܍Π ӈԵ၂ہ঩ቩࢗ೯ၸϷα၂Ϸ਱Ǵ੝Ԝ᛾ܴ

α၂ہ঩Ǻ

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ȐᛝӜȑ ȐࡰᏤ௲௤ȑ

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

سЬҺǵ܌ߏ

! ! ! ! ! ! ! ! ! ! ! ȐᛝӜȑ

ᇞ ᇞᖴ

གᖴֈػၰ௲௤ӧ೭ය໔ϣऐЈӦ๏ךࡰᏤکගٮཀـǴᡣךૈ໩ճֹԋࣴ

ز܌ޑᏢ཰کፕЎǶ

གᖴৎΓჹךޑЍ࡭کႴᓰǶЀځࢂࣁΑךޑ٣КךᗋाྠඊޑР҆ᒃǴך གྷᇥޑࢂǴᖴᖴգॺǴך౥཰ΑǶ

Ψགᖴךа߻ޑӕᏢॺǴૈόսܭ๏ךཀـکᐟᓰǴаϷӧፐ཰ፕЎǵϐࡕ

྽ծک҂ٰπբޑ࿶ᡍϩ٦Ƕ

നࡕᖴᖴ܌Ԗම࿶ӧౚ൑΢ഉՔךวࢻᓸΚޑ΋ဂౚ϶ॺǶᗋԖ൳Տ೭ࢤਔ ໔ϩ٦Ј٣ک΋ଆ࡫౥཰ޑӳ϶ॺǶ

ᄔ ᄔा

ᒿ๱ႝတၮᆉೀ౛ૈΚӦቚуǴӧຑሽߎᑼ़ғ܄୘ࠔਔΨૈၲډ׳Ԗਏ౗

Ъ҅ዴޑ่݀ǹԶΨӢࣁߎᑼ୘ࠔϐीᆉሽ਱Ԗਔ཮Ԗਔ໔ޑ࡚ॐ܄ǴӢԜӵՖ Ҕႝတฯᡏکᄽᆉݤቚ຾ीᆉೲࡋک҅ዴ܄൩ԋࣁख़ाޑፐᚒǶ

ӧҁጇፕЎύךॺ٬ҔЃᎿᒧ᏷៾کკ׎ೀ౛Ꮤٰ଺ٯηǴЃᎿᒧ᏷៾ࣁ΋

ᅿᏱԖምᛖᒧ᏷៾੝܄ޑ΋ᅿၡ৩࣬ᜢᒧ᏷៾ǶLi ک Zhao ගрճҔғԋڄኧٰ

ीᆉЃᎿᒧ᏷៾ޑሽ਱Ǵךॺӧύѧೀ౛Ꮤکკ׎ೀ౛Ꮤ΢ჴբр၀БݤǶ่݀

߄Ңҗܭკ׎ೀ౛ᏔԖ๱மεޑѳՉၮᆉૈΚǴӢԜӧ୺Չਔ໔΢Кύѧೀ౛Ꮤ Ͽ΢೚ӭǴЀځ྽යኧॶຫεຫܴᡉǴԶനࡕሽ਱کύѧೀ౛Ꮤၮᆉࡕޑሽ਱൳ คৡ౦ǶΨӢԜךॺёа׳זೲӦᕇளЃᎿᒧ᏷៾ޑሽ਱ԶόѨځᆒዴࡋǶ ᜢᗖӷǺЃᎿᒧ᏷៾Ǵምᛖᒧ᏷៾Ǵᒧ᏷៾ຑሽǴკ׎ၮᆉൂϡǴीᆉ᏾ӝး࿼

่ᄬǴѳՉၮᆉǴғԋڄኧǶ

Abstract

As computing power increases, we can get faster and more correct results in pricing derivatives. How to improve speed and correctness by computer hardware and algorithms is an important issue because pricing financial products is often a time-critical task.

In this thesis we use Parisian options and GPUs as an example. Parisian options are path-dependent options with barrier-like features. Binb-Qing Li and Hai-Jian Zhao proposed to price Parisian options by generating functions. We implement this method in both CPUs and GPUs. The results show that the execution time used by GPUs is much smaller than those by CPUs because of their powerful parallel-processing capabilities, especially when number of periods grows bigger. As a result, we can price Parisian options faster and with accuracy.

Keywords: Parisian options, barrier options, option pricing, GPU, CUDA, parallel processing, generating function.

Ҟᒵ

α၂ہ঩཮ቩۓਜ……….i

ᇞᖴ………ii

ᄔा... iii

Abstract ... iv

Ҟᒵ... v

კҞᒵ... vii

߄Ҟᒵ... viii

ಃ΋ക ᒧ᏷៾ᙁϟ... 1

1.1ᒧ᏷៾ ... 1

1.2 ምᛖᒧ᏷៾ ... 2

1.3 ЃᎿᒧ᏷៾ ... 2

1.3.1୷ҁཷۺ ... 2

1.3.2 ᅿᜪ׎Ԅ ... 2

1.3.3੝܄ ... 3

ಃΒക ୷ҁᢀۺаϷπڀ... 4

2.1 Β໨Ԅᒧ᏷៾ۓሽኳࠠ ... 4

2.2 ғԋڄኧ ... 5

2.3 GPUک CUDA... 6

2.3.1 GPUᙁϟکᐕў ... 6

2.3.2 GPGPUک CUDA ... 7

ಃΟക ճҔғԋڄኧٰीᆉЃᎿᒧ᏷៾ޑሽ਱... 11

3.1 ಔӝኧᏢࢎᄬ ... 11

3.2 ीᆉЃᎿᒧ᏷៾ޑሽ਱ ... 13

3.2.1 ೱុࠠЃᎿᒧ᏷៾ ... 13

3.2.2 ಕᑈࠠЃᎿᒧ᏷៾ ... 15

ಃѤക ჴբکኧᏵ่݀૸ፕ... 18

ಃϖക ่ፕ... 23

ୖԵЎ᝘... 24

კҞᒵ

კ 2-1. ΋යޑΒ໨Ԅᒧ᏷៾ۓሽኳࠠǶ ... 4

კ 2-2. n යޑΒ໨Ԅᒧ᏷៾ۓሽኳࠠǴS₀ࣁ߃ۈሽ਱Ƕ ... 5

კ 2-3. CPU ک GPU ϩձӧੌᗺኧၮᆉޑೲࡋ [8]Ƕ ... 7

კ 2-4. CUDA ࢎᄬ [8]Ƕ ... 8

კ 2-5. ThreadǵBlock ک Grid ޑᜢ߯ [8]Ƕ ... 9

კ 2-6. ૶Ꮻᡏቫԛ [8]Ƕ ... 10

߄Ҟᒵ

߄ 1-1. ምᛖᒧ᏷៾ϐϩᜪǶ ... 2 ߄ 4-1. CPU ک GPU ޑКၨǶ ... 18 ߄ 4-2. Кၨ Li ک Zhao аϷךॺӧ CPU ک GPU ΢ჴբीᆉೱុࠠЃᎿᒧ᏷៾

ޑሽ਱ޑ่݀Ƕ...19 ߄ 4-3. Кၨ Li ک Zhao аϷךॺӧ CPU ک GPU ΢ჴբीᆉಕᑈࠠЃᎿᒧ᏷៾

ޑሽ਱ޑ่݀Ƕ...19 ߄ 4-4. ӧ ୺Չᆣࣁ 1 ਔǴCPU ک GPU ΢ीᆉೱុࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ໔ КၨǶ...20 ߄ 4-5. ӧ ୺Չᆣࣁ 1 ਔǴCPU ک GPU ΢ीᆉಕᑈࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ໔ КၨǶ...20 ߄ 4-6. ӧ ୺Չᆣࣁ 100 ਔǴCPU ک GPU ΢ीᆉೱុࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ ໔КၨǶ...21 ߄ 4-7. ӧ ୺Չᆣࣁ 100 ਔǴCPU ک GPU ΢ीᆉಕᑈࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ ໔КၨǶ...21

ಃ

ಃ΋ക ᒧ᏷៾ᙁϟ

1.1ᒧ᏷៾

ᒧ᏷៾ࣁ΋ᅿ៾ճࠨऊǴວБЍб៾ճߎࡕǴߡԖ៾ճӧ҂ٰऊۓޑࢌ੝ۓ Вය(ډයВ)Ǵ٩ऊۓϐቬऊሽ਱(Strike Price)ǴວΕ܈፤р΋ۓኧໆޑऊۓ኱ޑ ނǶᒧ᏷៾ϩࣁວ៾ک፤៾Ǵ୺Չޑ୏բϩࣁວ຾Ϸ፤рǴаວ຾ວ៾ࣁٯǴວ Εວ៾ཀښ๱ӧډයВਔǴࠨऊᏱԖޣԖ៾ճ٩ቬऊሽ਱ວ຾ࠨऊኧໆޑ኱ޑނ

܈ࢂᒧ᏷ό୺ՉǴԶ྽ວ຾ວ៾ޑΓᒧ᏷୺ՉࠨऊਔǴ፤рວ៾ޑΓԖက୍ቬ ऊǴວΕວ៾ޣࢂჹѱ൑ޑ҂ٰو༈࣮ᅍǴ܌а׆ఈૈӧ҂ٰҔၨѱ൑եޑሽ਱

ວΕ኱ޑၗౢаᕇճǶ

ᒧ᏷៾٩ວБளा؃ቬऊϐයज़ǴΞёϩࣁȨऍԄȩᆶȨኻԄȩᒧ᏷៾Ǵऍ Ԅᒧ᏷៾ޑວБૈܭᒧ᏷៾ډය߻Һ΋Ϻ୺Չ៾ճǴኻԄᒧ᏷៾ޑວБѝૈӧډ යВωૈՉ٬៾ճ[4]Ƕ

ᒧ᏷៾ޑ੝܄ӧܭǺ 1. ᄫఎᏹբ

ᒧ᏷៾ޑວБѝሡЍбλᚐ៾ճߎǴࠅԖคज़ᕇճޑёૈǴ܌аԖаλཛεǴ Ҕၨեҁߎᕇڗၨଯ׫ၗൔၿޑ੝܄Ƕ

2. ᗉᓀ

׫ၗޣऩ࡭Ԗ౜೤Ǵӵ݀όዴۓѱ൑҂ٰوӛǴࣁΑೕᗉ॥ᓀǴёаҔᖼວ ᒧ᏷៾ޑБԄǶӳೀӧܭ܈౜೤ሽॶΠຳǴ߾ᒧ᏷៾ᕇճёаᔆံ౜೤ཞѨǴԶ ӵ݀౜೤ሽॶ΢ᅍǴ߾ѝཞѨλ೽ҽ៾ճߎǶ

3. ሀۯ׫ၗ،฼

җܭᒧ᏷៾வғਏВډ୺ՉВԖࢤਔ໔ǴӢԜວБёаԖၨӭޑਔ໔ᢀჸѱ

൑ޑوӛаϷղᘐǴԶऍԄᒧ᏷៾׳ӢࣁܭډයВ߻ࣣё୺ՉǴӢԜჹܭວБޑ

ၗߎፓࡋගٮ׳ଯޑቸ܄Ƕ ᒧ᏷៾Ξᆀࣁය៾Ƕ

1.2 ምምᛖᒧ᏷៾

ምᛖԄᒧ᏷៾ᆶ΋૓ᒧ᏷៾ޑനεৡ౦ӧܭǺምᛖᒧ᏷៾ନΑԖ΋૓ϐቬऊ ሽ਱ѦǴۘԖ೛ी΋੝ۓޑምᛖሽ਱(ሽ਱΢ज़܈Πज़)Ǵ྽኱ޑނሽ਱ӧࠨऊډ ය߻࿘᝻ډԜምᛖሽ਱Ǵᒧ᏷៾ࠨऊջҥڅғਏ܈ಖЗǶځϩᜪӵΠ߄܌ҢǺ

ғਏࠠ ಖЗࠠ

΢ज़ࠠ ΢ज़ғਏࠠວ៾

΢ज़ғਏࠠ፤៾

΢ज़ಖЗࠠວ៾

΢ज़ಖЗࠠ፤៾

Πज़ࠠ Πज़ғਏࠠວ៾

Πज़ғਏࠠ፤៾

Πज़ಖЗࠠວ៾

Πज़ಖЗࠠ፤៾

߄ 1-1. ምᛖᒧ᏷៾ϐϩᜪǶ

а΢ज़ಖЗࠠᒧ᏷៾(up-and-out)ࣁٯǴԜᒧ᏷៾Ԗঁεܭ߃ۈሽ਱ޑምᛖሽ

਱(barrierǴ܈ᆀࣚज़)Ǵ྽኱ޑނሽ਱ၲډԜምᛖሽ਱ਔǴԜᒧ᏷៾࠹֋ѨਏǶ Զ΢ज़ғਏࠠᒧ᏷៾߾ࢂ྽኱ޑނሽ਱᝻࿘ډምᛖሽ਱ਔω໒ۈғਏ[3]Ƕ 1.3 ЃᎿᒧ᏷៾

྽ምᛖᒧ᏷៾ޑғਏ(܈ಖЗ)చҹவȨ᝻࿘ȩډምᛖሽ਱ᡂԋȨ᝻࿘Ъ࡭ុ

΋ࢤਔ໔ȩਔǴԜምᛖᒧ᏷៾ߡᆀࣁЃᎿᒧ᏷៾[14]Ƕ 1.3.1୷ҁཷۺ

ЃᎿᒧ᏷៾ёᇥࢂ΋ᅿምᛖᒧ᏷៾ޑᡂ׎ǴѬڀഢምᛖᒧ᏷៾ޑ੝܄Ǵሽ਱

ᆶ኱ޑ୘ࠔၡ৩΢ޑሽ਱ࢂցම᝻Ϸምᛖሽ਱ԖᜢǴՠࢂచऊೕۓၨࣁᝄ਱Ǵё ගٮ׫ၗΓϐᒧ᏷܄ၨӭǶᙁقϐǴЃᎿᒧ᏷៾ࣁ΋ᅿሽॶڗ،ܭ኱ޑၗౢࢂց ܭࢌ੝ۓය໔ϣ࿘ډ܈ࢂຬၸࢌ੝ۓࣚज़ǴЪຬၸࣚज़ޑය໔ߏࡋၲډࢌ੝ۓ኱

ྗޑ୘ࠔǶ 1.3.2 ᅿᜪ׎Ԅ

(1)ಕᑈࠠЃᎿᒧ᏷៾ǺԜᒧ᏷៾ሽॶᆶ኱ޑނሽ਱᝻࿘܈ࢂຬၸምᛖሽ਱

ޑය໔ᕴߏࡋԖᜢǶၗౢ኱ޑނሽ਱᝻࿘܈ຬၸምᛖሽ਱ޑȨಕᑈȩය໔ߏࡋε ܭ฻ܭךॺӃႣۓޑߏࡋॶਔǴԜᒧ᏷៾ω཮ғਏ(܈ಖЗ)Ƕӵӕ΋૓ምᛖᒧ᏷

΋ኬǴԜᒧ᏷៾ӅԖΖᅿࠠԄǶ

(2)ೱុࠠЃᎿᒧ᏷៾ǺԜᅿᒧ᏷៾ӧၗౢ኱ޑނሽ਱᝻࿘܈ຬၸምᛖሽ਱

ޑȨೱុȩය໔ߏࡋεܭ฻ܭךॺӃႣۓޑߏࡋॶਔǴᒧ᏷៾ω཮ғਏ(܈ಖЗ)Ƕ ӵӕ΋૓ምᛖᒧ᏷΋ኬԜᒧ᏷៾ӅԖΖᅿࠠԄǶ

(3)షӝࠠЃᎿᒧ᏷៾Ǻࣁа΢ٿᅿࠠԄᒧ᏷៾ϐషӝᡏǴӧҁጇፕЎ္ό ӧ૸ፕϐӈǶ

1.3.3੝੝܄

ЃᎿᒧ᏷៾ޑӳೀӧܭдගٮΑ׫ၗΓ΋ঁ៾ճߎၨեԶЪߥៈၨӭޑ΋

ঁ़ғ܄୘ࠔǴЀܭѬࢂӧ኱ޑނሽᒲ࿘ډ܈ຬၸࢌ΋ࣚज़΋ࢤਔ໔ࡕջғਏ܈

ύЗǴӢԜ៾ճߎሽ਱཮К΋૓ද೯ᒧ᏷៾եǴԶΨό཮ӵӕ΋૓ምᛖᒧ᏷៾΋

ኬ΋ՠ኱ޑނሽ਱᝻࿘܈ຬၸࣚज़ଭ΢൩ѨਏǴԶૈගٮၨӭޑߥៈǶ

ԶऩаಕᑈࠠکೱុࠠЃᎿᒧ᏷៾଺КၨǴҗܭೱុࠠЃᎿᒧ᏷៾ޑచҹК

ၨό৒ܰၲԋǴ୘ࠔКၨό৒ܰѨਏǴ܌аځᒧ᏷៾ሽ਱཮КၨଯǶ

ЃᎿᒧ᏷៾ޑຑሽБԄନΑک΋૓ᒧ᏷៾΋ኬޑ୷ҁୖኧႽࢂǺቬऊሽ਱Ǵ ය໔ߏอǴค॥ᓀճ౗฻ǴᗋᆶΠӈ൳ঁୖኧԖᜢǺ

1. ډයВߏอ

2. Ⴃۓғਏ(ಖЗ)ය໔ߏอ 3. ምᛖሽ਱[14]

ಃ

ಃΒക ୷ҁᢀۺаϷπڀ

2.1 Β໨Ԅᒧ᏷៾ۓሽኳࠠ

Β໨Ԅᒧ᏷៾ۓሽኳࠠനԐҗ CoxǴRoss ک Rubinstein ΟΓӧ 1979 ԃගрǴ Ҟޑӧຑ՗ډයВ߻Չ٬ޑᒧ᏷៾ϐӝ౛ሽॶǶΒ໨Ԅᒧ᏷៾ۓሽኳࠠࣁ΋ᅿᚆ ණਔ໔ኳࠠǴԜѦ٠ଷ೛ިሽѝԖ΢ϲکΠຳǴԶЪ؂ԛϲکຳޑᐒ౗аϷ൯ࡋ ό཮ׯᡂǶԶިሽᡂ୏ࣁ΋ೱុਔ໔ޑᡂϯǴՠӧԜኳࠠύஒԜೱុਔ໔Ϫԋ΋

ኬߏࡋޑਔࢤǴ؂ঁਔࢤ೿Ԗঁᢀჸᗺ܈ᆀ࿯ᗺǴ٠ኳᔕр܌Ԗёૈޑว৖ၡ ৩Ǵӆჹ؂΋ঁၡ৩΢ޑ؂΋ঁ࿯ᗺीᆉр኱ޑނӧ၀࿯ᗺਔޑԏ੻аϷᒧ᏷៾

ӧ၀࿯ᗺޑሽ਱Ƕ

ଷ೛౜ӧިሽ S Ǵ΢ϲکΠຳ൯ࡋϩձࣁ u ک d Ǵ΢ϲکΠຳᐒ౗ϩձࣁ p ک q Ǵ๏ۓᢀჸਔ໔ t' ǴԋҬሽࣁ X Ǵᒧ᏷៾ሽ਱ࣁC (ӧԜଷ೛ࣁວ៾)Ǵ߾ё аҔΠკٰ߄Ң΋යޑΒ໨Ԅᒧ᏷៾ۓሽኳࠠ[13]Ǻ

კ 2-1. ΋යޑΒ໨Ԅᒧ᏷៾ۓሽኳࠠǶ ԶC_uǴC_dջࣁӚԾ܌ӧ࿯ᗺޑᒧ᏷៾ሽ਱Ƕ

ࣁΑБߡǴךॺஒ q ೛ԋ1pǴଷ೛࿶ၸ n යǴ߾ኳࠠᡂԋკ 2-2Ǵଷ೛ӧ

ࢌ΋҃ᆄ࿯ᗺύޑ၀ᗺިሽ࿶ᐕ j යࢂΠຳǴn යࢂ΢ϲǴ߾၀ᗺϐިሽࣁj

j j

n d

Su ^ Ǵډၲ၀ᗺᐒ౗ࣁ p^{n j} p ^j j

n (1 )

¸¸¹·

¨¨©§ _

Ƕஒש౜౗Եቾ຾ѐޑ၉Ǵךॺஒёа ளр၀ᒧ᏷៾ޑය߃ሽॶǺ

¦ ^{ }

 ¸¸¹·  

¨¨©§

n

j

j j n j

j n

rT p p Su d X

j e n

c

0

) 0 , max(

) 1 (

ځύrࣁค॥ᓀճ౗ǴTࣁᒧ᏷៾ޑӸុය໔Ƕ

ऩஒ u ॶۓࣁ ⁿ

T

e^V Ǵd ॶۓࣁ u

1Ǵ߾ԜΒ໨Ԅᒧ᏷៾ۓሽኳࠠջࣁচۈ CRR

ኳࠠǶ

კ 2-2. n යޑΒ໨Ԅᒧ᏷៾ۓሽኳࠠǴS₀ࣁ߃ۈሽ਱Ƕ

2.2 ғԋڄኧ

ଷ೛Ԗ΋ኧӈA_n {a₀,a₁,}Ǵ߾ڄኧ

¦

r r

rx a a x a x

a x

f _{0 1}

0

) (

ᆀࣁኧӈA_nϐғԋڄኧǶ

ғԋڄኧ೯தҔܭှ،ঁኧಔӝϐୢᚒǶᖐٯٰᇥǴԖٿঁόӕ೓ηǴϩձ Ԗٿᗭౚӧ္य़Ǵ߾܌ԖёૈڗౚޑБݤёаҔΠӈԄηٰ߄ҢǺ(ౚຎࣁόӕ)

2 2 2 2 2 2

2

2)(1 ) 1

1

( xx  y y xx  yxyx yy xy x y

ځύ1ж߄όڗǴ΋ԛБ߄Ңڗ΋ᗭǴٿԛБ߄ҢڗٿᗭǶऩౚຎࣁ࣬ӕǴ

߾΢Ԅёׯቪԋ12x3x² 2x³ x⁴Ǵ x ޑԛБ߄ҢౚޑঁኧǴ߯ኧ߄ҢБݤ ኧǴႽࢂ3x²߄Ңڗ 2 ᗭౚޑБݤԖΟᅿǺࡷځύ΋ঁ೓ηڗځ္य़ӄ೽ޑౚ܈

ࢂٿঁ೓ηϩձڗ΋ᗭౚ[2]Ƕ

2.3 GPUکک CUDA 2.3.1 GPUᙁϟکᐕў

GPU(Graphic Processing Unit)ǴύЎᙌԋკ׎ೀ౛ᏔǴࢂ΋ᅿ஑ߐҔٰೀ౛

ቹႽၮᆉπբޑ༾ೀ౛ᏔǶGPU നεфҔӧܭ٬ᡉҢь෧Ͽჹ CPU ޑ٩ᒘǴ٠ ЪϩᏼΑ CPU ޑπբǴЀځӧೀ౛ 3D კ׎ਔਏૈ׳уܴᡉǶ΋ᗭ CPU ཮Ԗ΋

ډѤᗭਡЈǴҔٰೀ౛ׇӈၮᆉ(Serial Processing)ǴԶ΋஭Ԗ GPU ޑᡉҢьύԖ ӭঁೀ౛ᏔǴ࣬྽ܭ CPU ޑၮᆉਡЈǴՠόӕϐೀӧܭ CPU ޑਡЈ΋ԛനӭѝ

ૈೀ౛ٿచ୺ՉᆣǴԶ GPU ޑࠅᕴӅёаЍජ΢ԭచа΢Ǵ܌аӧѳՉೀ౛ำ ԄѳՉၮᆉΠǴ٬Ҕ GPU ޑीᆉਏ౗཮К٬Ҕ CPU ז΢೚ӭ७ǶךॺВதғࢲ ύКၨதௗ᝻ډѳՉၮᆉޑҔ೼Ǻ1.ቹТᆶྣТǵ2.ຎ᝺ᆛઠǵ3.ၯᔍǶаྣТ

ٰᖐٯǴ΋஭ྣТࢂҗӭঁႽનಔԋǴCPU ᗨฅёаזೲӦ଺ၨፄᚇӦၮᆉǴ ՠࢂҗܭೀ౛Ꮤޑज़ڋǴCPU ѝૈ΋ঁႽન΋ঁ࣬નीᆉǹԶ GPU ߡёаӕਔ

ीᆉӭঁႽનǴӢԜёаӕਔೀ౛΋༧୔ୱޑՅறǴ܌аӧೀ౛ྣТ΢ GPU ᡉ ளԖၨεӦᓬ༈ǶԶךॺஒԜמೌၮҔӧᒧ᏷៾ޑीሽ΢ǴӢࣁӧΒ໨Ԅኳࠠ

ύǴऩ n ॶ(යኧ)ၨεǴ߾ᕴၡ৩ኧஒ཮ၲډ΋ঁёᢀޑኧӷǴऩҔ໺಍ޑ CPU

଺ၮᆉஒ཮઻ၗεໆޑਔ໔ǴӢԜךॺஒ౳Ӏܫӧ GPU ΢ǴයఈૈᙖҗѬޑѳ ՉၮᆉૈΚٰ෧Ͽ੃઻ޑਔ໔ǶӕኬӦǴεӭኧᔈҔำԄ໒วޣӧ଺ GPU ၮᆉ ਔǴࢂஒำԄޑೱុ೽ҽҬҗ CPU ଺ǴԶख़ᙟ܄ଯޑπբҬҗ GPU ೀ౛ǶӵԜ ϐѦǴGPU ӧೀ౛ੌᗺኧၮᆉБय़ޑԋߏೲࡋΨᇻεܭ CPUǴӵკ 2-3Ƕՠ࣬ჹ ޑԖ٤඲Т٠คϩ໒Ӧ᏾ኧၮᆉൂϡӢԜӧ᏾ኧၮᆉБय़ਏ౗ัৡ[12]Ƕ

კ 2-3. CPU ک GPU ϩձӧੌᗺኧၮᆉޑೲࡋ [8]Ƕ

Զ଺ GPU ਡЈനԖӜޑϦљ྽ኧ NVIDIAǴനԐܭ 1998 ԃวթ NV4ǴҔܭ Riva TNTᡉҢь΢ǹ႖ԃ 4 ДΞ௢р NV5ǴҔܭ Riva TNT5 ΢ǹӧ 1999 ԃޑ 9 Д߾ԖΑ NV10 ޑୢШǴNV10 Ξᆀ TNT3ǴԶ GPU ޑཷۺΨ൩ࢂவ೭ਔং໒ۈ ԖޑǶӧ 2000 ԃ 4 ДΞрΑ NV15Ǵးӧ NVIDIA ཥسӈ GeForce 2 GTS ΢Ǵځ ύ TS ж߄ޑࢂȨGigaTexture ShaderȩǴཀࡘࢂ΋ࣾёа༤кޑႽનঁኧࢂ 10 ሹ ભޑǶӧϐࡕΞഌុ௢рΑ NV10ǴҔӧ GeForce 2 MX ΢ǹNV20Ǵࣁ GeForce 3 سӈᡉьޑਡЈ٠Ъஒ๱Յൂϡ२ԛϩԋႽનکഗᗺٿ೽ҽǴЍ DirectX8ǹ2002 ԃ 2 ДӕਔวթΑ NV17 ک NV25Ǵ߻ޣҔܭ GeForce 4 MX440(ࣁ߈ԃٰεӭኧ ၯᔍޑനեा؃ᡉҢьଛഢ)ǴࡕޣҔܭၨଯ໘ޑࠠဦ GeForce 4 Ti4600Ƕ2004 ԃࣴวр NV40 ԋࣁ GeForce 6 سӈǴӕԃ 10 ДځᝡݾჹЋ ATI Ψ௢р Radeon 9700Ǵࣁӄౚಃ΋ঁЍජ Direct 9.0 ޑᡉҢьǶځࡕ NVIDIA ܭ 2005 ԃஒ NV ׯ Ӝࣁ G سӈǴࣴวр G70ǴG72 ฻฻Ǵ΋ޔ ډ 2006 ԃޑ G80ǴҔܭ GeForce 8800 سӈǴӕਔΨЍජ DirectX 10ǹаϷന߈ӧ 2008 ԃ௢рޑ G200[10]Ƕ

2.3.2 GPGPUکک CUDA

ᇥډ GPU ၮᆉ൩΋ۓाගډ CUDAǴCUDA ӄӜࣁ Compute Unified Device

ArchitectureǴࣁ NVIDIA Ϧљ௢рޑ΋ঁ GPU ᏾ӝמೌǴ೸ၸԜמೌǴ٬Ҕޣ ёаճҔ NVIDIA Ϧљрޑ GeForece 8 سӈаࡕޑ GPU ٰ຾ՉၮᆉǴΨࢂ၀Ϧ љჹܭ GPGPU ޑ҅ԄӜᆀǴ܌ᒏ GPGPU ύЎӄӜࣁ general-purpose computing on graphic processing unitsǴࢂ΋ᅿճҔ GPU ٰीᆉচҁҗ CPU ೀ౛ޑ೯Ҕीᆉ Һ୍ǴΨ൩ࢂஒ GPU Ҕӧߚ໺಍ޑ 3D კ׎ᡉҢ[9]Ƕ໺಍ޑ GPGPU ໒วБݤ

೿ࢂ೸ၸ OpenGL ܈ Direct3DǴаጓቪ shading language ޑБݤ௓ڋ shader ٰགྷ ᒤݤၮᆉǶԶ NVIDIA ගрޑ CUDA ࢂёа೸ၸ C ᇟقޑڄԄ৤ٰጓቪޑǴу

΢ ό٬Ҕკ׎ڄԄ৤ǴӢԜӧำԄ೛ी΢׳ࣁБߡǶCUDA εठ΢ϩࣁ libraryǵruntimeǵdriver Οঁ೽ҽǶځࢎᄬკӵკ 2-4Ǻ

კ 2-4. CUDA ࢎᄬ [8]Ƕ

Զӧቪ CUDA ޑำԄਔǴךॺ཮ஒำԄ୺Չޑ୔ୱϩԋٿঁ೽ҽǴ΋ঁࢂ

CPU୺Չޑ host ᆄǴќѦ΋೽ҽ൩ࢂ GPU ୺Չޑ device ᆄǶԶӧ CUDA ำԄ

ࢎᄬ္ǴЬाޑำԄᗋࢂ CPU ܌୺ՉޑǴѝԖӧၶډሡाѳՉೀ౛ޑਔংǴω

཮ஒำԄጓ᝿ԋ device ૈ୺ՉޑำԄӆҧ๏ GPU ୺ՉǴԶԜำԄӧ CUDA ύᆀ ϐࣁ kernelǶԶӧ device ύǴೀ౛ kernel ޑനλൂϡћ଺ threadǴύЎࣁ୺ՉᆣǴ

؂΋ঁ device ύԖӭঁ threadǴ؂ঁ thread ೿ࢂӕਔ୺Չ kernel ำԄǴԶךॺ ܌ ճҔޑ൩ࢂ؂ঁ thread ޑ index όӕǴԶԖόӕޑၗ਑ٰ຾ՉၮᆉǶኧঁ thread ёаಔԋ΋ঁ blockǴӕঁ block ္ޑ thread ёаӸڗӕ΋༧૶ᏫᡏǴӢԜёа

຾Չזೲޑӕ؁୏բǴԶόӕ block ޑ thread ߾คݤޔௗӸڗӕঁ૶ᏫᡏǴӭঁ

block ёаಔԋ΋ঁ gridǴ೸ၸ೭ᅿኳԄǴךॺёа׳уԖਏӦճҔ؂ঁ thread ޑфਏǴԶό཮೏ thread ኧҞ܌ज़ڋǶځᜢ߯ӵკ 2-5Ƕ

კ 2-5. ThreadǵBlock ک Grid ޑᜢ߯ [8]Ƕ

ӧቪ CUDA ำԄਔǴाᡣ thread ܌٬Ҕޑၗ਑೿཮Ӄவ host ໺຾ device ޑ

૶ᏫᡏύǴԶਥᏵόӕޑᔈҔΞԖϩόӕޑ૶ᏫᡏᜪࠠǶӵკ 2-6 ܌ҢǴ؂ঁ

threadԖԾρޑ local memoryǴԶӧӕ΋ঁ block ္ޑόӕ thread ߾ёӸڗӕ΋

ঁ shared memoryǴԶ܌Ԗޑ thread ջ٬ӧόӕ block ࣗԿόӕ gridǴ߾೿ૈ٬Ҕ ӕ΋ঁ୤᠐ޑ૶ᏫᡏΨ൩ࢂ global memory ္ޑၗ਑ǶନԜϐѦǴᗋԖٿᅿ୤᠐ ޑ૶Ꮻᡏޜ໔Ǵϩձᆀࣁ constant memory ک texture memoryǴΨࢂૈᡣ܌Ԗ thread

܌Ӹڗޑޜ໔Ƕ΢ॊޑ૶ᏫύǴglobalǵconstant ک texture ૶ᏫᡏޑӸុਔ໔ࢂ

ک kernel ำԄӸӧਔ໔΋ኬߏޑǶԶךॺதाݙཀޑࢂ thread ӧӸڗӕ΋༧૶Ꮻ ᡏਔޑӕ؁ϯǶ

კ 2-6. ૶Ꮻᡏቫԛ [8]Ƕ Զ CUDA ำԄ୺ՉਔޑࢬำࣁǺ

1. Host(೯தࣁЬᐒ)૶Ꮻᡏ໺ଌၗ਑کำԄዸډ device(೯தࣁᡉҢь)૶ᏫᡏǶ 2. Host଺ځдޑ٣܈໕࿼Ǵdevice ୺Չ kernel ำԄǶ

3. Device૶Ꮻᡏஒ୺Չ่݀໺ӣ host ૶ᏫǴำԄᝩុ୺ՉǶ

ಃ

ಃΟക ճҔғԋڄኧٰीᆉЃᎿᒧ᏷៾ޑሽ਱

ӧ೭ঁക࿯ύǴךॺ߾ჴբ Li ک Zhao ܌ගрҔғԋڄኧٰຑሽЃᎿᒧ᏷៾

ޑБݤǴ΋໒ۈӃࢂа CRR ኳٰࠠ଺΋٤ۓကکϟಏǴௗ๱ߡҔ؃рғԋڄኧ ޑ߯ኧБݤٰှрЃᎿᒧ᏷៾ӧ n යਔޑሽ਱[1]Ƕ

3.1 ಔӝኧᏢࢎᄬ

ӧ CRR ኳࠠύǴךॺࢂҔ lattice path ଺୷ᘵǴ٠ଷ೛ިሽѝԖ΢ᅍکΠຳ ٿᅿёૈǴךॺஒ؂΋ঁ΢ᅍ܈Πຳ΋යޑၡ৩ᆀࣁൂϡၡ৩ǶԶӵ݀ӧ০኱კ

΢Ǵ؂΋ঁ΢ᅍޑൂϡၡ৩൩ࢂவ(x,y)౽୏ډ(x y1, 1)ǴךॺҔ U ٰ߄ҢǶ

؂ঁΠຳޑൂϡၡ৩൩ࢂவ(x,y)౽୏ډ(x y1, 1)ǴךॺҔ D ٰ߄ҢǶ܌аӧ ԜኳࠠύǴ؂చ lattice path ךॺ೿ёаׯቪԋa1a2a_nǴځύa_i A {U, D}Ǵ ԶЪԜၡ৩ߏࣁ n ǶԜѦךॺᗋۓကޜၡ৩ࣁ1ǶԶٿచၡ৩ޑ४ᑈךॺຎࣁೱ

่ Ǵ ଷ ೛ ΋ చ ၡ ৩D a1a2a_n ^{Ǵ ќ ΋ చ}E b1b2bmǴ ߾ д ॺ ޑ ४ ᑈ ջ ࣁ

m

nbb b

a a

a1 2 1 2

DE ^{ǴΨࢂ΋చཥޑၡ৩Ƕ}

ௗ๱ךॺࣁΑפрၡ৩ࣁ n ޑ܌Ԗ lattice path ঁኧǴߡЇ຾Αғԋڄኧޑཷ

ۺǴଷ೛Lࣁ܌Ԗ CRR ኳࠠύ܌Ԗ lattice path ޑ໣ӝǴךॺۓကL_nࣁӧ܌ԖӧL ύԶЪߏࡋ฻ܭ n ޑၡ৩໣ӝǴځኧᏢԄࣁǺ^Ln

^

`

Ln ޑ ϡ ન ঁ ኧ Ǵ Ҕ |L |_n ଺ ߄ Ң Ǵ ߾ ך ॺ ё а ቪ р ׇ ӈ { f }_n ޑ ғ ԋ ڄ ኧ Ǻ

¦_t

0

) (

n

n nt L t

L ǴԶΞӢࣁ^L

^

`

ຎࣁL໣ӝޑғԋڄኧǶௗΠٰךॺۓကU {U}ǴΨ൩ࢂᇥӧU ໣ӝ္ѝԖ΋

ঁϡનћ଺ UǴࡐܴᡉёа࣮рU_nύନΑU₁ UаѦځдࣣࣁޜ໣ӝǴӢԜU ໣ ӝޑғԋڄኧࣁU(t) tǴӕ౛ٰᇥǴऩךॺۓကD {D}߾D(t) tǴךॺΨё аஒ t ຎࣁӧ lattice path ύޑ΋؁Ǵόࢂ۳΢൩ࢂ۳ΠǶ

೭ਔа΋ঁΠज़ಖЗࠠምᛖᒧ᏷៾ວ៾ٰբٯηǴ२ӃךॺӃஒ CRR ኳࠠ

኱ӧ০኱კ΢ǴԶ؂చᆶ x ືѳՉЪک x ືຯᚆࣁ k ޑጕᆀࣁಃ k ቫǴಃ 0 ቫջ

ࣁ x ືǶךॺёаགྷᕵືࣁ኱ޑނሽ਱Ǵᐉືࣁਔ໔Ǵଷ೛Ԝᒧ᏷៾வচᗺ(0,0) ໒ۈǴԶምᛖሽ਱ࣁಃ1ቫǴΨ൩ࢂᇥ྽ࢌచሽ਱ၡ৩࿘ډಃ1ቫਔ߾Ԝᒧ᏷

៾ѨਏǴӢԜ྽ךॺाीᆉԜᒧ᏷៾ሽॶਔǴሡाԵቾޑࢂ܌Ԗ҂᝻࿘ډಃ1^ቫ ޑၡ৩Ǵ؂చၡ৩ಖᗺࣁ( in, )ǴځύndidnǶࣁΑᙁϯीᆉǴךॺ೭ᜐѝሡ Եቾ܌Ԗ҂᝻࿘ډಃ 1 ቫЪಖᗺӧ( n2 ,0)ޑၡ৩Ƕ

ଷ೛΋໣ӝ C ࣁ܌Ԗ҂᝻࿘ډಃ1ቫЪಖᗺӧ x ື΢ޑ lattice pathǴC_nࣁ C

္Ъಖᗺӧ( n2 ,0)ޑᕴၡ৩ঁኧǴ߾ C ޑғԋڄኧࣁ ¦

t0

) 2

(

n

n nt C t

C ǴԶӧीᆉ

) (t

C ޑϦԄ߻ǴךॺӃϟಏ፦ኧ໣ӝǺӧ΋ lattice path ܌໣ӝԶԋޑ໣ӝ L ္Ǵ ऩL္य़؂΋చၡ৩೿ёа೏୤΋Ӧϩှԋኧঁόӕޑλၡ৩Ǵ߾Ԝᅿλၡ৩ޑ

܌ Ԗ ໣ ӝ ջ ࣁ ໣ ӝL ޑ፦ኧ໣ӝP Ƕ Զ ך ॺ ё а ٬ Ҕ ΋ ঁ ᇶ շ ۓ ౛ ࣁ Ǻ

) ( 1 ) 1

(t P t

L  ǶԜѦךॺᗋёаޕၰǴӧ C ໣ӝ္ޑ፦ኧ໣ӝ P ࢂҗ܌ӧ C ္Զ ЪѝԖ໒ۈکಖᗺӧ x ື΢ޑၡ৩܌ಔԋǴӢԜךॺёаஒҺՖӧP္ޑၡ৩ϩ

ှԋ 8ȕ'ǴځύEC^{ǴԶ P ޑғԋڄኧP(t)൩฻ܭU(t)C(t)D(t)ǴӆճҔখখ} ޑۓ౛Ǵ߾฻ԄᡂԋǺ

) ( 1 ) 1

( ₂

t C t t

C  Ǵ࿶ၸϯᙁکΒϡ΋ԛБำԄှਥޑϦԄǴ

ךॺёаᏤр ₂

2

2 4 1 ) 1

( t

t t

C  

Ǵӆ࿶ၸੀୌ৖໒ԄޑᡂඤǴךॺёаளډt²ⁿޑ

߯ኧ ¸¸¹·

¨¨©§

 n n

C_n n 2

1

1 Ǵ೭ΨࢂԖӜޑ Catalan ኧǶԶC_nࡽࣁ܌Ԗ҂࿶ၸ x ືۭΠ

Ъಖᗺӧ( n2 ,0)ޑၡ৩ঁኧǴѬޑॶஒךॺ฻฻ाीᆉЃᎿᒧ᏷៾ሽ਱ਔ཮Ҕ ډǶ

ନԜϐѦǴךॺҔ Lagrange ϸᆉϦԄёа؃рtⁿӧ(C ))(t ^rਔޑ߯ኧࣁ

 

¨¨©§

¸¸¹·





 n r r N

r n

r

n 1 ,

2 2

2

ӧךॺा໒ۈीᆉೱុࠠЃᎿᒧ᏷៾ሽ਱߻Ǵᗋाϟಏ΋ঁख़ाޑۓ౛Ǻ [ۓ౛΋]

з m ࣁ΋᏾ኧǴᒧ΋ঁၡ৩ȝǴۓကȨȝ ӧಃ m ቫ΢നӭޑೱុ؁ኧȩࣁmcm(P)Ǵ ϞϺ๏ۓ΋ঁ҅᏾ኧ l Ǵ߾܌Ԗmc_m(P)l^{٠Ъ࿘ډ܈ຫၸಃ}m¹ቫаϷಖᗺӧ

ಃ i ቫޑၡ৩ȝ ܌ԋޑ໣ӝࣁTǴځύi Ǵ߾m TޑғԋڄኧࣁǺ

2 2

1 0

2 2

2

) 2

(

, 2 2

, ) 1 ( ) ( ) (

) ( ) ) (

(

 

















l r r

t C t

C C t Q

i m r

t C t t

Q t Q t

t Q t t C

T

 l

Πय़ޑᇶշۓ౛߾ࢂӧךॺीᆉಕᑈࠠЃᎿᒧ᏷៾ਔ཮ҔډǶ

Chung-Feller TheoremǺவ(0,0)ډ( n2 ,0)٠ЪҗൂՏၡ৩ UǵD ܌ಔԋǴ٠ Ъ҅ӳԖ k2 ؁ӧಃ 0 ቫ΢ޑ܌Ԗၡ৩Ǵځঁኧک k ॶคᜢǴԶࢂکಃ n ঁ Catalan ኧ΋ኬǶ

3.2 ीᆉЃᎿᒧ᏷៾ޑሽ਱

ӧ೭കךॺ܌ाीᆉޑࢂ n ය΢ज़ಖЗࠠЃᎿᒧ᏷៾ວ៾ޑሽ਱Ǵӧीᆉϐ

߻Ӄჹ΋٤ኧॶ଺ۓကǴଷ೛ךॺޑምᛖሽ਱ H ࣁS₀u^mǴ m ࣁިሽ࿘ډምᛖሽ

਱ਔाوޑനϿӛ΢ԛኧǴw ࣁךॺ೛ۓޑ୔໔ǴΨ൩ࢂӵ݀ިሽӧምᛖሽ਱΢

ೱុӸӧ w ਔ໔ࡕԜᒧ᏷៾ѨਏǴl ࣁ࣬ჹᔈ w ޑǴӧምᛖሽ਱΢ޑਔ໔ᗺ؁ኧǴ T n

l t w Ǵךॺ೯தஒ l ଷ೛ԋଽኧǶ

3.2.1 ೱុࠠЃᎿᒧ᏷៾

ೱុࠠЃᎿᒧ᏷៾ѝԵቾӧምᛖሽ਱΢ೱុӸӧޑޑਔ໔ǴӢԜךॺҔখখ ۓကޑȨȝ ӧಃ m ቫ΢നӭޑೱុ؁ኧȩΨ൩ࢂmcm(P)Ǵ٠Ъӆۓက f( in, )ࣁ வচᗺ(0,0)໒ۈǴಖᗺӧ( in, )Ъmc_m(P)l^{ޑၡ৩ ȝ ϐঁኧǶࡐܴᡉ྽n} ࣁⁱ ڻኧਔǴf(n,i) 0ǴӢԜךॺޔௗஒniբࣁଽኧǶԶਥᏵiॶόӕǴϩԋΟঁ

௃ݩ૸ፕǺ

ಃ΋ᅿ௃ݩǴitlmǺӧ೭௃ݩύǴf(n,i) 0ǶӢࣁ྽Ԝၡ৩࿘ډಃ m ቫ ࡕǴѬᗋ໪ाԿϿimtlঁӛ΢؁ኧωၲډಃ i ቫǶ

ಃΒᅿ௃ݩǴi Ǻӧ೭ঁ௃ݩךॺΞाӆ٩Ᏽ೭చၡ৩ࢂցԖ࿘ډಃm

1

m ቫٰϩԋٿ೽ҽ૸ፕǶଷ೛g( in, )ࣁ܌Ԗmc_m(P)l^{ԶЪම࿶࿘ډ܈ຫၸಃ}

1

m ቫЪಖᗺӧ( in, )ޑ܌Ԗၡ৩ȝ ኧǶฅࡕҔখখޑۓ౛΋Ǵךॺёаᕇளа Πޑғԋڄኧ:

2 2

1 0 1

2 2

2 2

1 1

) (

, 2 2

, ) 1 ( ) ( ) (

) ( ) ) (

, (

 















¦

l r r n

t C t

C C t Q

i m r

t C t t

Q t Q t

t Q t t C

i n g

 l

ӆҔϐ߻ගډޑtⁿӧ(C ))(t ^rਔޑ߯ኧϦԄаϷၮᆉࡕǴёаளрtⁿޑ߯ኧ ࣁǺ

¦d

d 



 ¸¸¹·

¨¨©§  

 

¸¸¹·

¨¨©§ 









2 1 1 1

2 1

1 1 2

1

1 2 1

2 2 2

t j

t n l rj

r l

n C

j r j r j l r

n r

l n

r

ځύ , 2

2

2 ₁

2 1 1

r l t n

r

t n   

Ƕ

ӆ೛h( in, )ࣁவ҂࿘ډಃm1ቫޑၡ৩ኧǴਥᏵϸ৔চ౛Ǵךॺёаޔௗ௢

Ꮴр

¸¸¹·

¨¨©§



 

¸¸¹·

¨¨©§



 ( 1)

) , (

2

2 m

n i n

n

h n i n i

നࡕg(n,i)کh( in, )ٿޣ࣬уջࣁ f( in, )

ಃΟᅿ௃ݩǴmdilmǺ؂ঁၡ৩೿ёаϩԋٿঁ೽ҽǴಃ΋ঁ೽ϩࢂ

வচᗺډ(n mj, 1)Ǵځύim1d jdlǴ೭೽ҽޑၡ৩ኧໆࣁg(n mj, 1)Ƕ

ಃΒ೽ҽޑၡ৩ߏࡋࣁ j Ǵjd ǴёаགྷԋவচᗺрวǴಖᗺӧl i m( 1)ԶЪ೼

ύؒԖӆӣډচᗺޑၡ৩ǶԶځғԋڄኧёቪࣁ(tC(t))^{i m( 1)}ǶӆҔϐ߻ቪၸޑtⁿ ӧ(C ))(t ^rਔޑ߯ኧϦԄǴ߾߯ኧջࣁ೭೽ҽၡ৩ኧໆǴीᆉࡕࣁǺ

l j r m i j r

r

r

j   d d

¸¸¹·

¨¨©§ 

 ^{ 2 2}

2 1 , ( 1),

2

നࡕךॺӆஒٿ೽ҽ଺४ᑈ൩ёளр྽mdilmЪmc_m(P)l^ਔޑ܌Ԗ ၡ৩ȝ ϐᕴঁኧǺ

) 1 (

1 , ) 2

1 , ( )

, (

2

1 2 2

2

2





¸¸¹·

¨¨©§ 

˜ 



 _

d d



 ¦

m i r

j r j m r j n g i

n

f _{j r}

l j m i

Զӧךॺ଺നࡕޑуᕴ߻ǴࣁΑ࿯࣪ၮᆉԋҁǴךॺӃᆉр΋ঁനӭ۳Π؁

ኧ a ٬ளեܭԜኧޑᒧ᏷៾നࡕ೿཮ӧሽϣǺ »

¼

« »

¬

«

) / log(

) / log( ₀

d u

K u a S

n

നࡕǴךॺёаᏤр΢ज़ಖЗࠠޑೱុࠠЃᎿᒧ᏷៾ޑሽ਱Ǻ

¦ ^{ }

 ^a   

j

j j n j j

n

rT f n n j p p S u d K

e c

0

0 )

( ) 1 ( ) 2 , (

ځύ f(n,n2)ࣁn ය΢ᅍ j යΠຳЪj mc_m(P)l^{ޑၡ৩ኧǶ}

3.2.2 ಕᑈࠠЃᎿᒧ᏷៾

ಕᑈࠠЃᎿᒧ᏷៾܌Եቾޑࢂިሽၡ৩ӧምᛖሽ਱΢ಕीޑਔ໔ᕴኧǴӵӕ )

(P

mcm ǴךॺۓကȨȝ ӧಃ m ቫ΢ޑಕᑈ؁ኧȩࣁcsm(P)ǴԶ f_l( in, )ࣁவচᗺ(0,0) ໒ۈǴಖᗺӧ( in, )Ъcs_m(P)l^ޑၡ৩ȝ ϐঁኧǶn ҭբࣁଽኧǶԶΨਥᏵi ॶi ϩԋΟঁ௃ݩ૸ፕǺ

ಃ΋ᅿ௃ݩǴitlmǺکೱុࠠЃᎿᒧ᏷៾΋ኬǴ f_l(n,i) 0Ƕ

ಃΒᅿ௃ݩǴid ǺਥᏵၡ৩ԖؒԖຫၸಃ m ቫϩԋٿ೽ҽ૸ፕǴଷ೛m Rࣁ

܌Ԗၡ৩ȝ Ъcs_m(P)l٠Ъමຫၸಃ m ቫԶനࡕಖᗺӧ i ޑ໣ӝǴ߾؂ঁR္ޑ ȝ ೿ёϩှԋΟঁ೽ҽ ĮȕȖǴಃ΋೽ҽ Į ࣁ܌Ԗவচᗺ໒ۈѝ࿘ډಃ m ቫ΋ԛΨ ൩ࢂಖᗺӧಃ m ቫޑޑၡ৩Ǵځ໣ӝᆀࣁR₁ǹಃΒ೽ҽ ȕ ࣁ܌Ԗcs₀(E)l^ԶЪ ӧಃ 0 ቫ΢നϿٿ؁Ъಖᗺӧ΢य़ޑၡ৩Ǵځ໣ӝᆀࣁR₂ǹಃΟ೽ҽȖ ࣁவಃ 0 ቫрวǴಖᗺӧಃi ቫЪ೼ύؒԖӆӣډಃ 0 ቫޑၡ৩Ǵځ໣ӝᆀࣁm R₃ǶԶ ךॺёа؃р R ޑғԋڄኧǺR(t) R₁(t)R₂(t)R₃(t)ǴځύR₁(t) (tC(t))^mǴԶ

i

t m

tC t

R₃( ) ( ( )) ^ Ƕௗ๱ךॺाᆉR₂(t)ǴճҔখখϟಏޑ Chung-Feller ۓ౛Ǵך

ॺଷ೛A_n^kࣁಖᗺӧ(n,0)ԶЪ҅ӳԖ k ؁ӧಃ 0 ቫ΢ޑၡ৩ኧໆǴ߾

2

Cn

A_n^k Ǵך ॺ߾ૈቪрΠԄǺ

¦

¦

¦





!

 



!











 



 

 

















4

0

2 2

2 2 2

1

2

2 2

2 2 2 2

2 2

2 2 2

2

2

2 ) 2

2 ( 2

2 2 2

2

) (

) (

) (

l

j

j l n

n l

l

n l

n

l n n

l l l l

t jC t l

l C

t l C

t l C

t C

t A A

t A A

t A t R

j

 n







೛g_l( in, )ࣁӧRύಖᗺӧ( in, )ޑ܌Ԗၡ৩ኧǴ߾ךॺਥᏵ΢य़ૈளډRޑ ғԋڄኧࣁǺ

j j l

j i m i

m i m n

l l jC t

t tC t

C l t

t i n g

2 4

0 2 1

2 2

2 )) 2

( ( )

2 ( ) 2 ,

( ¦

¦ ^{       }

ӆଷ೛r3 2miǴฅࡕжΕtⁿӧ(C ))(t ^rਔޑ߯ኧϦԄǴёаᏤрǺ

¦^{  }

 ¸¸¹·

¨¨©§  







 

¸¸¹·

¨¨©§







 ⁴

0 3 2 2

3 2

3 3

3 3

) 1 2 ( 2

) 1 )(

2 ) (

, (

l

j

j r j n r

n

l n j C

r j n

j l n r

r n

r i l

n g

೛h_l( in, )ࣁவ҂࿘ډಃ m ቫޑၡ৩ǴӕኬӦΨό཮࿘ډಃm1ቫǴӢԜё а؃рǺ

¸¸¹·

¨¨©§



 

¸¸¹·

¨¨©§



 ( 1)

) , (

2

2 m

n i n

n

hl n i n i

നࡕךॺёаஒٿ೽ҽуଆٰ؃р f_l(n,i) g_l(n,i)h_l(n,i)

ಃΟᅿ௃ݩǴmilmǺ೭္ޑ f_l( in, )کখখӧᆉೱុࠠЃᎿᒧ᏷៾ޑ

ಃΟύ௃ݩᜪ՟ǴёҔΠԄ߄ҢǺ

m i r

j r j m r j n f i

n f

l j m i

r j j

l l

 ¸¸¹·

¨¨©§ 

 

¦d 

  

4

1 4 2

4 1 ,

) 2 , ( )

,

( 4

ځύ f_lj(n j,m)ࣁၡ৩ Ș ځύcs_m(K)l j^{Ъಖᗺӧ(n j,m)ޑၡ৩ኧ}

ໆǶԶഭΠவ(n j,m)ډ( in, )Ъவ҂࿘ډಃ m ቫޑၡ৩ኧࣁ

¸¸¹·

¨¨©§ 

 4 ^2 4

4

2 1

r j

j r j

r Ƕ

നࡕǴךॺёаᏤр΢ज़ಖЗࠠޑಕᑈࠠЃᎿᒧ᏷៾ޑሽ਱Ǻ

¦ ^{ }

 ^a   

j

j j n j j

n l

rT f n n j p p S u d K

e c

0

0 )

( ) 1 ( ) 2 , (

aޑۓကک΢य़ךॺӧीᆉೱុࠠЃᎿᒧ᏷៾ਔޑۓကޑ΋ኬǴࣁ΋ঁനӭ۳Π

؁ኧ٬ளեܭԜኧޑᒧ᏷៾നࡕ೿཮ӧሽϣǺ »

¼

« »

¬

«

) / log(

) / log( ₀

d u

K u a S

n

Ƕ

ಃ

ಃѤക ჴբکኧᏵ่݀૸ፕ

ךॺӧჴբಃΟകගډޑБݤਔǴࢂӧ Windows ᕉნΠǴ٬Ҕ༾೬ޑ Visual C++ǶԶ΋໒ۈךॺӃКၨךॺ܌ा٬Ҕޑ CPU ک GPU ฯᡏᕉნόӕǴௗ๱ך ॺεཷඔॊӵՖҔ C ᇟقٰቪЬำԄаϷ CUDA ำԄǴനࡕӧόӕޑᡂኧΠǴ

ٰᡍ᛾ךॺޑዴૈճҔ GPU ᕇள׳ӳޑਏૈǶ ךॺ܌٬Ҕޑ CPU ک GPUǺ

Intel Core2 Duo T7100 Nvidia GeForce 8400M GS

ਡЈኧҞ 2 16

୺ՉᆣኧҞ 2 Ծु

ਡЈᓎ౗(ೲࡋ) 1.8GHz 400MHz

߄ 4-1. CPU ک GPU ޑКၨǶ

ॶளݙཀޑࢂǴӢࣁךॺቪӧ CPU ΢ޑำԄ٠҂ԵቾډᚈਡЈǴӢԜ྽ CPU ӧ

ೀ౛ำԄਔࢂҔ΋ᗭਡЈаϷ΋చ୺Չᆣѐ୺ՉޑǶ

ӧ໒ۈჴᡍ߻Ǵךॺޕၰ྽යኧᡂӭਔǴाीᆉ኱ޑނሽ਱ޑၡ৩ᕴኧޑኧ ໆቚуࢂᚳεޑǴࣗԿຬၸ C ᇟق္ double ᡂኧ܌ૈ৒યޑനεॶǴӢԜךॺ ӧ೭٬ҔΑ Lyuu ک Wu ܌ቪޑፕЎ[6]္ޑБݤǴஒኧӷڗΑჹኧϐࡕӆीᆉǶ

ךॺ೛ޑୖኧॶࣁǺ

15 , 1 , 2 . 0 , 08 . 0 , 110 ,

95 ,

100 K H r T w

S V ^(days)

ځύ S ࣁ኱Ӧނޑ߃ۈሽ਱ǴKࣁቬऊሽ਱ǴHࣁምᛖሽ਱Ǵrࣁԃࡋค॥ᓀճ

౗ǴV ^{ࣁݢ୏ࡋǴ}Tࣁᒧ᏷៾Ӹុය໔(аԃٰᆉ)Ǵ w ࣁૈӧምᛖሽ਱΢നӭό

཮ᡣԜᒧ᏷៾ѨਏޑϺኧǶ

ךॺӃӧ CPU ΢୺ՉำԄǴځύයኧ n ॶךॺ೛ 100Ǵ200,…,1000Ǵ٠ीᆉ рਔ໔аϷ่݀ǶௗΠٰךॺඤԋҔ GPU ѐೀ౛ǴԶନΑයኧаѦǴךॺᗋ཮

೛ۓ GPU ୺Չᆣ(thread)ޑኧҞǴ၂კନΑ᛾ܴ GPU ޑਏૈѦᗋ࣮ૈցפрന

٫ϯޑБݤǶ

२ӃךॺӃᆉрٿᜐीᆉሽ਱ޑ่݀٠Ъک Li аϷ Zhao ӧдॺ܌ीᆉрޑ

่݀଺КၨǶ

යኧ Liک Zhao ޑ่݀ CPU ޑ่݀ GPUޑ่݀

100 1.1684 1.167366 1.167224

߄ 4-2. Кၨ Li ک Zhao аϷךॺӧ CPU ک GPU ΢ჴբीᆉೱុࠠЃᎿᒧ᏷៾

ޑሽ਱ޑ่݀Ƕ

යኧ Liک Zhao ޑ่݀ CPU ޑ่݀ GPUޑ่݀

100 1.0157 1.016322 1.015997

߄ 4-3. Кၨ Li ک Zhao аϷךॺӧ CPU ک GPU ΢ჴբीᆉಕᑈࠠЃᎿᒧ᏷៾

ޑሽ਱ޑ่݀Ƕ

ӧ೭ᜐךॺ࣮ډٿᜐ่݀ࢂৡόӭޑǴёૈޑᇤৡচӢϐ΋ࣁךॺڗჹኧࡕ

଺ၮᆉӆᙯӣѐਔӚ཮Ԗ٤ᇤৡǶќ΋চӢӧܭᗨฅ CUDA ౜Ϟς࿶ගٮᚈᆒ ዴੌᗺኧ(double)ၮᆉǴՠᗋࢂѸ໪࣮ฯᡏૈЍජޑीᆉૈΚހҁ(Compute Capability)ǴႽךॺӧ೭ጇፕЎ܌٬Ҕޑ Nvidia GeForce 8400M GS ೭஭ᡉҢь൩ ѝૈЍජډ 1.1ǴΨӢԜคݤҔᚈᆒዴੌᗺኧٰᆉǴԶѸ໪ҔൂᆒዴੌᗺኧǴӢ Ԝ཮Ԗ٤ᇤৡǶ

ௗΠٰךॺӧ GPU ΢೛ۓόӕኧҞޑ୺ՉᆣаϷόӕයኧٰीᆉ୺Չਔ ໔Ƕӧ೭ᜐךॺाݙཀޑࢂǴҗܭӧ Li ک Zhao ޑ่݀ύǴයኧࣣࢂ 100 ޑ७ኧǴ ӢԜךॺஒ thread Ψ൩ࢂ୺ՉᆣኧҞ೛ࣁ 100ǴԶӢࣁฯᡏज़ڋୢᚒǴӧ೭஭ᡉ Ңь΢ךॺനӭૈ೛ۓޑ୺ՉᆣኧҞࣁ 128Ƕ

྽୺ՉᆣኧҞࣁ 1 ਔǺ

යኧ CPUೀ౛ਔ໔(ms) GPUೀ౛ਔ໔(ms)

10 0.030171 0.046863

20 0.108813 0.400006

50 0.850527 12.498731

100 6.545664 73.193116

200 40.502705 396.019381

߄ 4-4. ӧ ୺Չᆣࣁ 1 ਔǴCPU ک GPU ΢ीᆉೱុࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ໔ КၨǶ

යኧ CPUೀ౛ਔ໔(ms) GPUೀ౛ਔ໔(ms)

10 0.019905 0.024478

20 0.025283 0.352692

50 0.090444 1.103192

100 0.433924 6.312448

200 2.361334 40.017239

߄ 4-5. ӧ ୺Չᆣࣁ 1 ਔǴCPU ک GPU ΢ीᆉಕᑈࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ໔ КၨǶ

ӧ೭ᜐךॺёа࣮рǴ྽୺ՉᆣኧҞࣁ 1 ਔǴGPU ӧೀ౛ԜᄽᆉݤБय़ޑ ਏ౗όӵ CPUǴЀځࢂ྽යኧຫεਔຫܴᡉǴаಕᑈࠠЃᎿᒧ᏷៾ࣁٯǴӵ߄ 4-5Ǵ྽යኧࣁ 10 ਔǴGPU ೀ౛܌޸ޑਔ໔ࣁ CPU ϐ 1.55 ७ǴډΑයኧࣁ 20 ਔ߾ࣁ 3.67 ७Ǵᒿ๱යኧቚу܌޸຤ޑਔ໔Ψа׳ଯޑ७౗ቚуǴ྽යኧࣁ 200 ਔǴGPU ܌޸ޑਔ໔ς࿶ৡόӭࢂ CPU ޑ 9.77 ७Ǵ࣬྽ܭਏ౗ࣁ 0.102 ७Ǵ೭

ࢂҗܭךॺؒԖҔډ GPU നεޑфૈΨ൩ࢂѳՉϯ܌ठǶ

྽୺ՉᆣኧҞࣁ 100 ਔǺ

යኧ CPUೀ౛ਔ໔(ms) GPUೀ౛ਔ໔(ms)

10 0.044140 0.035817

100 5.995385 2.331474

200 47.229465 8.078661

500 648.958618 44.038397

1000 7926.140625 113.346356

߄ 4-6. ӧ ୺Չᆣࣁ 100 ਔǴCPU ک GPU ΢ीᆉೱុࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ ໔КၨǶ

යኧ CPUೀ౛ਔ໔(ms) GPUೀ౛ਔ໔(ms)

10 0.013130 0.011041

100 0.423657 0.174266

200 2.388921 0.493527

500 33.134659 2.848529

1000 424.776459 7.332075

߄ 4-7. ӧ ୺Չᆣࣁ 100 ਔǴCPU ک GPU ΢ीᆉಕᑈࠠЃᎿᒧ᏷៾ޑሽ਱ޑਔ ໔КၨǶ

ௗ๱ךॺஒ୺ՉᆣኧҞ೛ࣁ 100Ǵҗ߄ 4-6Ǵ4-7 ૈ࣮рࡐܴᡉӦӧӭΑѳՉ ϯϐࡕޑׯ๓ǶаೱុࠠЃᎿᒧ᏷៾ٰᇥǴӧයኧࣁ 10 ਔǴGPU ᗨฅೲࡋၨז ՠόܴᡉǴ೭ࢂӢࣁ྽යኧϼλਔǴ଑୮ाख़ᙟ୺ՉޑԛኧၨϿǴԶ CPU ਡЈ ޑଯೀ౛ೲࡋ෧ϿΑӢࣁቚуೀ౛ԛኧ܌аӭ޸຤ޑਔ໔Ǵу΢ GPU ᗋाೀ౛

ၗ਑໺ሀୢᚒǴӢԜׯ๓ਏ੻όܴᡉǴԶӧයኧቚуډ 100 ࡕǴ߾໒ۈԖၨӳޑ ׯ๓ਏ੻Ǵς࿶Ԗ 2.571 ७ޑуೲǶӧයኧࣁ 500 ਔǴGPU ޑ୺Չೲࡋς࿶ࢂ

CPU୺Չೲࡋޑ 14.74 ७ǶԶයኧډ 1000 ࡕǴٿᜐ୺Չਔ໔ޑৡ౦ς࿶Ԗ 69 ७ ӭǴӧಕᑈࠠᒧ᏷៾Бय़ǴCPU ܌޸ޑਔ໔߾ࣁ GPU ܌޸຤ਔ໔ޑ 57 ७ϐӭǶ

ಃ

ಃϖക ่ፕ

ЃᎿᒧ᏷៾ຑሽࢂόϿ࿶ᔮᏢৎ܈ኧᏢᏢৎࣴزޑᒧ᏷៾ǴԶ Li ک Zhao ܌ ගрҔғԋڄኧٰीᆉЃᎿᒧ᏷៾ΞගٮΑ΋ঁှ،БݤǴฅԶǴӧԜᄽᆉݤ ύǴाीᆉрᒧ᏷៾ሽ਱܌ሡाޑਔ໔ໆӧයኧቚуਔΨ཮ε൯ӦቚуǴӢԜǴ ךॺߡགྷճҔ GPU ޑѳՉϯૈΚٰቚ຾ਏૈǶ

җჴᡍ่࣮݀рǴ྽ GPU ୺ՉᆣޑኧҞࣁ 1 ਔǴGPU ୺Չޑᕉნک CPU

΋ኬǶՠ GPU ୺Չೲࡋၨ CPU ᄌǴջ٬ ԜЃᎿᒧ᏷៾ीᆉၸำ٠όፄᚇԶ GPU

ೀ౛ੌᗺኧၮᆉޑਏૈΞК CPU ӳǴ྽යኧቚуਔǴGPU ӧೀ౛ޑਔ໔΢ϝฅ

ࢂӭܭ CPUǶ

Զ྽ךॺ೛ۓ GPU ୺ՉᆣޑኧҞࣁ 100 ਔǴࡐܴᡉૈ࣮рᏱԖѳՉϯೀ౛

ࡕǴGPU ޑਏૈᓬܭѝԖൂ΋୺Չᆣޑ CPUǴՠӢࣁךॺӧҬ๏ GPU ୺ՉਔǴ ᗋाೀ౛ၗ਑໺ሀǴ૶Ꮻᡏ฻ୢᚒԶ੃઻΋٤ਔ໔Ƕ܌а྽යኧࡐλਔǴGPU ѳՉϯޑᓬ༈٠όܴᡉǶՠ฻ډයኧε൯ቚуਔǴךॺёа࣮ډ GPU ୺Չ܌޸

ޑਔ໔ቚуޑόӭǴϸԶࢂ CPU ୺Չਔ໔ቚу൯ࡋᡂεǶ೭ࢂҗܭ྽ךॺஒය ኧॶቚεਔǴ୺ՉᆣኧҞࣁ 100 ޑ၉؂ঁ୺Չᆣѝाӭາ൳ԛǴԶൂ΋୺Չᆣ߾

ाೀ౛܌Ԗޑीᆉၸำ܌ԿǶ

ӢԜǴջ٬౜Ϟӧ٬Ҕ CUDA ΢ϝԖ΋٤લᗺǴКӵᇥ೏ฯᡏ(ᡉҢь)܌ज़ ڋՐǴаϷคݤೀ౛ሀ଑฻Ǵךॺϝฅૈ࣮р GPU ӧೀ౛ᜪ՟ୢᚒޑᓬ༈ǴԶ ନΑӧҁጇᄽᆉݤаѦǴGPU ӧೀ౛ᆾӦьᛥኳᔕǴ܈ࢂ΋٤ёаஒᏱԖᚳε ኧໆޑीᆉ೽ҽܨှԋ೚ӭ۶Ԝ࣬ᜢ܄όεޑीᆉ೽ҽୢᚒ฻Бय़ΨࢂԖ๱࣬

྽ޑਏ੻Ƕϐࡕᒿ๱ฯᡏמೌޑ຾؁Ǵךॺӧೀ౛εໆၗ਑ޑਏ౗ׯ๓཮׳уܴ

ᡉǶ

ୖ

ୖԵЎ᝘

[1] Bing-Qing Li , Hai-Jian Zhao, Pricing Parisian Options by Generating Functions, The Journal of Derivatives, 2009, 72–81.

[2] H. S. Wilf, Generatingfunctionology, Academic Press, 2^nd edition, 1994.

[3] Costabile, M., A combinatorial approach for pricing Parisian options, Decisions in Economics and Finance, 2002, 25(2), 111–125.

[4] Hull, J.C., Options, Futures, and Other Derivatives, 6th edition, Upper Saddle River, NJ: Prentice-Hall, 2006.

[5] Yuh -Dauh Lyuu and Yi-Chun Wu, Performance of GPU for a Tree Model for Convertible Bonds Pricing with Stock Price, Interest Rate, and Default Risks, 2008.

[6] Yuh -Dauh Lyuu and Cheng-Wei Wu, Pricing Parisian Options: Combinatorics, Simulation, and Parallel Processing, 2008.

[7] Yuh -Dauh Lyuu and Cheng-Wei Wu, An Improved Combinatorial Approach for Pricing Parisian Options, Decisions in Economics and Finance, 33(2010) , 49–61.

[8] NVIDIA Corporation, NVIDIA_CUDA_Programming_Guide_2.2.1, 2009.

[9] Heresy’ Space, http://heresy.spaces.live.com/blog/.

[10] NVIDIA GPUਡЈᐕўӣ៝৖, http://www.fevernet.com/thread-6501-1-1.html.

[11] CUDA ZONE, http://www.nvidia.com.tw/object/cuda_home_new_tw.html.

[12] CUDA Wiki, http://zh.wikipedia.org/zh-tw/CUDA.

[13] ഋЎሱ, ᒧ᏷៾Β໨Ԅुሽ.

[14] ู໡Ⴀ, ЃᎿᒧ᏷៾ϟಏ, ᝊٰߎᑼബཥۑтಃΒΜϖය^{, 2003.}