隨機波動度Heath-Jarrow-Morton模型下之利率衍生性商品評價

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(1)୯ҥᆵ᡼εᏢᆅ౛Ꮲଣ଄୍ߎᑼᏢࣴ‫܌ز‬ ᅺγፕЎ Graduate Institute of Finance College of Management. National Taiwan University Master Thesis. ᒿᐒ‫ ࡋ୏ݢ‬Heath-Jarrow-Morton ኳࠠΠ ϐճ౗़ғ‫܄‬୘ࠔຑሽ Pricing Interest Rate Derivatives in Heath-Jarrow-Morton Model with Stochastic Volatility. ᛿ҥཧ Lap Fai, Tam ࡰᏤ௲௤Ǻ‫׵‬፣ྍ റγ! ҡԭၲ റγ Advisor: Shyan Yuan, Lee, Ph.D., Pai Ta, Shih, Ph.D. ύ๮҇୯ΐΜΐԃϤД June, 2010.

(2) ୯ҥᆵ᡼εᏢᅺγᏢՏፕЎ. α၂‫ہ‬঩཮ቩ‫ۓ‬ਜ ᒿᐒ‫ ࡋ୏ݢ‬Heath-Jarrow-Morton ኳࠠΠ ϐճ౗़ғ‫܄‬୘ࠔຑሽ Pricing Interest Rate Derivatives in Heath-Jarrow-Morton Model with Stochastic Volatility ҁፕЎ߯᛿ҥཧȐR97723058ȑӧ୯ҥᆵ᡼εᏢ଄୍ߎᑼ Ꮲࣴ‫ֹ܌ز‬ԋϐᅺγᏢՏፕЎǴ‫҇ܭ‬୯ΐΜΐԃϤДΜΎВ‫܍‬ ΠӈԵ၂‫ہ‬঩ቩࢗ೯ၸϷα၂Ϸ਱Ǵ੝Ԝ᛾ܴ. α၂‫ہ‬঩Ǻ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ȐᛝӜȑ ȐࡰᏤ௲௤ȑ. ‫س‬ЬҺǵ‫ߏ܌‬. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! !. ! ! ! ! ! ! ! ! ! ! ! ȐᛝӜȑ. i.

(3) ii.

(4) ᖴᜏ ಖ‫ܭ‬Ǵाቪ೭ፕЎ‫ޑ‬നࡕ೽ҽΑǶ २ӃǴགᐟ‫ך‬ৎΓ‫ޑ‬Ѝ࡭ᆶႴᓰǴவ௴ᆾԿϞΒΜӭԃǴ‫ޔ‬Կࣴ‫܌ز‬౥཰Ǵ ᡣ‫ך‬ค៝ቾӦӼЈֹԋᏢ཰Ƕ ‫ך‬ӕኬߚதགᖴ‫ٿ‬ՏࡰᏤ௲௤Ǵ‫׵‬፣ྍԴৣ‫ک‬ҡԭၲԴৣǶӧ‫׵‬ԴৣࡰᏤΠǴ ‫ך‬όѝள‫ډ‬ᙦ൤‫ޑ‬Ꮲೌ‫ک‬ჴ୍‫ޕ‬᛽ǴΨᏢ཮ΑࡐӭΓғεၰ౛ǶҡԴৣҭৣҭ϶ ‫ޑ‬௲ᏤǴόჇ‫ྠځ‬Ӧഉ‫૸ך‬ፕ‫ډ‬ఁ΢Ƕ‫ך‬Ψࡐགᖴ‫ٿ‬ՏԴৣӧ‫ך‬Вதғࢲύ‫ޑ‬ᔅ շǶΨߚதᖴᖴӧࣴ‫࠻ز‬ӕᏢॺ‫ޑ‬Ѝ࡭‫ک‬ႴᓰǴ੝ձࢂӕืঋ࠻϶ЎᡣǴ๏‫ך‬ό ϿፕЎ‫ࡌޑ‬᝼ǴΨததᔅ‫ך‬ວϱᓓǶ നࡕाᖴᖴ‫ޑך‬ζܻ϶ Amy ‫ޑ‬Ѝ࡭‫ک‬ऐЈǴӧ‫ך‬೭‫ٿ‬ԃࣴ‫܌ز‬ғࢲ္‫ޑ‬όᘐ Ⴔᓰ‫ک‬х৒Ǵջ٬ӧҬඤᏢғਔΨதத஌ܴߞТ๏‫ך‬Ƕ. ҥཧ Β႟΋႟ԃϤД Ѡч. iii.

(5) ᄔा ҁЎග‫ٮ‬Α΋ঁᡫࢲ‫ޑ‬ӭӢηᒿᐒ‫ ࡋ୏ݢ‬Heath–Jarrow–Morton ኳࠠǴԜኳࠠ ᡣᇻයճ౗ᆶ‫ڀࡋ୏ݢځ‬Ԗ࣬ᜢ‫܄‬ǴЪԖ N ঁᒿᐒӢη཮ቹៜճ౗่ᄬǴќԖᚐ Ѧ N ঁᒿᐒӢη཮ቹៜ‫(ࡋ୏ݢ‬Ϸճ౗़ғ‫܄‬୘ࠔ)ǶԜኳࠠ‫ׯ‬຾Α Trolle and Schwartz (2009)‫ޑ‬ኳࠠǴᡣճ౗‫ࡋ୏ݢ‬ᆶอයճ౗(short rate)НྗԖ҅Кᜢ߯ǶԜ ኳࠠૈ୼ᙯඤԋԖज़‫ރ‬ᄊᡂኧ(finite number of state variables)‫ޑ‬ଭёϻ߄౜ (Markov representation)‫س‬಍ǴࡺૈᇸܰӦ٬ҔᆾӦьᛥኳᔕ‫ٰݤ‬ຑሽӚᅿճ౗़ғ ‫܄‬ౢࠔǴӵճ౗΢ज़ᒧ᏷៾ǵճ౗Ҭඤᒧ᏷៾฻Ƕനࡕ‫཮ॺך‬ϩ‫݋‬Ӛୖኧჹຑሽ ่݀‫ޑ‬ቹៜǶ. ᜢᗖຒǺHeath–Jarrow–Morton ኳࠠǵᒿᐒ‫ࡋ୏ݢ‬ǵ‫ރ‬ᄊ٩ᒘ‫ࡋ୏ݢ‬ǵᆾӦьᛥኳ ᔕ‫ݤ‬ǵճ౗΢ज़ᒧ᏷៾. iv.

(6) Abstract This. article. provides. a. flexible. stochastic. volatility. multi-factor. Heath–Jarrow–Morton term structure model, which allows forward rate correlative with its volatility, and there are N random factors affect the interest rate structure, while additional N random factors would affect volatilities (and also interest rate derivatives). This model improves the Trolle and Schwartz (2009) model, so that interest rate volatility is proportional to the short rate. This model can be converted into a finite-state variables Markov representation system, so under this model, Monte Carlo simulation can be easily used to evaluate the various interest rate derivatives, such as interest rate cap, swaption, etc.. Finally, we will analyze the impact of various parameters on the pricing result.. Keywords: Heath–Jarrow–Morton model, Stochastic volatility, State dependent volatility, Monte Carlo simulation, Caps. v.

(7) Ҟ. ᒵ. α၂‫ہ‬঩཮ቩ‫ۓ‬ਜ ........................................................................................................... i ᖴᜏ ................................................................................................................................. iii ᄔा ................................................................................................................................. iv Abstract............................................................................................................................. v Ҟ. ᒵ ............................................................................................................................. vi. კҞᒵ ............................................................................................................................ vii ߄਱Ҟᒵ ........................................................................................................................ vii ΋ǵ. ᙁϟ .................................................................................................................. 1. 1.. Ў᝘ӣ៝ .......................................................................................................... 1. 2.. ࣴ‫୏ز‬ᐒ .......................................................................................................... 2. Βǵ. ኳࠠ೛‫ ۓ‬.......................................................................................................... 3. 1.. ᇻයճ౗ϐ୏ᄊၸำ ...................................................................................... 3. 2.. อයճ౗ϐ୏ᄊၸำ ...................................................................................... 6. 3.. ႟৲໸‫چ‬ϐ୏ᄊၸำ ...................................................................................... 7. 4.. ճ౗΢ज़ᒧ᏷៾ (Interest Rate Cap) .............................................................. 8. Οǵ. ኧॶ‫ٯ‬η .......................................................................................................... 9. 1.. ᆾӦьᛥኳᔕ‫ ݤ‬.............................................................................................. 9. 2.. ຑሽ႟৲໸‫چ‬፤៾ ........................................................................................ 10. 3.. ຑሽճ౗΢ज़ᒧ᏷៾ .................................................................................... 12. Ѥǵ. ‫׳‬΋૓ϯ‫ޑ‬ኳࠠ ............................................................................................ 14. ϖǵ. ᕴ่ ................................................................................................................ 15. ߕᒵ ................................................................................................................................ 16 A. ‫ۓ‬౛ 1.ϐ᛾ܴ ................................................................................................ 16 B. ‫ۓ‬౛ 2.ϐ᛾ܴ ................................................................................................ 17. vi.

(8) C. AJD ‫ ک‬LQJD చҹ ........................................................................................ 18 ୖԵЎ᝘ ........................................................................................................................ 21. კҞᒵ კ 1. Ꭺঢ়‫ ࡋ୏ݢ׎‬.................................................................................................. 3. კ 2. V i ᡂ୏ჹ႟৲໸‫چ‬፤៾ሽ਱‫ޑ‬ቹៜ .............................................................11. კ 3. ኳᔕ 10000 ԛࡕǴ P TC , T

(9) ပӧόӕ୔ୱ‫ޔޑ‬Бკ................................... 12. კ 4. ᇻයճ౗Ԕጕ΢ΠѳՉ౽୏ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ .................... 13. კ 5. U ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ ................................................................ 14. ߄਱Ҟᒵ ߄਱ 1. V i ᡂ୏ჹ႟৲໸‫چ‬፤៾ሽ਱‫ޑ‬ቹៜ .........................................................11. ߄਱ 2. ᇻයճ౗Ԕጕ΢ΠѳՉ౽୏ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ ................ 13. ߄਱ 3. U ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ ............................................................ 14. vii.

(10) ΋ǵ ᙁϟ 1.. Ў᝘ӣ៝. Heath, Jarrow and Morton (1992) (ᙁᆀ HJM)௢рࡕǴჹճ౗ኳࠠࣴ‫ز‬Ԗߚதε ‫ޑ‬ଅ᝘Ƕಃ΋ǴѬࢂ΋ঁᇻයճ౗Ԕጕኳࠠ(forward rate curve model)Ǵճ౗Ԕጕ཮ ‫ډڙ‬ค঺ճచҹ(no arbitrage condition)‫܌‬ज़‫ڋ‬ǹಃΒǴѬૈ୼ࡐ৒ܰӦᘉк‫ډ‬ӭӢ ηኳࠠǴٰਂਆόӕ‫ޑ‬ѱ൑੝‫܄‬ǹಃΟǴѬૈ୼఼ᇂε೽ҽ‫ޑ‬໺಍ኳࠠǴӵ Vasicek (1977)ǵCox-Ingersoll-Ross (1985)ǵHo and Lee (1986) ‫ ک‬Hull and White (1990)฻೿ ࢂ HJM ‫ޑ‬η໣ӝǶ HJM ኳࠠӧค঺ճచҹ(no arbitrage condition)ΠǴᇻයճ౗‫ޑ‬ᅆੌ໨(drift term) ࢂ೏‫(ࡋ୏ݢ‬volatility term)୤΋،‫ޑۓ‬Ƕ‫܌‬а‫ޑࡋ୏ݢ‬೛‫ۓ‬ǴԋࣁΑ HJM ኳࠠനख़ ा‫زࣴޑ‬ჹຝǶ߈ԃόϿࣴ‫֡ز‬ଞჹ HJM ‫ࡋ୏ݢޑ‬բ௖૸Ǵ‫׆‬ఈૈਂਆѱ൑ჴ‫ݩ‬ ‫ޑ‬ၗૻǴ٠ᅰёૈள‫ډ‬ຑሽ़ғ‫܄‬୘ࠔ‫࠾ޑ‬ഈှǶ‫ٯ‬ӵ Bhar and Chiarella (1997) ԵቾΑ‫ࡋ୏ݢ‬ᆶอයճ౗ԋ҅Кᜢ߯ǴMercurio and Moraleda (2000)‫(ک‬2001)Եቾ Α‫ޑࡋ୏ݢ‬Ꭺঢ়౜ຝǴCollin-Dufresne and Goldstein (2003)‫ ک‬Trolle and Schwartz (2009)Եቾᒿᐒ‫ࡋ୏ݢ‬ǴճҔ Duffie, Pan and Singleton (2000)‫ ޑ‬Affine Jump-Diffusions ளрຑሽճ౗୘ࠔ‫࠾ޑ‬ഈှǶ ќѦǴCollin-Dufresne and Goldstein (2002)ǵHeidari and Wu (2003)‫ک‬Li and Zhao (2006)ว౜ճ౗़ғ‫܄‬୘ࠔ‫ࢌڙ‬٤ᒿᐒӢηቹៜǴՠ೭٤Ӣη٠όቹៜճ౗ ය໔่ᄬǴ‫܌‬аᆀѬࣁߚճ౗යज़่ᄬᒿᐒӢη(unspanned stochastic volatility factor; USV factor)ǶAndersen and Benzoni (2008)ҭว౜੿ჴճ౗‫ڙ཮ࡋ୏ݢ‬USVӢ ηቹៜǶ. 1.

(11) 2.. ࣴ‫୏ز‬ᐒ. ന߈‫ޑ‬Ў᝘ගрόϿᜢ‫ܭ‬ճ౗‫ޑࡋ୏ݢ‬੝‫܄‬Ǻಃ΋Ǵճ౗‫ޑࡋ୏ݢ‬ዴࢂᒿᐒ ‫ޑ‬ǶಃΒǴճ౗‫֖ࡋ୏ݢ‬Ԗख़ा‫ߚޑ‬ճ౗යज़่ᄬၗૻ1ǶಃΟǴอයճ౗‫ک‬Ѭ‫ޑ‬ ‫ڀࡋ୏ݢ‬Ԗॄ࣬ᜢ‫ޑ‬౜ຝǶಃѤǴߚచҹ(unconditional)‫่ࡋ୏ݢ‬ᄬ‫ڀ‬ԖᎪঢ়‫׎‬ (hump-shaped)౜ຝǶ ଞჹಃΟᗺǴAndersen and Lund (1997)‫ ک‬Ball and Torous (1999)ᘉкЪᡍ᛾ Chan et al. (1992)‫ޑ‬อයճ౗ኳࠠ. dr t

(12) N1 P1 r t

(13)

(14) dt X t

(15) r t

(16) dW1 t

(17). (1). d log X t

(18) N 2 P 2 log X t

(19)

(20) dt V dW2 t

(21). (2). J. W1 t

(22) ‫ ک‬W2 t

(23) ᐱҥǶдॺว౜࣬ჹճ౗‫کࡋ୏ݢ‬ճ౗և౜ॄ࣬ᜢ‫ޑ‬ᜢ߯ǹ๊ჹճ ౗‫ک୏ݢ‬ճ౗և౜҅࣬ᜢ‫ޑ‬ᜢ߯ǶԶЪǴAndersen and Lund (1997)՗ीр J Զ Ball and Torous (1999)՗ीр J. 0.544Ǵ. 0.754 ֡ᆶ Chan et al. (1992)‫ޑ‬ჴ᛾่݀࣬಄Ƕ. Trolle and Schwartz (2009)ගр΋ঁᡫࢲ‫ޑ‬ӭӢη HJM ኳࠠǴЪߚచҹ‫ࡋ୏ݢ‬ ԖᎪঢ়౜ຝǴ٠‫ ڙ‬USV ӢηቹៜǺ. df t , T

(24). N. P f t , T

(25) dt ¦ V f ,i t , T

(26) Xi t

(27) dWi Q t

(28). (3). i 1. dXi t

(29) N i Ti Xi t

(30)

(31) dt V i Xi t

(32) Ui dWi Q t

(33) 1 Ui 2 dZi Q t

(34) i 1,! , N Ǵ‫ځ‬ύ V f ,i t , T

(35). D. 0,i.

(36). (4). D1,i T t

(37)

(38) e J i T t

(39) Ǵ dWi Q t

(40) ‫ ک‬dZi Q t

(41) ࢂᐱҥ‫ޑ‬. ኱ྗѲਟၮ୏ǶԜ೛‫ۓ‬಄ӝ(i) ‫ࢂࡋ୏ݢ‬ᒿᐒ‫ޑ‬ǹ(ii) ‫ڀࡋ୏ݢ‬Ԗ USV Ӣηǹ(iii) ߚ చҹ‫ࡋ୏ݢ‬ԖᎪঢ়౜ຝ(ӵკ 1ǴᒿᐒӢηჹόӕϺය‫ޑ‬ᇻයճ౗ቹៜำࡋόӕǴ อϺයϷߏϺය‫ڙ‬ቹៜၨλǴԶύය‫ڙ‬ቹៜၨε)ǶԶЪૈճҔ Bhar and Chiarella (1997)‫ޑ‬ᙯඤǴᡣԜ୏ᄊၸำ಄ӝ Duffie, Pan and Singleton (2000) (ᙁᆀ DPS)ගр 1. ၁‫ ـ‬Collin-Dufresne and Goldstein (2002)ǵHeidari and Wu (2003)ǵCasassus, Collin-Dufresne,. and Goldstein (2005)‫ ک‬Li and Zhao (2006)Ƕ. 2.

(42) ‫ ޑ‬Affine Jump-Diffusions (ᙁᆀ AJD)‫ޑ‬చҹǴૈᕇள໸‫چ‬ᒧ᏷៾ຑሽϦԄ‫࠾ޑ‬ഈှ (closed form solution)Ƕҗ‫ࣁܭ‬Αा಄ӝ AJDǴ‫܌‬а Trolle and Schwartz (2009)‫ܫ‬క Α‫ࡋ୏ݢ‬ᆶ r(t)‫ޑ‬ሀቚᜢ߯ǶࡺҁЎଞჹԜୢᚒբ‫ؼׯ‬Ǵз‫ࡋ୏ݢ‬. V. V t , T ,X t

(43) , r t

(44)

(45) Ǵᡣ‫ࡋ୏ݢ‬ӕਔ‫ ڙ‬USV Ӣη‫ک‬྽ਔ‫ޑ‬ճ౗Нྗ r(t)‫ޔޑ‬ௗቹ. ៜǴ‫ॺך‬ᆀѬࣁ‫ރ‬ ‫ރ‬ᄊ٩ᒘ‫(ࡋ୏ݢ‬state dependent volatility)Ƕ ‫ࡋ୏ݢ‬. T t კ 1. Ꭺঢ়‫ࡋ୏ݢ׎‬. ҁЎኳү Trolle and Schwartz (2009)‫ޑ‬೛‫ۓ‬Ǵ௖૸‫׳‬΋૓ϯ HJM ճ౗ኳࠠϐ੝ ‫܄‬ǴаϷຑሽ่݀Ƕ ҁЎ‫ࢎޑ‬ᄬӵΠǺಃ΋࿯ࣁᙁϟǹಃΒ࿯ඔॊ‫ޑॺך‬ᒿᐒ‫ࡋ୏ݢ‬ᇻයճ౗ኳ ࠠǴ٠௢ᏤрӚᡂኧ‫୏ޑ‬ᄊၸำ(diffusion)ǹಃΟ࿯ஒ৖Ңኳࠠ‫ޑ‬ኧॶ่݀ǹಃѤ ࿯཮ϟಏ‫׳‬΋૓ϯ‫ޑ‬ኳࠠǴ٠ᆶ‫ځ‬дኳࠠբКၨǹനࡕ΋࿯ࢂ่ፕǶ. Βǵ ኳࠠ೛‫ۓ‬ 1.. ᇻයճ౗ϐ୏ᄊၸำ. ‫ۓॺך‬က f t , T

(46) ࣁӧ t ਔᗺᢀჸǴT ਔᗺॷ(‫܈‬ສ)‫ޑ‬ᕓ໔ᇻයճ౗(instantaneous forward interest rate)Ƕ‫୏ځ‬ᄊၸำ೛‫ۓ‬ӵΠǺ df t , T

(47). N. P f t , T

(48) dt ¦ V f ,i t , T

(49) Xi t

(50) r t

(51) dWi Q t

(52) i 1. 3. (5).

(53) . dXi t

(54) N i Ti Xi t

(55)

(56) dt V i Xi t

(57) Ui dWi Q t

(58) 1 Ui 2 dZ i Q t

(59).

(60). (6). i 1,! , N Ǵ‫ځ‬ύ dWi Q t

(61) ‫ ک‬dZi Q t

(62) ࢂӧ॥ᓀύҥෳࡋ(risk neutral measure)Q Π‫ޑ‬. ᐱҥ኱ྗѲਟၮ୏ǹ r t

(63). f t , t

(64) ࢂᕓ໔อයճ౗(short rate ‫ ܈‬instantaneous spot. ᒿᐒ‫୏ݢ‬Ӣη2Ƕ‫ځ‬ᓬᗺ rate)ǶԜኳࠠࢂ‫ڀ‬ᒿᐒ‫ ޑࡋ୏ݢ‬HJM ኳࠠǴ‫ॺך‬ᆀ Xi t

(65) ࣁᒿ ӵΠǺ(i) ‫ڀ‬Ԗᒿᐒ‫ࡋ୏ݢ‬ǹ(ii) N ঁӢη(ճ౗่ᄬӢη dWi Q t

(66) )཮ቹៜճ౗่ᄬǴ ќԖᚐѦ N ঁӢη(ߚቹៜճ౗යज़่ᄬᒿᐒ‫ࡋ୏ݢ‬Ӣη dZi Q t

(67) )཮ቹៜ‫ࡺ(ࡋ୏ݢ‬ ҭቹៜճ౗़ғ‫܄‬୘ࠔ)ǹ(iii) ᇻයճ౗‫ߚޑ‬చҹ‫ࡋ୏ݢ‬ᆶ྽යճ౗Нྗ r t

(68) ԋ҅ КǺ(iv) Ϣ೚ f t , T

(69) ᆶ Xi t

(70) ‫ڀ‬Ԗ࣬ᜢ‫܄‬Ƕ ਥᏵ HJM (1992)Ǵӧค঺ճచҹΠǴ(1)ύ‫ޑ‬ᅆੌ໨ሡ಄ӝǺ. P f t,T

(71). N. ¦X t

(72) r t

(73) V t , T

(74) ³ i. f ,i. i 1. T. t. V f ,i t , u

(75) du. (7). ӧ΋૓௃‫ݩ‬ΠǴf(t,T)‫ ک‬r(t)٠ό཮ࢂଭёϻ(Markovian)ၸำǴόᆅӧ‫؃‬ှ‫ࢂ܈‬ ኳᔕ‫ޑ‬ၸำ္೿ࢂ΋εምᛖǶ‫܌‬аόϿЎ᝘ଞჹ HJM ‫ޑ‬ଭёϻ‫܄‬፦բࣴ‫ز‬Ǵ Carverhill (1994)‫ ک‬Jeffrey (1995)ග‫ٮ‬ΑкҽѸाచҹǴRitchken and Sankarasubramanian (1995)ǵBhar and Chiarella (1997)ǵBhar, Chiarella, El-Hassan and Zheng (2000)ǵChiarella and Kwon (2001)ǵBjörk and Svensson (2001)‫ ک‬Björk, Landén and Svensson (2004)֡ჹ HJM ‫ޑ‬Ԗज़ᆢଭёϻၸำᙯඤբ௖૸ǶBhar and Chiarella (1997)ࡰрǴऩ‫ࡋ୏ݢ‬ය໔่ᄬև V f ,i t , T

(76). pn T t

(77) e J i T t

(78) ( pn W

(79) ж߄ n ԛӭ໨. Ԅ)Ǵ߾ HJM ኳࠠૈᙯඤԋԖज़‫ރ‬ᄊᡂኧ(finite number of state variables)‫سޑ‬಍Ǵ Ъ‫่ࡋ୏ݢ‬ᄬԖਔ໔ሸ΋‫(܄‬time-homogeneous)‫܄ޑ‬፦Ƕऩ n 2 Ǵ߾. V f ,i t , T

(80) 2. D. 0i. D1i T t

(81)

(82) e J i T t

(83) Ǵ. (8). Xi t

(84) ࢂ Cox-Ingersoll-Ross ၸำǴ‫ڀ‬Ԗѳ֡ኧൺᘜ(mean reverting)੝‫܄‬Ǵ၁‫ ـ‬Heston (1993). ‫ިޑ‬ሽᒿᐒ‫ࡋ୏ݢ‬ኳࠠǶ. 4.

(85) D1i ! J i Ǵ߾‫཮ࡋ୏ݢ‬և౜Ꭺঢ়‫่׎‬ᄬǶௗΠٰ‫ॺך‬ஒ٬Ҕ(8)Ԅբࣁ(5)Ԅύ‫ޑ‬ D 0i. Ъ. V f ,i t , T

(86) Ƕ ਥᏵ(5)ǵ(7)‫(ک‬8)Ǵ f t , T

(87) ё߄ҢԋǺ. ‫ۓ‬౛ 1.. f t,T

(88). N. N. i 1. i 1 j 1. 6. f 0, T

(89) ¦ E x T t

(90) xi t

(91) ¦¦ EI j ,i T t

(92) I j ,i (t ). (9). ‫ځ‬ύ. E x W

(93) i. D. 0i.

(94). +D1iW e J iW e J iW. (10). EI W

(95) D1i e J W i. (11). 1,i. EI. 2,i ( t ). W

(96) =. D1i § 1 D 0i · J W ¨ ¸ D +D W

(97) e J i © J i D1i ¹ 0i 1i. ªD D EI3,i (t ) W

(98) = « 0i 1i «¬ J i. EI W

(99) = 4,i. EI. 5,i ( t ). EI. (12). i. § 1 D 0i · D1i ¨ ¸+ © J i D1i ¹ J i. § D1i · D12i 2 º 2J iW +2D 0i ¸W W » e ¨ J i »¼ © Ji ¹. D12i § 1 D 0i · J W ¨ ¸e J i © J i D1i ¹. (W )= 6 ,i ( t ). (14). i. (W )= . · D1i § D1i +2D1iW ¸ e 2J W ¨ 2D 0i Ji © Ji ¹. (15). i. D12i. Ji. (13). e 2J iW. (16). ‫ރ‬ᄊᡂኧࣁǺ. J i xi (t ) dt Xi (t ) r t

(100) dWi Q t

(101). (17). d I1,i (t ). x (t ) J I. (18). d I2,i (t ). X (t )r t

(102) J I. d I3,i (t ). X (t )r t

(103) 2J I. d I4,i (t ). I. dxi (t ). i. i 1,i. i. i 2,i. i. 2,i. (t )

(104) dt (t )

(105) dt. i 3,i. (t )

(106) dt. (t ) J iI4,i (t )

(107) dt. 5. (19) (20) (21).

(108) d I5,i (t ). I. dI6,i (t ). 2I. 3,i. (t ) 2J iI5,i (t )

(109) dt. 5,i. (t ) 2J iI6,i (t )

(110) dt. (22) (23). ଆ‫ۈ‬చҹࣁ xi (0) I1,i (0) I2,i (0) I3,i (0) I4,i (0) I5,i (0) I6,i (0) 0 Ƕ ᛾ܴǺୖ‫ߕـ‬ᒵ AǶ வ‫ۓ‬౛ 1.ё‫ـ‬Ǵቚу 7 u N ঁ‫ރ‬ᄊᡂኧǴ٬ኳࠠૈ߄Ңԋଭёϻ߄౜(Markov. representation)Ƕ‫ځ‬ύ I1,i t

(111) ,!, I6,i t

(112) ࢂֽ೽ዴ‫(ޑۓ‬locally deterministic)ǴѬॺॄೢ ԏ໣ xi (t ) ‫ ک‬Xi (t ) ‫ޑ‬ᐕўၗૻǴ‫܌‬а‫ॺך‬ᆀࣁȸᇶշ‫ޑ‬ȹ‫ރ‬ᄊᡂኧǶӢࣁኳࠠԖଭ ёϻ‫܄‬፦Ǵ‫܌‬а‫ॺך‬ӧ٬ҔᆾӦьᛥኳᔕ‫(ݤ‬Monte Carlo simulation method)ѐຑሽ ፄᚇ‫ޑ‬ճ౗୘ࠔਔ཮Кၨ৒ܰǶ. 2.. อයճ౗ϐ୏ᄊၸำ. ӧ(5)ԄύǴอයճ౗ r t

(113) ࢂ f t , T

(114) ‫ځޑ‬ύ΋ঁ‫ރ‬ᄊᡂኧǴ‫܌‬а‫ॺך‬௢Ꮴр r t

(115) ‫୏ޑ‬ᄊၸำǺ ‫ۓ‬౛ 2.. t ਔᗺϐอයճ౗ r t

(116) ୏ᄊၸำࣁǺ N § 6 ªw ·º dr t

(117) « f 0, t

(118) ¦ ¨ J i f 0, t

(119) J i r t

(120) D1i x t

(121) ¦ D j ,iI j ,i t

(122) ¸ » dt i 1© j 1 «¬ wt ¹ »¼ N. ¦ D 0i Xi t

(123) r t

(124) dWi. Q. (24). t

(125). i 1. ‫ځ‬ύ D1,i. D2,i D3,i. 0. (25). D1i 2 D 0iD1iJ i J i2 D 0i 2J i 2 D1i 2 D 0iD1iJ i J i2. (26) (27). 6.

(126) D 0iD1iJ i 2 D1i 2J i D1i 2 D 0iD1iJ i J i2 D 2 2D 0iD1i 1i Ji. D4,i D5,i. (28) (29). D1i 2. D6,i. (30). ᛾ܴǺୖ‫ߕـ‬ᒵ BǶ ӧ(24)ԄύǴ‫ॺך‬ว౜ r t

(127) ‫ڀ‬Ԗѳ֡ኧൺᘜ‫ޑ‬੝‫܄‬Ǵᅆੌ໨཮ԖΚໆᡣ r t

(128) ӣ ‫ډ‬ѳ֡ॶǶ೭္‫ޑ‬ѳ֡ॶࢂঅ҅ࡕ‫ޑ‬ᇻයճ౗Ǵ཮ᒿ๱ਔ໔Զ‫ׯ‬ᡂǶќѦǴอය ճ౗‫ࡋ୏ݢ‬Ψ཮ᆶ྽ය‫ޑ‬ճ౗Нྗԋ҅КǶԜ‫ٿ‬੝‫܄‬೿ᜪ՟‫ ܭ‬Cox-Ingersoll-Ross. (1985)Ƕ. ႟৲໸‫چ‬ϐ୏ᄊၸำ. 3.. ‫ۓ‬က P t , T

(129) ࣁ t ਔᗺᢀჸǴT ਔᗺ‫ډ‬යϐ႟৲໸‫چ‬Ǵ߾. ^. `. P t , T

(130) { exp ³ f t , u

(131) du T. t. P 0, T

(132). N 6 N ½ exp ®¦ % xi T t

(133) xi t

(134) +¦¦ %I j ,i T t

(135) I j ,i t

(136) ¾ P 0, t

(137) i 1 j 1 ¯i 1 ¿. (31). ‫ځ‬ύ % xi W

(138). · D1i § § 1 D 0i · J W J W ¨¨ ¨ ¸ e 1

(139) W e ¸¸ J i © © J i D1i ¹ ¹. (32). %I1,i W

(140). D1i J W e 1

(141) Ji. (33). i. i. i. 2. § D · § 1 D ·§§ 1 D · %I2 ,i W

(142) = ¨ 1i ¸ ¨ 0 i ¸ ¨ ¨ 0i ¸ e J iW 1 W e J iW ¨ ¸¨ © J i ¹ © J i D1,i ¹ © © J i D1i ¹ %I3,i W

(143). .

(144). · ¸¸ ¹. (34). · §D · D1i § § D1i D 0i D 02i · 2J W D 1

(145) ¨ 1i D 0i ¸W e 2J W 1i W 2 e 2J W ¸ e. ¨ ¸ 2 ¨ 2 ¸ J i ¨© © 2J i J i 2D1i ¹ 2 © Ji ¹ ¹ i. 2. i. §D · § 1 D · %I4 ,i W

(146) = ¨ 1i ¸ ¨ 0i ¸ e J iW 1 © J i ¹ © J i D1i ¹.

(147). i. (35). (36). 7.

(148) %I5,i W

(149) = . · · D1i § § D1i D 0i ¸ e 2J W 1

(150) D1iW e 2J W ¸ ¨ 2 ¨ ¸ J i ¨© © J i ¹ ¹ i. 2. i. 1 §D · %I6,i W

(151) = ¨ 1i ¸ e 2J iW 1 2 © Ji ¹.

(152). (37). (38). Ҕ Itô Lemma ৖໒Ǵ߾ P t , T

(153) ‫୏ޑ‬ᄊၸำࣁ dP t , T

(154) P t, T

(155). 4.. N. r t

(156) dt ¦ % xi T t

(157) Xi t

(158) r t

(159) dWi Q t

(160) Ƕ. (39). i 1. ճ౗΢ज़ᒧ᏷៾ (Interest Rate Cap). ճ౗΢ज़ᒧ᏷៾ࢂҗ΋‫س‬ӈόӕ‫ډ‬යВ‫ޑ‬ճ ճ౗ວ៾(caplets)ಔӝԶԋǴΠय़ஒ ཮ϟಏճ౗ວ៾ϐຑሽБ‫ݤ‬Ƕ ӧ॥ᓀύҥෳࡋΠǴԵቾ΋ճ౗ວ៾Ǵа 3 ঁДϐ LIBOR բࣁୖԵճ౗Ǵൂ ճीᆉǴӜҞҁߎ 1 ϡǴt ࢂວ៾ᛝऊਔᗺǴғਏය໔ࣁ t0 ‫ ډ‬t1Ǵճ౗΢ज़ࣁ xǶ з 't { t1 t0 Ǵа L ж߄ t0 ਔ 3 ঁДϐ LIBORǴ߾ວ៾࡭ԖΓӧ t1 ள‫ډ‬. max ^ L x, 0` u 't Ƕρ‫ ޕ‬P t0 , t1

(161) 1 1 L't

(162) Ǵ‫܌‬а max ^ L x, 0` u 't. °1 P t0 , t1

(163) ½° max ® x, 0 ¾ u 't °¯ P t0 , t1

(164) 't °¿ ° 1 ½° max ® 1 x't

(165) , 0 ¾ ¯° P t0 , t1

(166) ¿° ° 1 °½ max ® k , 0¾ ¯° P t0 , t1

(167) ¿°. ‫ځ‬ύ k { 1 x't Ƕ‫܌‬аճ౗ວ៾ӧ t ਔ‫ޑ‬ሽ਱ࣁ. 8.

(168) ª 1 r s

(169) ds º EtQ «e ³t max ^ L x, 0` u 't » ¬ ¼ ª t1 r s

(170) ds ° 1 ½° º max ® k , 0¾» EtQ « e ³t °¯ P t0 , t1

(171) °¿ »¼ «¬ t. ª 0 r s

(172) ds 1 ½º max ® P t0 , t1

(173) , 0 ¾ » k u EtQ « e ³t ¯k ¿¼ ¬ t. 1 ‫܌‬аǴճ౗ວ៾࣬྽‫ ܭ‬k ൂՏኻԄ໸‫چ‬፤៾Ǵ‫ځ‬኱‫ޑ‬ၗౢࣁ P t , t1

(174) Ǵ୺Չሽࣁ Ǵ k ‫ډ‬යВࣁ t0Ƕ ӧߕᒵ C ύǴ‫עॺך‬Ԝᒿᐒ༾ϩБำಔ(SDE system) (6)ǵ(17) – (23)‫(ک‬39)ӈ рٰ3Ǵ٠ό಄ӝ DPS ‫ۓ‬ကϐ Affine Jump-Diffusion model(ᙁᆀ AJD)Ǵҭό಄ӝ. Leippold and Wu (2002)‫ ܈‬Cheng and Scaillet (2007)ගрϐ Linear-Quadratic Jump-Diffusion model(ᙁᆀ LQJD)Ǵ‫܌‬аค‫঺ݤ‬Ҕдॺ‫ޑ‬ճ౗ວ៾ຑሽБ‫ݤ‬Ƕᗨฅ ӵԜǴՠԜኳࠠ‫ڀ‬Ԗଭёϻ‫܄‬፦Ǵૈ୼Ԗਏ౗Ӧ٬ҔᆾӦьᛥኳᔕ‫ٰݤ‬ຑሽӚᅿ ճ౗़ғ‫܄‬୘ࠔǶ‫܌‬аௗΠٰ‫ॺך‬ஒ཮ҔԜБ‫ٰݤ‬ຑሽ໸‫چ‬፤៾‫ک‬ճ౗΢ज़ᒧ᏷ ៾Ƕ. Οǵ ኧॶ‫ٯ‬η ҁ࿯ஒ཮ճҔ߻य़௢Ꮴр‫ޑ‬໸‫୏چ‬ᄊၸำ(31)ǴҔᆾӦьᛥኳᔕ‫ݤ‬Ǵຑሽ໸‫چ‬ ፤៾‫ک‬ճ౗΢ज़ᒧ᏷៾Ƕ‫ॺך‬ஒ཮٬ҔൂӢηኳٰࠠीᆉ(ջ N. 1.. 1 )Ƕ. ᆾӦьᛥኳᔕ‫ݤ‬. ‫ୖॺך‬Ե Chiarella, Clewlow and Musti (2005)‫ݤ଺ޑ‬Ǵӧ॥ᓀύҥෳࡋ Q ϐΠǴ ΋ኻԄ፤៾а႟৲໸‫ چ‬P t , T

(175) բࣁ኱‫ޑ‬ၗౢǴ‫ډ‬යВࣁ TC Ǵ 0 d TC d T Ǵ୺Չሽ਱. 3. ୖ‫ ـ‬Trolle and Schwartz (2009) Proposition 2.Ƕ. 9.

(176) ࣁ KǶ߾Ԝ፤៾ӧ t ਔᗺሽॶࣁǺ.

(177). TC EtQ ªexp ³ r s

(178) ds max ^ K P TC , T

(179) , 0`º Ƕ «¬ »¼ t TC ‫ עॺך‬TC Ϫԋ N C ฻ϩǴ 'tC Ǵ‫ॺך‬ख़ፄኳᔕ 3 ԛǴ‫܌‬а፤៾ሽॶࣁ NC. Put t , TC , T

(180). Put MC t , TC , T

(181). (40). º § n 1 · 1 3 ª exp « ¨ ¦ ri j 'tC

(182) 'tC ¸ max ^ K Pi nC 'tC , T

(183) , 0`» Ƕ(41) ¦ 3 i 1 «¬ »¼ © j 0 ¹. ӕኬǴວ៾ҭૈҔᜪ՟Б‫ٰݤ‬ຑሽǶௗΠٰ‫ॺך‬ஒҔԜБ‫ٰݤ‬ຑሽ໸‫چ‬፤៾Ƕࣁ Αቚуྗዴ‫܄‬Ǵ‫ॺך‬٬ҔΑ 1. Antithetic-variates Б‫ٰݤ‬у‫ז‬ᆾӦьᛥ‫ޑ‬ԏᔙೲࡋǹ. 2. Milstein scheme ٰफ़ե SDE ᚆණϯ‫ޑ‬ᇤৡ(ୖ‫ ـ‬Glasserman (2004) p.343)Ƕ. 2.. ຑሽ႟৲໸‫چ‬፤៾. ๏‫ۓ‬΋႟৲໸‫چ‬Ǵय़ᚐ 100 ϡǴ‫ډ‬යВ T 1 ǶаԜ໸‫چ‬բࣁ኱‫ޑ‬ၗౢϐኻԄ 0.5 Ǵ୺Չሽ਱ࣁ K. ፤៾Ǵ‫ډ‬යВࢂ TC. D0. 0.0302, D1 =0.0879, J. X 0

(184) 0.7542, T. 0.3341, U. 98.5 Ƕ๏‫ۓ‬аΠୖኧǺ. 0.4615, N. 2.1476, V. 0.3325. 0.7542. ‫ॺך‬٬Ҕ Nelson and Siegel (1989)‫ޑ‬ᇻයճ౗ԔጕǺ f t, T

(185). ‫ځ‬ύ E0. 0.0053, E1. E 0 E1e J T t

(186) E 2 T t

(187) e J. 0.0169, E 2. 1. 0.0079, J 1. 0.0585, J 2. 2. T t

(188). (42). 0.0585 Ƕ‫ॺך‬ख़ፄኳᔕ. P t , T

(189) 10000 ԛǴջ 3 10000 Ƕ २ӃǴ‫ॺך‬ᢀჸ V i ‫ޑ‬ᡂ୏ჹ፤៾ሽ਱‫ޑ‬ቹៜǶ‫ׯॺך‬ᡂ V i. 0,1,!,10 Ǵ‫ځ‬Ꭹ. ୖኧߥ࡭όᡂ4Ƕவკ 2 Ϸ߄਱ 1Ǵჹ‫ܭ‬ԜಔୖኧǴ‫ॺך‬ёа࣮р፤៾ሽ਱ᒿ๱ V i. 4. ऩჹ‫܌‬Ԗ‫ ޑ‬iǴ V i. 0 Ǵ߾ଏϯࣁߚᒿᐒ‫ࡋ୏ݢ‬ኳࠠǶ. 10.

(190) ຫεԶຫ຦Ƕӧკ 3Ǵ‫ॺך‬Кၨ V i. 0 ‫ک‬Vi. 5 ‫ٿ‬ಔୖኧǴόӕኳᔕၡ৩Π‫ޑ‬. P TC , T

(191) ‫ޑ‬ԛኧ‫ޔ‬Бკ(ёа‫ע‬Ѭ࣮ԋࢂᐒ౗ϩଛკ)Ǵว౜྽ V i ၨεǴ཮з P TC , T

(192) ‫ޑ‬ϩଛև౜ࠆ‫׀‬౜ຝǴ P TC , T

(193) ပΕሽϣ(<98.5)‫ޑ‬ᐒ౗ᡂεǴ‫܌‬а፤៾ሽ ਱ຫ຦Ƕ೭ᆶ Black-Scholes ኳࠠΠި౻ᒧ᏷៾ሽ਱ᆶ V ԋ҅К‫ۺཷޑ‬ᜪ՟Ƕ. კ 2. ߄਱ 1. V i ᡂ୏ჹ႟৲໸‫چ‬፤៾ሽ਱‫ޑ‬ቹៜ. V i ᡂ୏ჹ႟৲໸‫چ‬፤៾ሽ਱‫ޑ‬ቹៜ. Vi 0 Put Price ኱ྗৡ. 2. 4. 6. 8. 10. <10-5. 0.0055. 0.0211. 0.0316. 0.0418. 0.0488. -4. 0.0003. 0.0008. 0.0011. 0.0014. 0.0017. <10. 11.

(194) კ 3. 3.. ኳᔕ 10000 ԛࡕǴ P TC , T

(195) ပӧόӕ୔ୱ‫ޔޑ‬Бკ. ຑሽճ౗΢ज़ᒧ᏷៾. ௗΠٰ‫ॺך‬ຑሽճ౗΢ज़ᒧ᏷៾ǴԵቾ΋ঁΟԃය‫ޑ‬ճ౗΢ज़ᒧ᏷៾Ǵ‫ۑ؂‬ б৲΋ԛ(Tenor = 1/4 ԃ)Ǵճ౗΢ज़ࣁ 2%Ƕ‫ॺך‬΢ΠѳՉ౽୏ᇻයճ౗Ԕጕ(Ҕ(42) Ԅу x%)Ǵ‫ځ‬Ꭹୖኧᆶ΢΋࿯࣬ӕǴኳᔕ 5000 ԛǴᢀჸճ౗΢ज़ᒧ᏷៾ሽ਱‫ׯޑ‬ ᡂǶᢀჸკ 4 Ϸ߄਱ 2Ǵ΋ӵ‫ॺך‬ႣයǴճ౗΢ज़ᒧ᏷៾ᆶި౻ᒧ᏷៾‫ڀ‬Ԗ࣬ӕ ‫ޑ‬੝‫܄‬Ǻ(i) ሽ਱ࢂ x ‫ޑ‬с‫ڄ‬ኧǹ(ii) ӧుࡋሽϣǴሽ਱൳Яࢂ x(‫ި܈‬ሽ)‫ޑ‬ጕ‫܄‬ሀ ቚ‫ڄ‬ኧǹ(iii) ӧుࡋሽѦǴሽ਱൳Яࢂ႟Ƕ. 12.

(196) კ 4. ߄਱ 2. ᇻයճ౗Ԕጕ΢ΠѳՉ౽୏ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ. ᇻයճ౗Ԕጕ΢ΠѳՉ౽୏ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ. x. -2%. -1%. 0%. 1%. Cap Price (Basis points). 6.93ʳ. 71.36ʳ. 301.28ʳ. 560.04ʳ. ኱ྗৡ. 0.04. 0.26. 0.31. 0.44. 2%. 3%. 4%. 809.26ʳ 1053.97ʳ 1289.46 0.50. 0.54. 0.56. ௗ๱‫ॺך‬ෳ၂ U ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜǴߥ࡭‫ځ‬дୖኧόᡂǴ‫ׯ‬ᡂ U Ƕ ‫ॺך‬ϩձаճ౗΢ज़ Cap = 2%ǵ3%‫ ک‬4%Ǵኳᔕ 5000 ԛǴவ߄਱ 3 ‫ک‬კ 5 ள‫ޕ‬Ǵ ӧൂӢηኳࠠΠǴ U ᆶሽ਱Ԗ΋‫࣬ޑۓ‬ᜢ‫܄‬Ƕՠ҅࣬ᜢ‫࣬ॄ܈‬ᜢǴሡຎЯୖኧа ϷԜᒧ᏷៾‫ޑ‬ሽϣำࡋǶ೭ΨᡉҢр U ӧԜኳ္ࠠ‫ޑ‬ख़ा‫܄‬Ǵж߄Ԝኳࠠૈਂਆ ߚճ౗යज़่ᄬᒿᐒӢη‫ޑ‬ቹៜΚǶ. 13.

(197) კ 5. ߄਱ 3. U ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ. U ჹճ౗΢ज़ᒧ᏷៾ሽ਱‫ޑ‬ቹៜ. Cap = 2%. Cap Price(bps) ኱ྗৡ. U -1. -0.6. -0.2. 0.2. 0.6. 1. 309.2ʳ. 307.0ʳ. 303.0ʳ. 301.3ʳ. 300.1ʳ. 297.9ʳ. 0.403ʳ. 0.414ʳ. 0.403ʳ. 0.398ʳ. 0.386ʳ. 0.366ʳ. Cap = 3%. Cap Price(bps) ኱ྗৡ. U -1. -0.6. -0.2. 0.2. 0.6. 1. 85.88ʳ. 84.94ʳ. 82.83ʳ. 80.60ʳ. 78.14ʳ. 74.71ʳ. 0.304ʳ. 0.317ʳ. 0.327ʳ. 0.336ʳ. 0.333ʳ. 0.341ʳ. Cap = 4%. Cap Price(bps) ኱ྗৡ. U -1. -0.6. -0.2. 0.2. 0.6. 1. 6.176ʳ. 6.529ʳ. 6.810ʳ. 7.042ʳ. 7.326ʳ. 7.494ʳ. 0.087ʳ. 0.101ʳ. 0.114ʳ. 0.125ʳ. 0.130ʳ. 0.142ʳ. Ѥǵ ‫׳‬΋૓ϯ‫ޑ‬ኳࠠ ‫ॺך‬ёа‫ע‬ಃΒ࿯္(5)ԄᘉкǴ‫္ࡋ୏ݢע‬य़‫ ޑ‬r t

(198) ‫ ࣁׯ‬r t

(199) Ǵ‫ ע‬X t

(200) ‫ׯ‬ O. 14.

(201) G. ࣁX t

(202). (‫ځ‬ύ O , G ! 0 )Ǵᡣ O ‫ ک‬G ԋࣁё՗ी‫ୖޑ‬ኧǶ df t , T

(203). N. G. O. P f t , T

(204) dt ¦ V f ,i t , T

(205) Xi t

(206) r t

(207) dWi Q t

(208). (43). i 1. ऩG. 1 ǴO 2. 0 Ǵ߾ኳࠠଏϯԋ Trolle and Schwartz (2009)ǹऩ G. 0 ǴN. 1Ǵ. ߾ଏϯԋ Bhar and Chiarella (1997)‫ ޑ‬HJM ኳࠠǶ. ϖǵ ᕴ่ Ҟ߻ς࿶ԖߚதӭЎക௖૸ HJM ኳࠠΠ‫ۓޑ‬ሽ᝼ᚒǴԶ᝼ᚒЬाࢂ(i) ኳࠠ ‫ޑ‬ёᏹբ‫(܄‬хࡴԖؒԖ࠾ഈှǴаϷኧॶБ‫ޑݤ‬ीᆉೲࡋ)ǹ(ii) ኳࠠჹѱ൑੝‫܄‬ ‫ਂޑ‬ਆૈΚǶ‫ؼׯॺך‬౜Ԗ‫ޑ‬ኳࠠǴᡣኳࠠ‫ڀ‬ԖаΠ੝‫܄‬Ǻ1. ӭӢη‫ ޑ‬HJM ᇻය ճ౗ኳࠠǹ2. ԜኳࠠԖ N ঁӢη཮ቹៜճ౗่ᄬǴќԖᚐѦ N ঁӢη཮ቹៜ‫୏ݢ‬ ࡋ‫ک‬ճ౗़ғ‫܄‬୘ࠔǹ3. ‫୏ݢ‬౗ᆶอයճ౗Ԗ҅Кᜢ߯ǹ4. ߚచҹ‫ࡋ୏ݢ‬ևᎪঢ় ‫ރ‬ǹ5. ճ౗Ϸ໸‫ރޑچ‬ᄊᡂኧ಄ӝ΋ঁଭёϻ‫ޑ‬ၸำǴӢԜ‫ॺך‬ӧ٬ҔᆾӦьᛥ ‫ݤ‬ѐຑሽፄᚇ‫ޑ‬ճ౗୘ࠔਔ཮Кၨ৒ܰीᆉǶ ҁኳࠠനεᓬᗺࢂᡣճ౗‫ࡋ୏ݢ‬ᆶอයճ౗Ԗ҅Кᜢ߯Ǵ೭ᆶ Chan et al.. (1992)‫ޑ‬ჴ᛾่݀࣬಄Ƕ‫ॺך‬ᆀϐࣁ‫ރ‬ᄊ٩ᒘ‫ࡋ୏ݢ‬Ƕҁኳࠠᗨค‫ݤ‬಄ӝ DSP (2000) ‫ ޑ‬AJD ‫ ܈‬Cheng and Scaillet (2007)‫ ޑ‬LQJD ϐచҹǴՠ‫ૈॺך‬೸ၸᙯඤԋԖज़‫ރ‬ ᄊᡂኧ‫ޑ‬ଭёϻၸำǴᡣ‫ॺך‬ҔᆾӦьᛥኳᔕ‫ݤ‬ਔ‫׳‬Ԗਏ౗ǶҁኳࠠѝԵቾค॥ ᓀ‫ޑ‬ճ౗යज़่ᄬ(Ϧ໸ճ౗)Ǵՠ‫ॺך‬ёаᘉкѬࣁၢ៌ኳࠠǴᡣᇻයճ౗ନΑ‫ڙ‬ ೱុ‫ޑ‬Ѳਟၮ୏ቹៜǴҭ཮‫ڙ‬όೱុ‫ޑ‬ၢ୏‫܌‬ቹៜǴҔаਂਆ॥ᓀ‫܄‬ճ౗(ӵϦљ ໸ճ౗)ӧߞҔ٣ҹਔ‫ޑ‬ቃਗ਼ၢ୏Ƕ. 15.

(209) ߕᒵ ‫ۓ‬౛ 1.ϐ᛾ܴ. A.. ๏‫(ۓ‬8)ԄǴ߾(7)ԄǺ N. ¦. P f (t , T ). ª D 0iD1i § 1 D 0i · J i T t

(210) 2J i T t

(211) § D 0,iD1,i · 2J i T t

(212) e ¨ ¨ ¸ e ¸ T t

(213) e «¬ J i © J i D1i ¹ © Ji ¹. Xi t

(214) r t

(215) «. i 1.

(216). § D12i · D12i § 1 D 0i · 2 2J T t

(217) º J T t

(218) 2J T t

(219) e ¨ » ¨ ¸ T t

(220) e ¸ T t

(221) e J i © J i D1i ¹ © Ji ¹ ¼». i. i.

(222). ‫܌‬аǴ N. f 0, T

(223) ³ P f s, T

(224) ds ¦ ³ V f ,i s, T

(225) Xi t

(226) r t

(227) dWi Q s

(228) t. f (t , T ). 0. i 1. t. 0. N. N. i 1. i 1 j 1. 6. f 0, T

(229) ¦ E xi T t

(230) xi t

(231) ¦¦ EI j ,i T t

(232) I j ,i t

(233). xi t

(234). ³. Xi t

(235) r t

(236) eJ t s

(237) dWi Q s

(238). t. i. 0. I1,i t

(239). ³. I2,i t

(240). ³. t. I3,i t

(241). ³. t. I4,i t

(242). ³. t. I5,i t

(243). ³. t. I6,i (t ). ³. t. 0. 0. 0. 0. 0. t. 0. Xi t

(244) r t

(245) t s

(246) eJ t s

(247) dWi Q s

(248) i. Xi t

(249) r t

(250) eJ t s

(251) ds i. Xi t

(252) r t

(253) e2J t s

(254) ds i. Xi t

(255) r t

(256) t s

(257) eJ t s

(258) ds i. Xi t

(259) r t

(260) t s

(261) e2J t s

(262) ds i. Xi t

(263) r t

(264) t s

(265) e2J t s

(266) ds 2. i. ‫ע‬Ӛᡂኧ঺Ҕ Itô’s Lemma ջёள(17) – (23)Ƕ. 16. i.

(267) ‫ۓ‬౛ 2.ϐ᛾ܴ. B.. ኳү Bhar and Chiarella (1997) Appendix 3.‫ޑ‬ीᆉၸำǴவ(9)ԄǴ r t

(268). f t, t

(269). N. N. i 1. i 1 j 1. 6. f 0, t

(270) ¦ E x 0

(271) xi t

(272) ¦¦ EI j ,i 0

(273) I j ,i (t ). ճҔ Itô’s LemmaǴ dr t

(274). N t w t ªw f 0, t « wt

(275) ¦ wt ³0 Xi (u )r u

(276) V i u , t

(277) ³u V i u , y

(278) dydu i 1 ¬ N N t w º Q ¦ ³ Xi u

(279) r u

(280) V i u , t

(281) dWi u

(282) » dt ¦Xi (t )r t

(283) V i (t , t )dWi Q t

(284) 0 wt i 1 i 1 ¼ N t t w ªw f t 0, X (u )r u

(285) ªV i u, t

(286) ³ V i u, y

(287) dy ºdu.

(288) ¦ « wt ³ « »¼ 0 i u wt ¬ i 1 ¬ N N t w º ¦ ³ Xi u

(289) r u

(290) V i u , t

(291) dWi Q u

(292) » dt ¦Xi (t )r t

(293) V i t , t

(294) dWi Q t

(295) 0 wt i 1 i 1 ¼ N t ªw J i ( t u ) 0, f t J i D 0i D1i t u

(296)

(297) e J i ( t u ) « wt

(298) ¦ ³0 Xi (u )r u

(299) D1i e i 1 ¬. N.

(300) ³ V t. u. ¦ ³ Xi (u )r u

(301) V i u , t

(302) du i 1 N. t. 0. ¦ ³ i 1. 2. t. 0. º. Xi u

(303) r u

(304) D1i e J (t u ) J i D 0i D1i t u

(305)

(306) e J (t u )

(307) dWi Q u

(308) » dt i. i. ¼. N. ¦ D 0i Xi t

(309) r t

(310) dWi Q t

(311) i 1. N § 6 ªw ·º « f 0, t

(312) ¦ ¨ J i f 0, t

(313) J i r t

(314) D1i x t

(315) ¦ D j ,iI j ,i t

(316) ¸ » dt i 1© j 2 ¹ ¼» ¬« wt N. ¦ D 0i Xi t

(317) r t

(318) dWi Q t

(319) i 1. ‫ځ‬ύ. w V i t , T

(320) D1i eJ i (T t ) J i D 0i D1i T t

(321)

(322) eJ i (T t ) Ǵ wT D j ,i ӵ(25) – (30)Ԅ‫܌‬ҢǶ. 17. i. u, y

(323) dydu.

(324) C.. AJD ‫ ک‬LQJD చҹ. Linear-Quadratic Jump-Diffusion ‫ۓޑ‬က. Cheng and Scaillet (2007)‫ۓ‬က n ᆢ‫ރ‬ᄊӛໆ X t

(325) (ջ n ঁ‫ރ‬ᄊᡂኧ)‫ޑ‬ Linear-Quadratic Jump-Diffusion (LQJD)ࣁǺ dX t

(326). P X t

(327) , t

(328) dt V X t

(329) , t

(330) dW t

(331) dJ t

(332). ‫ځ‬ύǺ. (i). W t

(333) ࢂ኱ྗ n ᆢѲਟၮ୏ંତǹ. (ii) J t

(334) ࢂၢ៌ၸำǴ dJ t

(335) ࢂᐱҥቚໆ(independent increments)Ǵ‫ځ‬ၢ៌ελ‫ޑ‬ϩ ଛࢂ 3 dy, t

(336) Ǵ‫ډ‬ၲமࡋ(arrival intensity)ࢂ O X t

(337) , t

(338) ǹ. (iii) H , F , P

(339) ࢂ‫ځ‬ᐒ౗‫ޜ‬໔Ǵ W, J

(340) ፾ᔈ‫( ܭ‬adapt to)ୱࢬ(filtration) ^Ft `t t0 Ƕ Ъ P X t

(341) , t

(342) ǵ V Xt , t

(343) ǵ : X t

(344) , t

(345) V X t

(346) , t

(347) V X t

(348) , t

(349) ‫ ک‬O X t

(350) , t

(351) ࣣ಄ӝ T. regularity conditions (ୖ‫ ـ‬DPS (2000))Ǵ٠Ѹሡૈቪԋ. X t

(352) , t

(353). 1 T X t

(354) X bT t

(355) X c t

(356) 2 0º ªA « » ¬ 0 0¼. A ᆢࡋ฻‫ܭ‬Βԛ‫׎‬Ԅ(quadratic form)‫ރޑ‬ᄊᡂኧঁኧǶ ‫ॺך‬ёа‫ ע‬X t

(357) ‫ ک‬b ቪԋ. X. ª X t

(358) º « » Ǵ b ¬« X t

(359) ¼». ª k º «l » ¬ ¼. ‫܌‬а X t

(360) , t

(361) ૈ୼߄Ңࣁ X t

(362) , t

(363). 1 T T T l t

(364) X X A t

(365) X k t

(366) X c t

(367) . 2 . ጕ‫܄‬೽ҽ. Βԛ೽ҽ. 18.

(368) ‫܌‬аԜ‫׎‬Ԅᆀࣁ Linear-Quadratic (LQ)‫ڄ‬ኧ(ջ X t

(369) ‫ޑ‬ጕ‫ڄ܄‬ኧǴ X t

(370) ‫ޑ‬Βԛ‫ڄ‬ ኧ)Ƕ Ҕ LQqm n

(371) ߄Ң n Ӣη‫ ޑ‬LQJD ኳࠠǴ‫ځ‬ύ q ঁ X t

(372) ‫ޑ‬ϡન(ԿϿр౜΋ԛ)ࢂ Βԛ‫׎‬Ԅ(ջ X t

(373) ᆢࡋࢂ q)Ǵm ঁ X t

(374) ‫ޑ‬ϡનр౜ӧ O ‫ ܈‬: Ƕ‫ۓॺך‬က X t

(375) ‫ޑ‬ᘉ кᙯඤ(extended transform)ࣁ.

(376). T g X t ,t I g ; X, t , T

(377) E ª«exp ³ R X s

(378) , s

(379) ds e

(380)

(381) | Ft º» Ƕ t ¬ ¼. ӧ LQqm n

(382) ΠǴ. 1 T T g X t

(383) , t

(384) l t

(385) X X A t

(386) X k t

(387) X c t

(388) 2 ‫ځ‬ύ A t

(389) ࢂ rank. q ‫ޑ‬ჹᆀંତǶᆶϐ߻‫ޑ‬ኳࠠКၨǴ྽ q. Leippold and Wu (2002)‫ ޑ‬Quadratic Gaussian (QG)ኳࠠǹ྽ q. n ਔǴ߾཮ଏϯԋ. 0 ਔǴ߾཮ଏϯԋ. DPS (2000)‫ ޑ‬ADJǶ Linear-Quadratic Jump-Diffusion ‫ޑ‬ज़‫ڋ‬చҹ g X t

(390) , t

(391) ‫ ک‬R X t

(392) , t

(393) ࣣࣁ X t

(394) ‫ ޑ‬LQ ‫ڄ‬ኧǴLQqm n

(395) ኳࠠѸሡ಄ӝаΠΟঁ. ज़‫ڋ‬చҹǺ. 1. J t

(396) ‫ ߻ޑ‬q ঁϡનࣣࣁ႟Ƕ 2. X t

(397) ‫ޑ‬ᅆੌ໨ࣁ. P X, t

(398).

(399). ª P X t

(400) , t « «P X t

(401) , X t

(402) , t «¬.

(403).

(404). º » » »¼ nu1.

(405). ‫ځ‬ύ(a) P X t

(406) , t Ѹ໪ࣁ X t

(407) ‫ޑ‬ጕ‫ڄ܄‬ኧǹ(b) P X t

(408) , X t

(409) , t ࢂ X t

(410) ‫ ޑ‬LQ ‫ڄ‬ኧǶ. 3. ࣬ᜢંତ. 19.

(411) : X, t

(412). ª : t

(413) « T «i «¬: X, t.

(414).

(415). i X, t º : » » : X, t

(416) » ¼ nu n. ‫ځ‬ύ(a) : t

(417) ࢂ t ‫ޑ‬ዴ‫ڄۓ‬ኧ(deterministic function)ǹ(b) : X,t

(418) ࢂ X ‫ ޑ‬LQ ‫ڄ‬.

(419). i X,t ࢂ X t

(420) ‫ޑ‬ጕ‫ڄ܄‬ኧǶ ኧǹ(c) : ҁኳࠠ‫୏ޑ‬ᄊၸำ όѨ΋૓‫܄‬ΠǴԵቾ N. 1 ௃‫׎‬Ƕኳү Collin-Dufresne and Goldstein (2003)‫଺ޑ‬. ‫ݤ‬Ǵ‫ځ‬ຑሽϦԄሡा 11 ঁ‫ރ‬ᄊᡂኧǴ‫ע‬ѬॺቪԋӛໆǺ. Y { ª¬ln P t , T0

(421) , ln P t , T1

(422) ,X , r , x, I1 , I2 , I3 , I4 , I5 , I6 º¼ Ǵ߾‫ ځ‬SDE ࣁ T. ª « d ln P0 « « « d ln P1 « « dX « « dr « dx « « dI1 « dI 2 « « dI3 « « dI4 « dI5 « ¬« dI6. º » » » » » » » » » » » » » » » » » » ¼». ª ª 1 N º 2 % r T t X r.

(423) «% x1 T0 t

(424) X r ¦ 0 x « 2 » i i 1 « « » « « 1 N » 2 « % x1 T1 t

(425) X r « r ¦ % xi T1 t

(426) X r » 2 1 i « « » « «N T X

(427) » V XU « « » 6 « « f 0, t

(428) D 0 x ¦ j 2 D j I j » Xr « « » Xr « J x » dt « « « x JI » 0 « « » « « rX JI2 » 0 « « rX 2JI » 0 3 « « » « «I2 JI4 » 0 « « » 0 « «I3 2JI5 » « «¬« 2I5 2JI6 »»¼ 0 ¬«. º » » » 0 » » 2 » V v 1 U » » 0 Q » ª dW t

(429) º » »« Q 0 dZ t.

(430) « »¼ »¬ 0 » » 0 » 0 » » 0 » 0 » » 0 ¼» 0. ೭္а Pi ж߄ P t , Ti

(431) Ƕவ΢Ԅё‫ޕ‬Ǵᗨฅ Y ࢂଭёϻၸำǴՠ‫ځ‬ᅆࢬ໨٠ߚ Y ‫ޑ‬ ጕ‫ڄ܄‬ኧǴ‫܌‬аόૈ٬Ҕ DPS (2000)‫ ޑ‬ADJ ຑሽϦԄǶฅࡕǴ‫ॺך‬ᢀჸ LQJD ‫ޑ‬ ‫ۓ‬ကǴ߾ q. 2 ( X t

(432). >X , r @. T. )Ǵՠ : t

(433) ٠όࢂ t ‫ޑ‬ዴ‫ڄۓ‬ኧ(deterministic function)Ǵ. ‫܌‬а Y ٠ό಄ӝ LQJD ‫ޑ‬ज़‫ڋ‬చҹǴ‫܌‬аҭค‫ݤ‬٬Ҕ Cheng and Scaillet (2007)‫ޑ‬ຑ ሽϦԄǶ. 20.

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