Calibration to swaptions in the libor market model

❈❆▲■❇❘❆❚■❖◆ ❚❖ ❙❲❆P❚■❖◆❙ ■◆ ❚❍❊ ▲■❇❖❘ ▼❆❘❑❊❚ ▼❖❉❊▲ P■❊❘❘❊ ❇❊❘❊❚ ◆❆❚■❖◆❆▲ ❯◆■❱❊❘❙■❚❨ ❖❋ ❙■◆●❆P❖❘❊ ✷✵✵✼ ❈❆▲■❇❘❆❚■❖◆ ❚❖ ❙❲❆P❚■❖◆❙ ■◆ ❚❍❊ ▲■❇❖❘ ▼❆❘❑❊❚ ▼❖❉❊▲ P■❊❘❘❊ ❇❊❘❊❚ ✭■♥❣❡♥✐❡✉r✱ ❊❝♦❧❡ ❈❡♥tr❛❧❡ P❛r✐s ✮ ❆ ❚❍❊❙■❙ ❙❯❇▼■❚❚❊❉ ❋❖❘ ❚❍❊ ❉❊●❘❊❊ ❖❋ ▼❆❙❚❊❘ ❖❋ ❙❈■❊◆❈❊ ❉❊P❆❘❚▼❊◆❚ ❖❋ ▼❆❚❍❊▼❆❚■❈❙ ◆❆❚■❖◆❆▲ ❯◆■❱❊❘❙■❚❨ ❖❋ ❙■◆●❆P❖❘❊ ✷✵✵✼ ✐ ◆❛♠❡ ✿ P✐❡rr❡ ❇❡r❡t ❉❡❣r❡❡ ✿ ▼❛st❡r ♦❢ ❙❝✐❡♥❝❡ ❙✉♣❡r✈✐s♦r ✿ ❉r ❖❧✐✈❡r ❈❤❡♥ ❉❡♣❛rt♠❡♥t ✿ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❚❤❡s✐s ❚✐t❧❡ ✿ ❈❛❧✐❜r❛t✐♦♥ t♦ s✇❛♣t✐♦♥s ✐♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❆❜str❛❝t ■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✐s ♣r❡s❡♥t❡❞ ❛♥❞ ✐ts ❝❛❧✐❜r❛t✐♦♥ ♣r♦❝❡ss ✐s ❞❡r✐✈❡❞✳ ❲❡ ❛ss✉♠❡ t❤❡ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡s ❢♦❧❧♦✇ ❧♦❣✲♥♦r♠❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✇✐t❤ ❛ d✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❛♥❞ ❜✉✐❧❞ ❛♥ ✐♥✲ t❡r❡st r❛t❡s ♠♦❞❡❧ ❛❜❧❡ t♦ ♣r✐❝❡ ✐♥t❡r❡st r❛t❡ ❞❡r✐✈❛t✐✈❡s✳ ❲❡ ❡♠♣❤❛s✐③❡ ❤♦✇ ❞✐✛❡r❡♥t ✐t ✐s ❢r♦♠ t❤❡ ✉s✉❛❧ s❤♦rt✲t❡r♠ ✐♥t❡r❡st r❛t❡s ♠♦❞❡❧s ✭❍✉❧❧✲❲❤✐t❡✮✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤✐s ♣r✐❝✐♥❣ ♠♦❞❡❧ ♦♥❧② ♠❛❦❡s s❡♥s❡ ✐❢ ✈❛♥✐❧❧❛ ♣r♦❞✉❝ts✱ ♥❛♠❡❧② ❝❛♣s ❛♥❞ ❊✉r♦♣❡❛♥ s✇❛♣t✐♦♥s✱ ❝❛♥ ❜❡ ✇❡❧❧ ♣r✐❝❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡✐r ♠❛r❦❡t ✈❛❧✉❡✳ ❚♦ ❝❤❡❝❦ t❤✐s✱ ✇❡ ♣r♦♣♦s❡ ❞✐✛❡r❡♥t ♣❛r❛♠❡tr✐❝ ❢♦r♠s ♦❢ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t✐❡s σi (t) ❛♥❞ ❝♦rr❡❧❛t✐♦♥s ρij t♦ ♦❜t❛✐♥ t❤❡ ❜❡st r❡s✉❧ts✳ ❚❤❡♥✱ ✇❡ s❤♦✇ ❛ ♠❡t❤♦❞ t♦ r❡❞✉❝❡ t❤❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ♠♦❞❡❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❋♦r✇❛r❞ r❛t❡s ✐♥✈♦❧✈❡❞ ❜② ✉s✐♥❣ ❘❡❜♦♥❛t♦ ❆♥✲ ❣❧❡s ❛♥❞ ❋r♦❜❡♥✐✉s ♥♦r♠✳ ❋✐♥❛❧❧②✱ ✇❡ ❞❡r✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r♠✉❧❛ ❢♦r ❊✉r♦♣❡❛♥ s✇❛♣t✐♦♥s ❛♥❞ s❤♦✇ ✇❡ ❝❛♥ ❛✈♦✐❞ ▼♦♥t❡✲❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❢♦r t❤❡ ❝❛❧❝✉❧❛t✐♦♥s ♦❢ t❤❡ s✇❛♣t✐♦♥s ❞✉r✐♥❣ t❤❡ ❝❛❧✐❜r❛t✐♦♥✳ ❙♦♠❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ❛r❡ ❣✐✈❡♥ ♦♥ ❛ 3 ❢❛❝t♦rs ♠♦❞❡❧✳ ❲❡ ❞✐s❝✉ss t❤❡♥ ❞✐✛❡r❡♥t ✐ss✉❡s r❛✐s❡❞ ❛♥❞ ❝✉rr❡♥t ❞❡✈❡❧♦♣♠❡♥ts✱ ♠♦r❡ s♣❡❝✐❢✲ ✐❝❛❧❧② t❤❡ ❙❆❇❘ s❦❡✇ ❢♦r♠ ❛♥❞ ❝r♦ss✲❛ss❡t ♣r♦❞✉❝ts✳ ❑❡②✇♦r❞s ✿ ■♥t❡r❡st ❘❛t❡ ❉❡r✐✈❛t✐✈❡s✱ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✱ ❈❛❧✐✲ ❜r❛t✐♦♥✱ ❘❛♥❦ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞s✱ ❙✇❛♣t✐♦♥ ❆♣♣r♦①✐✲ ♠❛t✐♦♥s✳ ✐✐ ❆❝❦♥♦✇❧❡❞❣♠❡♥t ■ ❝♦♥s✐❞❡r ♠②s❡❧❢ ❡①tr❡♠❡❧② ❢♦rt✉♥❛t❡ t♦ ❤❛✈❡ ❜❡❡♥ ❣✐✈❡♥ t❤❡ ♦♣♣♦rt✉♥✐t② ❛♥❞ ♣r✐✈✐❧❡❣❡ ♦❢ ❞♦✐♥❣ t❤✐s r❡s❡❛r❝❤ ✇♦r❦ ❛t t❤❡ ◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙✐♥❣❛♣♦r❡✳ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ❛❧❧ t❤❡ ♣❡♦♣❧❡ ✇❤♦ ❤❛✈❡ ❤❡❧♣❡❞ ♠❡ ❞✉r✐♥❣ ♠② ▼❛st❡r✬s ❞❡❣r❡❡ ♣r♦❣r❛♠✳ ❆❧❧ ♠② ❣r❛t✐t✉❞❡ t♦ ❉♦❝t♦r ❖❧✐✈❡r ❈❤❡♥ ✇❤♦ ❛❝❝❡♣t❡❞ t♦ ❜❡ ♠② s✉♣❡r✈✐s♦r ❛♥❞ ♣r♦✈✐❞❡❞ ✇❛r♠ ❛♥❞ ❝♦♥st❛♥t ❣✉✐❞❛♥❝❡ t❤r♦✉❣❤♦✉t ♣r♦❣r❡ss ♦❢ t❤✐s ✇♦r❦✳ ▼② ✇❛r♠❡st t❤❛♥❦s t♦ t❤❡ ❘♦②❛❧ ❇❛♥❦ ♦❢ ❙❝♦t❧❛♥❞ ✇❤♦ ✇❡❧❝♦♠❡❞ ♠❡ ✐♥ ✐ts ❊①♦t✐❝ ❘❛t❡s ❙tr✉❝t✉r✐♥❣ ❚❡❛♠ ❢♦r ✻ ♠♦♥t❤s✳ ❚❤✐s ❡①♣❡r✐❡♥❝❡ ✇❛s ✈❡r② r✐❝❤ ❛♥❞ ■ ❧❡❛r♥❡❞ ❛ ❧♦t ✇✐t❤ ❙❡r❣❡ P♦♠♦♥t✐✳ ■ ❛♠ ❤❛♣♣② t♦ ❝♦♥t✐♥✉❡ t❤✐s ❝♦❧❧❛❜♦r❛t✐♦♥ ✐♥ ❏❛♥✉❛r②✳ ■ ❛♠ ❛❧s♦ t❤❛♥❦❢✉❧ ❢♦r t❤❡ ❣r❛❞✉❛t❡ r❡s❡❛r❝❤ s❝❤♦❧❛rs❤✐♣ ♦✛❡r❡❞ t♦ ♠❡ ❜② t❤❡ ◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙✐♥❣❛♣♦r❡ ✇✐t❤♦✉t ✇❤✐❝❤ t❤✐s ▼❛st❡r✬s ❞❡❣r❡❡ ♣r♦❣r❛♠ ✇♦✉❧❞ ♥♦t ❤❛✈❡ ❜❡❡♥ ♣♦ss✐❜❧❡✳ ❋✐♥❛❧❧②✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② ❞❡❡♣ ❛✛❡❝t✐♦♥ ❢♦r ♠② ❢❛♠✐❧② ❛♥❞ ♠② ❢r✐❡♥❞s ✐♥ ❙✐♥❣❛♣♦r❡ ✇❤♦ ❤❛✈❡ ❡♥❝♦✉r❛❣❡❞ ♠❡ t❤r♦✉❣❤♦✉t t❤✐s ✇♦r❦ ❛♥❞ ❢♦r ❈❛♠✐❧❧❡ ✇❤♦ s✉♣♣♦rt❡❞ ♠❡ ❡✈❡r②❞❛②✳ ▼❛r❝❤ ✸✱ ✷✵✵✼ ❈♦♥t❡♥ts ✶ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✶ ✶✳✶ ■♠♣♦rt❛♥t ❝♦♥❝❡♣ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✶ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷ ❙❤♦rt✲❚❡r♠ ✐♥t❡r❡st r❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✸ ❚❤❡ ❆r❜✐tr❛❣❡ ❢r❡❡ ❛ss✉♠♣t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✹ ❋♦r✇❛r❞ ■♥t❡r❡st r❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✺ ▲■❇❖❘ ✐♥t❡r❡st r❛t❡ ❛♥❞ s✇❛♣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✻ ❙t♦❝❤❛st✐❝ t♦♦❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✶ ❙❤♦rt t❡r♠ ✐♥t❡r❡st r❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✷ ❍❡❛t❤ ❏❛rr♦✇ ❛♥❞ ▼♦rt♦♥ ❋r❛♠❡✇♦r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✸ ❚❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷✳✹ ▲✐❜♦r ▼❛r❦❡t ♠♦❞❡❧ s✉♠♠❛r② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✸ Pr✐❝✐♥❣ ❱❛♥✐❧❧❛ ❉❡r✐✈❛t✐✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✸✳✶ ■♥t❡r❡st r❛t❡ ♦♣t✐♦♥s✿ ❝❛♣ ❛♥❞ ✢♦♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✸✳✷ ❙✇❛♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✸✷ ✷✳✶ ❚❤❡ s❡tt✐♥❣s✿ ▼❛✐♥ ♣✉r♣♦s❡ ♦❢ t❤❡ ❈❛❧✐❜r❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✶ ❚♦t❛❧ ♣❛r❛♠❡t❡r✐③❡❞ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✷ ●❡♥❡r❛❧ P✐❡❝❡✇✐s❡✲❈♦♥st❛♥t P❛r❛♠❡t❡r✐③❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✸✺ ✐✈ ❈❖◆❚❊◆❚❙ ✷✳✷✳✸ ▲❛❣✉❡rr❡ ❢✉♥❝t✐♦♥ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ t②♣❡ ✈♦❧❛t✐❧✐t② ✳ ✸✻ ✷✳✸ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❛♠♦♥❣ t❤❡ ❋♦r✇❛r❞ ❘❛t❡s ✳ ✳ ✳ ✳ ✹✵ ✷✳✸✳✶ ❍✐st♦r✐❝ ❝♦rr❡❧❛t✐♦♥ ✈s ♣❛r❛♠❡tr✐❝ ❝♦rr❡❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸✳✷ ❘❛♥❦ ❘❡❞✉❝t✐♦♥ ♠❡t❤♦❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✹ ❙✇❛♣t✐♦♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✷✳✹✳✶ ❘❡❜♦♥❛t♦ ❋♦r♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✷✳✹✳✷ ❍✉❧❧ ❛♥❞ ❲❤✐t❡ ❋♦r♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✷✳✹✳✸ ❆♥❞❡rs❡♥ ❛♥❞ ❆♥❞❡r❡❛s❡♥ ❋♦r♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✷✳✺ ▼♦♥t❡ ❈❛r❧♦ ❙✐♠✉❧❛t✐♦♥ ❛♥❞ ❘❡s✉❧ts ♦♥ ✸ ❋❛❝t♦rs ❇●▼ ✳ ✳ ✳ ✻✻ ✷✳✺✳✶ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✷✳✺✳✷ ◆✉♠❡r✐❝❛❧ ❘❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✸ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ✼✶ ✸✳✶ ❙t♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ❛♣♣❧✐❡❞ t♦ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✳ ✼✶ ✸✳✶✳✶ ❙t♦❝❤❛st✐❝ α β ρ ♠♦❞❡❧ ✲ ❙❆❇❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✸✳✷ ❍②❜r✐❞s Pr♦❞✉❝ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✸✳✸ ■ss✉❡s r❛✐s❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✸✳✶ ❈❤♦✐❝❡ ❜❡t✇❡❡♥ ❍✐st♦r✐❝❛❧ ❛♥❞ ■♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ✳ ✳ ✳ ✼✻ ✸✳✸✳✷ ■♥t❡r❡st✲r❛t❡s s❦❡✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✸✳✸ ❆♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✸✳✸✳✹ ▼❛r❦❡t ❧✐q✉✐❞✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹ ●❡♥❡r❛❧ ▼❡t❤♦❞♦❧♦❣② ♣r♦♣♦s❡❞ ❢♦r ❝❛❧✐❜r❛t✐♦♥ ✼✽ ✹✳✶ ❆ss✉♠♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✷ ▼♦❞❡❧✐♥❣ ❝❤♦✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✸ ▼❛r❦❡t ❞❛t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✹✳✹ ❈❛❧✐❜r❛t✐♦♥ ♣r♦❝❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✹✳✺ ❈♦♥❝❧✉s✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ▲✐st ♦❢ ❋✐❣✉r❡s ✶✳✶ ❩❡r♦✲❝♦✉♣♦♥ ❜♦♥❞ ♠❡❝❤❛♥✐s♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❙✇❛♣ ♠❡❝❤❛♥✐s♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✶ ▲❛❣✉❡rr❡✲t②♣❡ ✈♦❧❛t✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷ ❍✐st♦r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ❛♠♦♥❣ ❋♦r✇❛r❞ r❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✸ ❙✐♠♣❧❡ ❊①♣♦♥❡♥t✐❛❧ P❛r❛♠❡t❡r✐③❡❞ ❝♦rr❡❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✹ ▼♦❞✐✜❡❞ ❊①♣♦♥❡♥t✐❛❧ P❛r❛♠❡t❡r✐③❡❞ ❝♦rr❡❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✺ ❙❝❤♦❡♥♠❛❦❡rs ❈♦✛❡② ❝♦rr❡❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✻ ❊✐❣❡♥✈❡❝t♦rs ❝♦♠♣❛r✐s♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✷✳✼ ✷❨ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡ ❈♦rr❡❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✷✳✽ ✺❨ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡ ❈♦rr❡❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✷✳✾ ✶✵❨ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡ ❈♦rr❡❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ▲✐st ♦❢ ❚❛❜❧❡s ✷✳✶ ●❡♥❡r❛❧ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷ P✐❡❝❡✇✐s❡✲❝♦♥st❛♥t ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✸ ▲❛❣✉❡rr❡ t②♣❡ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✹ ❊✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✺ ❙✇❛♣t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❝❝✉r❛❝② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ❈❤❛♣t❡r ✶ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❆t t❤❡ ❡♥❞ ♦❢ t❤❡ ✼✵✬s✱ ❛❢t❡r ❇❧❛❝❦ ❛♥❞ ❙❝❤♦❧❡s ❜r❡❛❦t❤r♦✉❣❤ ✇✐t❤ t❤❡✐r ❢♦r♠✉❧❛ t♦ ✈❛❧✉❡ ❛ ❊✉r♦♣❡❛♥ ♦♣t✐♦♥✱ ❇❧❛❝❦ ❛❧s♦ ♣r♦♣♦s❡❞ t❤❡ ❛❧t❡r ❡❣♦ ♦❢ t❤✐s ❢♦r♠✉❧❛ ✐♥ t❤❡ ✇♦r❧❞ ♦❢ ✐♥t❡r❡st r❛t❡s✳ ❚❤✐s ✇❛s t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✐♥t❡r❡st r❛t❡s ❞❡r✐✈❛t✐✈❡s✳ ❙✐♥❝❡ ✶✾✼✻ ❛♥❞ ❇❧❛❝❦✬s ❢♦r♠✉❧❛ ❬✷❪✱ ❛ ❧♦t ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ♦♥ t❤❡ ✐♥t❡r❡st r❛t❡s t♦♣✐❝✳ ❋✐rst ✇❡r❡ ♣r❡s❡♥t❡❞ ♠♦❞❡❧s t❤❛t tr✐❡❞ t♦ ❛❞❛♣t t❤❡ ❢r❛♠❡✇♦r❦s ❝♦♠✐♥❣ ❢r♦♠ t❤❡ ❡q✉✐t② ✇♦r❧❞ ✿ t❤♦s❡ ✉s❡❞ ❛ st♦❝❤❛st✐❝ ❡q✉❛t✐♦♥ t♦ ❞❡s❝r✐❜❡ ❛ s❤♦rt✲t❡r♠ r❛t❡ ❛s ✐t ✇❛s ❞♦♥❡ ❢♦r ❛ st♦❝❦✳ ❋r♦♠ t❤✐s ❜❛s✐❝ ✐❞❡❛ ❞✐✛❡r❡♥t ❡✈♦❧✉t✐♦♥s r♦s❡ ❜② ❝❤❛♥❣✐♥❣ t❤❡ ❢♦r♠ ♦❢ t❤✐s st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ t♦ ✜t t❤❡ ❡❝♦♥♦♠✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ✐♥t❡r❡st r❛t❡s ❣❡♥❡r❛❧❧② ♦❜s❡r✈❡❞ ✲ ❢♦r ✐♥st❛♥❝❡ t❤❡ ♠❡❛♥ r❡✈❡rs✐♦♥ ♣❤❡♥♦♠❡♥♦♥✳ ❋✐♥❛❧❧② ✐♥ ✶✾✾✼✱ ❇r❛❝❡✱ ●❛t❛r❡❦ ❛♥❞ ▼✉s✐❡❧❛ ♣r♦♣♦s❡❞ ❛ ♥❡✇ ❝♦♥❝❡♣t ✇❤❡r❡ ♦❜s❡r✈❛❜❧❡ r❛t❡s ✇❡r❡ ♠♦❞❡❧❡❞ ✉s✐♥❣ t❤❡ ✇♦r❦ ♦❢ ❍❡❛t❤✱ ❏❛rr♦✇ ❛♥❞ ▼♦rt♦♥ ✐♥ ✶✾✾✷✳ ❚❤✐s ❝♦♠♣❧❡t❡❧② r❡❞❡✜♥❡❞ t❤❡ ✈✐s✐♦♥ ♦❢ ♣r✐❝✐♥❣ ❛♥❞ ❡✈❡r②t❤✐♥❣ ♥❡❡❞s t♦ ❜❡ ❞♦♥❡ ✐♥ t❤✐s ✜❡❧❞✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♠♦❞❡❧ ✐s ✉♥❞♦✉❜t❡❞❧② t♦ ❜❡ ❛❜❧❡ t♦ ✜t t❤❡ ♠❛r❦❡t✳ ❍❡♥❝❡✱ ✇❡ ❝❛❧❧ ❝❛❧✐❜r❛t✐♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❞✐✛❡r❡♥t ❛ss✉♠♣t✐♦♥s ❛♥❞ ✐♥♣✉ts s♦ t❤❛t ✇❡ ♦❜t❛✐♥ t❤❡ ❜❡st ✜t t♦ t❤❡ ♠❛r❦❡t✳ ❈❛❧✐❜r❛t✐♦♥ ✐s ❛❧✇❛②s ❛ ❤✉❣❡ ✐ss✉❡ ❢♦r ♠❛r❦❡t ♦♣❡r❛t♦rs ❛s t❤❡② ♠❛② ❢❛❝❡ s❡✈❡r❡ ♠✐s♣r✐❝❡s ✐❢ t❤❡ ♠♦❞❡❧ t❤❡② ✉s❡ ✐s ♥♦t ✇❡❧❧ ❝❛❧✐❜r❛t❡❞ ❛♥❞ ■ ✇✐❧❧ ❜❡ ✷ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ♣r❡s❡♥t✐♥❣ ❤♦✇ t❤✐s ❝❛♥ ❜❡ ❤❛♥❞❧❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ♣❛rt❀ ❜❡❢♦r❡ ❡①♣❧❛✐♥✐♥❣ ✇❤❛t ❛r❡ t❤❡ ♠❛✐♥ ✐ss✉❡s ❛♥❞ ❤♦✇ s♦♠❡ ❛r❡ ♠❛♥❛❣❡❞ ✭s❦❡✇✴s♠✐❧❡✱ ❧✐q✉✐❞✐t②✳✳✮ ❛♥❞ ✇❤❛t ❛r❡ t❤❡ ♥❡①t ❝❤❛❧❧❡♥❣❡s ❢❛❝❡❞ ❜② t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✭❈r♦ss✲❛ss❡t ❤②❜r✐❞ ♣r♦❞✉❝ts✮✳ ■♥ t❤✐s ✜rst ❝❤❛♣t❡r t❤❡ ♠❛✐♥ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡ ♠♦❞❡❧s ❝✉rr❡♥t❧② ✉s❡❞ ✐♥ t❤❡ ✇♦r❧❞ ♦❢ ✐♥t❡r❡st r❛t❡s ❛r❡ ❞❡✜♥❡❞ ❛♥❞ ❡①♣❧❛✐♥❡❞✳ ✶✳✶ ■♠♣♦rt❛♥t ❝♦♥❝❡♣ts ✶✳✶✳✶ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ❚❤❡ ✜rst ❝♦♥❝❡♣t ✇❡ ❤❛✈❡ t♦ ❞❡✜♥❡ ✇❤❡♥ ❞✐s❝✉ss✐♥❣ ✐♥t❡r❡st r❛t❡s ♣r♦❞✉❝ts ✐s t❤❡ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ✭❩✳❈✳✮✳ ■♥ t❤✐s t❤❡s✐s✱ t❤❡ ✉♥❞❡r❧②✐♥❣ ❛ss❡ts ❛r❡ ♥♦t st♦❝❦s ❧✐❦❡ ✐♥ ❇❧❛❝❦✲❙❝❤♦❧❡s ♦r✐❣✐♥❛❧ ❢r❛♠❡✇♦r❦ ✐♥ ✶✾✼✸ ✐♥ ❬✶❪ ❜✉t ❜♦♥❞s✳ ❙❡✈❡r❛❧ ❜♦♥❞s ❝❛♥ ❜❡ ❞❡✜♥❡❞✱ ♣❛②✐♥❣ ✈❛r✐♦✉s ❝♦✉♣♦♥s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ s♦♠❡ ❝♦♥❞✐t✐♦♥s. . .✶ ❍❡♥❝❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❞❡✜♥❡ ❛ s✐♠♣❧❡st ✉♥❞❡r❧②✐♥❣✿ t❤✐s ♦♥❡ ✐s t❤❡ s❡t ♦❢ ❞✐s❝♦✉♥t ❢❛❝t♦rs ❢♦r ❞✐✛❡r❡♥t ♠❛t✉r✐t✐❡s✳ ❲❡ ✇✐❧❧ ❞❡♥♦t❡ t❤❡♠ ❜② B(t, T )✳ ❚❤✐s ❜♦♥❞ r❡♣r❡s❡♥ts ❛t t✐♠❡ t t❤❡ ♣r✐❝❡ ♦❢ ✶ ♣❛✐❞ ❛t t✐♠❡ T ✱ t❤❡ ♠❛t✉r✐t② ♦❢ t❤❡ ❜♦♥❞✳ ❙❡❡ ❋✐❣✉r❡ ✶✳✶ ❢♦r ❛ ♠♦r❡ ✈✐s✉❛❧ ❡①♣❧❛♥❛t✐♦♥✳ ❋✐❣✉r❡ ✶✳✶✿ ❩❡r♦✲❝♦✉♣♦♥ ❜♦♥❞ ♠❡❝❤❛♥✐s♠ ✶ ❋♦r ✐♥st❛♥❝❡✱ ❛ ❞❛✐❧② r❛♥❣❡ ❛❝❝r✉❛❧ ❝♦✉♣♦♥✿ ■ ♣❛② X% Nn ✇❤❡r❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ✸✲♠♦♥t❤s ▲■❇❖❘ r❛t❡ st❛②s ❜❡❧♦✇ 6.5% ❛♥❞ N t❤❡ ♥✉♠❜❡r ♦❢ ❞❛②s ✐♥ t❤❡ ❛❝❝r✉❛❧ ♣❡r✐♦❞✳ ✸ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❖♥❡ ❝❛♥ ♦❜s❡r✈❡ t❤❛t ❛t ❛♥② ❞❛t❡ t✱ t❤♦s❡ ♣r✐❝❡s ❛r❡ ♥♦t ❛❧❧ q✉♦t❡❞ ♦♥ t❤❡ ♠❛r❦❡t ❜✉t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ♦t❤❡r ③❡r♦ ❝♦✉♣♦♥s ❜♦♥❞s✳ ❚❤✐s ❜♦♥❞ ❞♦❡s ♥♦t ♣❛② ❛♥② ❝♦✉♣♦♥✱ t❤❛t ✐s ✇❤② ✇❡ ❣❡♥❡r❛❧❧② ❝❛❧❧ t❤❡ ❞✐s❝♦✉♥t ❢❛❝t♦rs B(t, T ) t❤❡ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ✭❩✳❈✳✮✳ ❲❡ ✐♥tr♦❞✉❝❡ ✈❡r② ❣❡♥❡r❛❧❧② t❤❡ ❧♦❣✲♥♦r♠❛❧ ❞②♥❛♠✐❝ ❢♦r ❛ ❩❡r♦ ❈♦✉♣♦♥ ❜♦♥❞ ❛s✿ dB(t, T ) = m(t, T )tB(t, T )dt + σ B B(t, T )dWt , B(T, T ) = 1 ✭✶✳✶✮ ❲✐t❤ m(t, T )✱ t❤❡ ❞r✐❢t✱ ❡q✉❛❧ t♦ t❤❡ s❤♦rt t❡r♠ ✐♥t❡r❡st r❛t❡ rt ✐♥ ❛ r✐s❦✲ ♥❡✉tr❛❧ ✇♦r❧❞✱ σ B ✱ t❤❡ ✈♦❧❛t✐❧✐t② ❡✈❡♥t✉❛❧❧② st♦❝❤❛st✐❝ ♦r t✐♠❡✲❞❡♣❡♥❞❡♥t ❛♥❞ Wt ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ✶✳✶✳✷ ❙❤♦rt✲❚❡r♠ ✐♥t❡r❡st r❛t❡ ❲❡ ❥✉st ♠❡♥t✐♦♥❡❞ t❤❡ s❤♦rt t❡r♠ ✐♥t❡r❡st r❛t❡ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ❚r❛✲ ❞✐t✐♦♥❛❧ st♦❝❤❛st✐❝ ✐♥t❡r❡st r❛t❡s ♠♦❞❡❧s ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ❡①♦❣❡♥♦✉s s♣❡❝✐✜✲ ❝❛t✐♦♥ ♦❢ ❛ s❤♦rt✲t❡r♠ ✐♥t❡r❡st r❛t❡ ❛♥❞ ✐ts ❞②♥❛♠✐❝✳ ❲❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② rt t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✐♥t❡r❡st r❛t❡ ♦r s❤♦rt✲t❡r♠ ✐♥t❡r❡st r❛t❡ t❤❡ r❛t❡ ♦♥❡ ❝❛♥ ❜♦rr♦✇ ✐♥ ❛ r✐s❦ ❢r❡❡ ❧♦❛♥ ❜❡❣✐♥♥✐♥❣ ❛t t ♦✈❡r t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ♣❡r✐♦❞ dt✳ ■♥ ❣❡♥❡r❛❧✱ ✇❡ ❛ss✉♠❡ t❤❛t rt ✐s ❛♥ ❛❞❛♣t❡❞ ♣r♦❝❡ss ♦♥ ❛ ✜❧t❡r❡❞ ♣r♦❜❛✲ ❜✐❧✐t② s♣❛❝❡✳ ❚❤❡ ✐♠♣♦rt❛♥t t❤✐♥❣ ❛❜♦✉t s❤♦rt t❡r♠ ✐♥t❡r❡st r❛t❡ ✐s t❤❛t ❜② ❝♦♥s✐❞❡r❛t✐♦♥ ♦✈❡r t❤❡ ❛❜s❡♥❝❡ ♦❢ ❛r❜✐tr❛❣❡ ✐♥ t❤❡ ♠❛r❦❡t ✇❡ ❝❛♥ ❝r❡❛t❡ ❧✐♥❦s ❜❡t✇❡❡♥ rt ❛♥❞ B(t, T )✳ ✶✳✶✳✸ ❚❤❡ ❆r❜✐tr❛❣❡ ❢r❡❡ ❛ss✉♠♣t✐♦♥ ❚❤✐s ❝❧❛ss✐❝ ❛ss✉♠♣t✐♦♥ ✐♥tr♦❞✉❝❡s ❝♦♥str❛✐♥ts ♦♥ t❤❡ ♣❛②♦✛ ♦❢ ❞❡r✐✈❛t✐✈❡s✳ ❍❡r❡ ✇❤❡♥ ✇❡ st✉❞② r❛t❡ ✐ss✉❡s✱ t❤✐s ❛ss✉♠♣t✐♦♥ ✐s ♠❛❞❡ ♦♥ t❤❡ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ❛s ✇❡ ❝❛♥ ❧✐♥❦ ❧♦♥❣ ♠❛t✉r✐t✐❡s ✭♠♦r❡ t❤❛♥ ✶ ②❡❛r✮ ❜♦♥❞s ✇✐t❤ ❝♦✉♣♦♥s ✇✐t❤ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❆r❜✐tr❛❣❡ ❢r❡❡ ❛ss✉♠♣t✐♦♥✳ ✹ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❚❤❡ ♣r✐❝❡ ♦❢ ❛♥ ❛ss❡t ❞❡❧✐✈❡r✐♥❣ ✜①❡❞ ❝❛s❤✲✢♦✇s ✐♥ t❤❡ ❢✉t✉r❡ ✐s ❣✐✈❡♥ ❜② t❤❡ s✉♠ ♦❢ ✐ts ❝❛s❤✲✢♦✇s ✇❡✐❣❤t❡❞ ❜② t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ♦❢ t❤❡ s❡tt❧❡♠❡♥t ❞❛t❡s✳ ❲❡ ♠❛❦❡ t❤❡ ✉s✉❛❧ ♠❛t❤❡♠❛t✐❝❛❧ ❛ss✉♠♣t✐♦♥✿ ❛❧❧ ♣r♦❝❡ss❡s ❛r❡ ❞❡✜♥❡❞ ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, {Ft ; t ≥ 0}, Q0 )✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ Q0 ✐s ❛♥② r✐s❦ ♥❡✉tr❛❧ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ✇❤♦s❡ ❡①✐st❡♥❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ♥♦✲❛r❜✐tr❛❣❡ ❛s✲ s✉♠♣t✐♦♥ ✭❙❡❡ ❚❤❡ ●✐rs❛♥♦✈ tr❛♥s❢♦r♠❛t✐♦♥ ✐♥ s❡❝t✐♦♥ ✶✳✶✳✻✮✳ ❚❤❡ ✜❧tr❛t✐♦♥ {Ft ; t ≥ 0}✷ ✐s t❤❡ ✜❧tr❛t✐♦♥ ❣❡♥❡r❛t❡❞ ✐♥ Q0 ❜② ❛ d✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ W Q0 = {W Q0 (t); t ≥ 0}✳ ◆♦✇✱ ✇❡ ✐♥❢❡r t❤❛t ♦♥❡ ❝❛♥ ✐♥✈❡st ✐♥ ❛ s❛✈✐♥❣s ❛❝❝♦✉♥t ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞❡❞ ✇✐t❤ t❤❡ st♦❝❤❛st✐❝ s❤♦rt r❛t❡ rs ♣r❡✈❛✐❧✐♥❣ ❛t t✐♠❡ s ♦✈❡r t❤❡ t✐♠❡ [s; s + ds]✳ ❚❤❡ ✈❛❧✉❡ ♦❢ 1 ✐♥✈❡st❡❞ ❛t t✐♠❡ t ❛t t✐♠❡ T ✐s βT ✿ T βT = exp rs ds t ❚❤❡r❡❢♦r❡✱ ✐❢ ✇❡ ✐♥✈❡st B(t, T ) ✐♥ ❛ ❩✳❈✳ ♦❢ ♠❛t✉r✐t② T ❛♥❞ t❤❡ s❛♠❡ ❛♠♦✉♥t ✐♥ ♦✉r s❛✈✐♥❣ ❛❝❝♦✉♥t✱ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❛ss❡t ♣r✐❝✐♥❣ ✭t❤✐s ✇✐❧❧ ❜❡ ❞❡t❛✐❧❧❡❞ ✐♥ ✶✳✶✳✻✮ ❡♥s✉r❡s t❤❛t t❤❡② ♣r♦❞✉❝❡ ♦♥ ❛✈❡r❛❣❡ ♦✈❡r ❛❧❧ t❤❡ ♣❛t❤s t❤❡ s❛♠❡ ❛♠♦✉♥t ♥❛♠❡❧② 1✳ ❚❤✐s ❡q✉❛❧✐t② ❛t t✐♠❡ t ❝❛♥ ❜❡ ✇r✐tt❡♥✿ 0 B(t, T ) = EQ exp t T −rs ds |Ft t ■♥ t❤❡ ❝❛s❡ ♦❢ ❛ ❞❡t❡r♠✐♥✐st✐❝ r❛t❡ rs ✱ ❛s B(T, T ) = 1✿ T −rs ds B(t, T ) = exp t ✷ ■♥ ❛ ✜♥❛♥❝✐❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ t❤❡ ✜❧tr❛t✐♦♥{Ft ; t ≥ 0} r❡♣r❡s❡♥ts t❤❡ str✉❝t✉r❡ ♦❢ ❛❧❧ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❦♥♦✇♥ ❜② ❡✈❡r② ♠❛r❦❡t ❛❣❡♥t✳ ✺ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❆♥❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ❝♦♥st❛♥t ❞❡t❡r♠✐♥✐st✐❝ r❛t❡ r ❝♦♠♣♦✉♥❞ n✲t✐♠❡s ♣❡r ②❡❛r✿ B(t, T ) = 1 (1 + nr )(T −t) ✭✶✳✷✮ ✶✳✶✳✹ ❋♦r✇❛r❞ ■♥t❡r❡st r❛t❡s ❲❡ ❝❛♥ ❞❡✜♥❡ ❋♦r✇❛r❞ ■♥t❡r❡st ❘❛t❡s ❢♦r ❛❧❧ t❤❡ ♣r❡✈✐♦✉s r❛t❡s ✇❡ s❛✇✿ ❼ Bt (T, T + δ) ✐s t❤❡ ❢♦r✇❛r❞ ✈❛❧✉❡ ❛t t ♦❢ ❛ ❩✳❈✳ ✐♥✈❡st❡❞ ❛t T ✇❤✐❝❤ ✇✐❧❧ ♣❛② ✶ ❛t T + δ ✳ ❇② ❛r❜✐tr❛❣❡ ✇❡ ❦♥♦✇ ✐t ✐s ✇♦rt❤✿ Bt (T, T + δ) = B(t, T + δ) B(t, T ) ❼ ❚❤❡ ❡q✉✐✈❛❧❡♥t r❛t❡ s✐♠♣❧② ❝♦♠♣♦✉♥❞❡❞ t♦ t❤✐s ❩❡r♦ ❈♦✉♣♦♥ ❇♦♥❞ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✇r✐t✐♥❣✿ Fδ (t, T ) = 1 δ B(t, T ) −1 B(t, T + δ) ✭✶✳✸✮ ❚❤✐s r❛t❡ ✐s ♥❛♠❡❞ t❤❡ ❋♦r✇❛r❞ ❘❛t❡ ❛♥❞ ✐s t❤❡ ❝♦♥st❛♥t r❛t❡ s✐♠♣❧② ❝♦♠✲ ♣♦✉♥❞❡❞ t♦ ❜❡ ♣❛✐❞ ✐❢ ②♦✉ ✇❛♥t t♦ ❜♦rr♦✇ ♠♦♥❡② ❛t t✐♠❡ t ❢♦r ❛ ❢✉t✉r❡ t✐♠❡ ♣❡r✐♦❞ ❜❡t✇❡❡♥ T ❛♥❞ T + δ ✳ ❲❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ f (t, T ) t❤❡ ✐♥st❛♥t❛♥❡♦✉s ❢♦r✇❛r❞ ✐♥t❡r❡st r❛t❡✱ t❤❡ ❢♦r✲ ✇❛r❞ ✈❡rs✐♦♥ ♦❢ rt ✳ ❋♦r♠❛❧❧②✱ f (t, T ) ✐s t❤❡ ❢♦r✇❛r❞ r❛t❡ ❛t t ♦♥❡ ❝❛♥ ❜♦rr♦✇ ✐♥ ❛ r✐s❦ ❢r❡❡ ❧♦❛♥ ❜❡❣✐♥♥✐♥❣ ❛t T ♦✈❡r t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ♣❡r✐♦❞ dt✳ ❚❤✐s ❝♦♥✲ ❝❡♣t ✐s r❛t❤❡r ❛ ♠❛t❤❡♠❛t✐❝❛❧ ✐❞❡❛❧✐③❛t✐♦♥ ❛s ✐t ❝❛♥ ♥♦t ❜❡ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ♠❛r❦❡t ❜✉t ✐s ✉s❡❢✉❧ t♦ ❞❡s❝r✐❜❡ ❜♦♥❞ ♣r✐❝❡ ♠♦❞❡❧s✳ ❖♥❡ ❝❛♥ ✇r✐t❡✿ T B(t, T ) = exp − f (t, u)du , t ∀t ∈ [0, T ] ✭✶✳✹✮ ✻ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✶✳✶✳✺ ▲■❇❖❘ ✐♥t❡r❡st r❛t❡ ❛♥❞ s✇❛♣s ▲✐❜♦r ✐♥t❡r❡st r❛t❡ ❉✉r✐♥❣ t❤❡ ✽✵✬s✱ ▲✐❜♦r ✭✇❤✐❝❤ st❛♥❞s ❢♦r ▲♦♥❞♦♥ ■♥t❡r ❇❛♥❦ ❖✛❡r❡❞ ❘❛t❡s✮ ✐♥t❡r❡st r❛t❡s ❤❛✈❡ ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ tr❛❞❡❞✳ ❚❤✐s r❛t❡ ✐s ❞❡❝❧✐♥❡❞ ❢♦r ❞✐✛❡r❡♥t s❤♦rt ♠❛t✉r✐t✐❡s ✭✐♥❢❡r✐♦r t♦ ♦♥❡ ②❡❛r✮ ❛♥❞ ✐s ❛ ❜❡♥❝❤♠❛r❦ ♦❢ t❤❡ ♠❛✐♥ ❜❛♥❦s ♦❢ t❤❡✐r ❧♦❛♥ r❛t❡ ❢♦r t❤♦s❡ ♠❛t✉r✐t✐❡s✳ ■t ✐s ✜①❡❞ ❡✈❡r②❞❛② ❛t ✶✶❤✵✵ ❛♠✱ ▲♦♥❞♦♥ ❚✐♠❡✳ ■t ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ ❣❡♥❡r❛❧ ❛s t❤❡ r✐s❦✲❢r❡❡ ✐♥t❡r❡st r❛t❡ ❜② t❤❡ ✐♥✈❡st♦rs✿ ❡✈❡♥ ❝r❡❞✐t ❞❡❢❛✉❧t s✇❛♣s ✈❛❧✉❡s ❛r❡ ❣✐✈❡♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲■❇❖❘ ❝✉r✈❡✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ♥♦t tr✉❡✱ t❤♦s❡ ✜♥❛♥❝✐❛❧ ✐♥st✐t✉t✐♦♥s ❤❛✈❡ ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞❡❢❛✉❧t ❛♥❞ ❤❡♥❝❡ t❤✐s ❞❡❢❛✉❧t r✐s❦ ✐s q✉❛♥t✐✜❡❞✳ ■♥ t❤❡ ♠❛r❦❡ts✱ t❤❡ r✐s❦ ❢r❡❡ ❞♦❡s ♥♦t r❡❛❧❧② ❡①✐st ❜✉t ✐t ❝❛♥ ❜❡ ❛ss✉♠❡❞ t❤❛t t❤❡ ♠❛✐♥ ❝❡♥tr❛❧ ❜❛♥❦s ✭▼♦r❡ s♣❡❝✐✜❝❛❧❧②✿ ❯❙ ❋❡❞✱ ❊❈❇✱ ❈❇❊✮ ❤❛✈❡ ❛♥ ❛❧♠♦st ♥✐❧ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞❡❢❛✉❧t ❛s t❤❡② ❝❛♥ ❧✐t❡r❛❧❧② ♣r✐♥t t❤❡✐r ♠♦♥❡② ❛♥❞ ❤❡♥❝❡ t❤❡ ❜♦♥❞s t❤❡② ✐ss✉❡ ❝❛❧❧❡❞ tr❡❛s✉r✐❡s ❤❛✈❡ ❛❧♠♦st ♥♦ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞❡❢❛✉❧t✸ ✳ ❚❤❡ s♣r❡❛❞ ❜❡t✇❡❡♥ t❤❡ ▲■❇❖❘ ❛♥❞ t❤❡ tr❡❛s✉r② r❛t❡ r❡♣r❡s❡♥ts t❤✐s r✐s❦ t♦ ❞❡❢❛✉❧t✳ ❋♦r t❤❡ ❯❙❉ ▼❛r❦❡t✱ ▲■❇❖❘ r❛t❡s tr❛❞❡ ❛r♦✉♥❞ 50 ❜❛s✐s ♣♦✐♥ts ❛❜♦✈❡ tr❡❛s✉r② r❛t❡s✳ ❲❡ ❝❛❧❧ Lδ (t, t)✱ t❤❡ ▲■❇❖❘ ■♥t❡r❡st r❛t❡ ❛t t✐♠❡ t ❢♦r ❛ ♠❛t✉r✐t② ♦❢ δ ✿ 1 = B(t, t + δ) 1 + δLδ (t, t) ✭✶✳✺✮ ✇✐t❤ δ ✐s t❤r❡❡ ♦r s✐① ♠♦♥t❤s ✉s✉❛❧❧②✳ ❯s✐♥❣ t❤❡ ❛r❜✐tr❛❣❡ ❢r❡❡ r✉❧❡ ❛♥❞ ❛♣♣❧②✐♥❣ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ❛❜♦✉t ❋♦r✇❛r❞ ■♥t❡r❡st r❛t❡s t♦ ▲✐❜♦r ■♥t❡r❡st ❘❛t❡s ❛♥❞ t❤❡✐r ❋♦r✇❛r❞s Lδ (t, T ) t❤❡ ▲✐❜♦r r❛t❡ ❛t t✐♠❡ t ❛t ✇❤✐❝❤ ♦♥❡ ❝❛♥ ❜♦rr♦✇ ♠♦♥❡② ❛t t✐♠❡ T ❢♦r ❛ ♠❛t✉r✐t② ♦❢ δ ✇❡ ❝❛♥ ✇r✐t❡✿ 1 B(t, T + δ) = 1 + δLδ (t, T ) B(t, T ) ✸ ■t s❤♦✉❧❞ ❜❡ ❡♠♣❤❛s✐③❡❞ t❤❛t t❤❡ s♦✈❡r❡✐❣♥ r✐s❦ ✐s r❡❛❧✿ ✐♥ ❏✉❧② ✶✾✾✽✱ ❘✉ss✐❛ ❞❡❢❛✉❧t❡❞ ♦♥ ✐ts ❜♦♥❞s ❝❛✉s✐♥❣ t❤❡ ❢❛❧❧ ♦❢ t❤❡ ❢❛♠♦✉s ❤❡❞❣❡✲❢✉♥❞ ▲❚❈▼✳ ✼ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❚❤❛t ✐s✱ Lδ (t, T ) = B(t, T ) − B(t, T + δ) δB(t, T + δ) ✭✶✳✻✮ ❲❡ ✇✐❧❧ s❦✐♣ t❤❡ ✐♥❞❡① δ ✇❤❡♥ t❤❡r❡ ✇✐❧❧ ❜❡ ♥♦ ❛♠❜✐❣✉✐t✐❡s ❛❜♦✉t t❤❡ ♠❛t✉✲ r✐t②✳ ❙✇❛♣ r❛t❡ ❚❤❡ ✜rst s✇❛♣ ❝♦♥tr❛❝ts ✇❡r❡ ❛❧s♦ ♥❡❣♦t✐❛t❡❞ ✐♥ t❤❡ ❡❛r❧② ✶✾✽✵s✳ ❙✐♥❝❡✱ ✐t ❤❛s s❤♦✇♥ ❛♥ ❛♠❛③✐♥❣ ❣r♦✇t❤ ❜❡❝♦♠✐♥❣ ♠♦r❡ ❛♥❞ ♠♦r❡ ✐♠♣♦rt❛♥t ✐♥ t❤❡ ❡①♦t✐❝ ❞❡r✐✈❛t✐✈❡s ♠❛r❦❡t✳ ❆ s✇❛♣ ✐s ❛ ❝♦♥tr❛❝t ❜❡t✇❡❡♥ t✇♦ ❝♦♠♣❛♥✐❡s t♦ ❡①❝❤❛♥❣❡ ❛ ♣r❡❞❡✜♥❡❞ ❝❛s❤ ✢♦✇ ✐♥ t❤❡ ❢✉t✉r❡✳ ❚❤❡ s❝❤❡❞✉❧❡ ♦❢ t❤❡ ❝❛s❤ ✢♦✇s ❛♥❞ t❤❡ ✇❛② t❤❡② ❛r❡ ❝❛❧❝✉❧❛t❡❞ ✐s s♣❡❝✐✜❡❞ ✐♥ t❤✐s ❛❣r❡❡♠❡♥t✳ ❆t t❤❡ ❜❡❣✐♥♥✐♥❣✱ s✇❛♣s ✇❡r❡ t❛✐❧♦r❡❞ ❢♦r ❝♦♠♣❛♥✐❡s ✇❤♦ ✇❛♥t❡❞ t♦ ❤❡❞❣❡ t❤❡✐r ❧♦❛♥s ❡①♣♦s✉r❡ ❛♥❞ ❧♦❝❦ ✐♥ ❛ ❣♦♦❞ ❧❡✈❡❧ ♦❢ ✐♥t❡r❡st r❛t❡✳ ❍❡♥❝❡ ♦♥❡ ❝❛♥ ❞❡❝✐❞❡ t♦ ❡♥t❡r ❛ s✇❛♣ ✇❤❡r❡ ❤❡ ✇✐❧❧ ❡①❝❤❛♥❣❡ ❤✐s s❡♠✐✲❛♥♥✉❛❧ ✜①❡❞ r❛t❡s ❝❛s❤✲✢♦✇s ❛t x% ❛❣❛✐♥st ❛ ✢♦❛t✐♥❣ r❛t❡✱ ❢♦r ✐♥st❛♥❝❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✻✲♠♦♥t❤s ▲■❇❖❘ r❛t❡ ✇✐t❤ ✜①✐♥❣ ❞❛t❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✻✲♠♦♥t❤s ♣❡r✐♦❞ ✭❋✐①✐♥❣ ✐♥ ❛❞✈❛♥❝❡ ✹ ✮ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❋✐❣✉r❡ ✶✳✷ ❡①♣❧❛✐♥s ❤♦✇ ✐s ❜✉✐❧t t❤❡ ❡①❝❤❛♥❣❡ ♦❢ ❝❛s❤✲✢♦✇s ❢r♦♠ t❤❡ ❝✉st♦♠❡r ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❚❤✐s t②♣❡ ♦❢ ❋✐❣✉r❡ ✶✳✷✿ ❊①❝❤❛♥❣❡ ♦❢ ❝❛s❤✲✢♦✇s ❢♦r ❛ P❛②❡r ❙✇❛♣ ✹ ❙❡✈❡r❛❧ ✐ss✉❡s ❛r❡ ♥♦t ♠❡♥t✐♦♥❡❞ ❤❡r❡ ❛❜♦✉t t❤❡ ✜①✐♥❣ ❞❛t❡s ❛♥❞ t❤❡ ❝♦♥✈❡①✐t② ❛❞✲ ❥✉st♠❡♥t t❤❛t ❛r❡ ♥❡❝❡ss❛r② ✇❤❡♥ ♣r✐❝✐♥❣ ♥♦♥ ♣❡r❢❡❝t❧② s❝❤❡❞✉❧❡❞ str✉❝t✉r❡ ♦r ✐♥ ❛rr❡❛rs ✜①✐♥❣ str✉❝t✉r❡s✱ ❢♦r ✐♥st❛♥❝❡ s❡❡ ❬✸❪ ✽ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s s✇❛♣ ✐s ❝❛❧❧❡❞ ✐s ❝❛❧❧❡❞ ❛ ♣❛②❡r s✇❛♣✳ ❚❤❡ s②♠♠❡tr✐❝ ✈❡rs✐♦♥ ✐s ❝❛❧❧❡❞ r❡❝❡✐✈❡r s✇❛♣✳ ❆s ❛ ♠❛tt❡r ♦❢ ❢❛❝t✱ ❢r♦♠ t❤✐s ❞❡✜♥✐t✐♦♥ ❛♣♣❡❛rs t❤❡ s✇❛♣ r❛t❡ Sp,n (t)❞❡✜♥❡❞ ❛s t❤❡ r❛t❡ ✇❤✐❝❤ ❣✐✈❡s ❛ ♥❡t ♣r❡s❡♥t ✈❛❧✉❡ ♦❢ 0 ❛t t✐♠❡ t t♦ t❤❡ s✇❛♣ ✇❤✐❝❤ ❡①❝❤❛♥❣❡ t❤✐s s✇❛♣ r❛t❡ ❛❣❛✐♥st ❛ ✢♦❛t✐♥❣ ♦♥❡ ✭δ ✲♠♦♥t❤s ▲✐❜♦r Lδ (t, Ti )✮ ♦♥ ❛ s❝❤❡❞✉❧❡ Ti , i = p, . . . , n✳ ❲❡ ❝❛♥ ❝♦♠♣✉t❡ t❤✐s s✇❛♣ r❛t❡ Sp,n (t) ❜② ❛r❜✐tr❛❣❡ ❝♦♥s✐❞❡r❛t✐♦♥s ❛♥❞✱ ✐t ✐s ✇♦rt❤ ♥♦t✐❝✐♥❣ ✐t✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ ❛♥② ♠♦❞❡❧ ❛ss✉♠♣t✐♦♥✳ ❚❤❡ ✜①❡❞ ❧❡❣ ✐s✿ n−p F ixedp,n (t) = Sp,n (t)δB(t, Tp+i ) i=0 ❆♥❞ t❤❡ ✢♦❛t✐♥❣ ❧❡❣ ✐s✿ n−p F loatingp,n (t) = B(t, Ti+p )δL(t, Ti−1+p ) i=1 n−p = B(t, Ti+p ) i=1 n−p B(t, Ti−1+p ) −1 B(t, Ti+p ) B(t, Ti−1+p ) − B(t, Ti+p ) = i=1 = B(t, Tp ) − B(t, Tn ) ❚❤❡ s✇❛♣ r❛t❡ ✐s ❜② ❞❡✜♥✐t✐♦♥ t❤❡ ♦♥❡ t❤❛t ❡q✉❛❧✐③❡ ❜♦t❤ ❧❡❣s✿ F ixedp,n (t) = F loatingp,n (t) Sp,n (t) = B(t, Tp ) − B(t, Tn ) n−p i=0 δB(t, Tp+i ) ❚❤✐s s✇❛♣ ✇❛s ♠♦r❡ ♣r❡❝✐s❡❧② ❛ ❢♦r✇❛r❞ st❛rt ✐♥t❡r❡st r❛t❡ s✇❛♣ ✇❤✐❝❤ ✜rst s❡tt❧❡♠❡♥t ❞❛t❡ ✐s Tp ✳ ❖♥❝❡ t❤✐s ♣r♦❞✉❝t ✇❛s ✇❡❧❧ ✉♥❞❡rst♦♦❞ ❜② ❡✈❡r② ♦♥❡ ♦♥ t❤❡ ♠❛r❦❡ts✱ ✐t ♥❛t✉r❛❧❧② ❣❛✈❡ r✐s❡ t♦ ✐ts ✜rst ♠♦st ♥❛t✉r❛❧ ❞❡r✐✈❛t✐✈❡✿ ✾ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s t❤❡ ❊✉r♦♣❡❛♥ s✇❛♣t✐♦♥ ✺ ✳ ❆ ❊✉r♦♣❡❛♥ s✇❛♣t✐♦♥ ✐s ❛ ♦♥❡✲t✐♠❡ ♦♣t✐♦♥ ♦♥ ❛ s✇❛♣ r❛t❡✳ ❋r♦♠ ♥♦✇✱ ✇❡ ✇✐❧❧ ❛❧✇❛②s r❡❢❡r t♦ ❊✉r♦♣❡❛♥ s✇❛♣t✐♦♥s ✇❤❡♥ ✇❡ ❞❡s❝r✐❜❡ s✇❛♣t✐♦♥s✳ ❲❤❡♥ ♦♥❡ ✐s ❧♦♥❣ ❛ s✇❛♣t✐♦♥ str✐❦❡ Sp,n ✱ ❤❡ ♦✇♥s t❤❡ r✐❣❤t ❛♥❞ ♥♦t t❤❡ ♦❜❧✐❣❛t✐♦♥ t♦ ❡♥t❡r ❛ s✇❛♣ ♦❢ t❡♥♦r Tn ❛t ♠❛t✉r✐t② Tp ✳ ❆ s✇❛♣t✐♦♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ t❤r♦✉❣❤ ❞✐✛❡r❡♥t ♠❡t❤♦❞s ❜✉t t❤❡ ♠❛r❦❡t ✐♥ ❣❡♥❡r❛❧ q✉♦t❡s t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ s✇❛♣t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❇❧❛❝❦ ❢♦r♠✉❧❛ ✭❙❡❡ s❡❝t✐♦♥ ✶✳✸✳✷✮✳ ❖♥ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ s✐❞❡ t❤✐s ❛r✐s❡ ✐ss✉❡s ❛s ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t s✇❛♣ r❛t❡s ❛♥❞ ❢♦r✇❛r❞ r❛t❡s ❝❛♥ ♥♦t ❜❡ ❧♦❣ ♥♦r♠❛❧ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss ❧❛t❡r t❤✐s ♣♦✐♥t ✐♥ s❡❝t✐♦♥ ✷✳✹✳ ✶✳✶✳✻ ❙t♦❝❤❛st✐❝ t♦♦❧s ❚❤✐s s✉❜s❡❝t✐♦♥ ✐s ❣♦✐♥❣ t♦ ♣r❡s❡♥t ❛ ❢❡✇ st♦❝❤❛st✐❝ t♦♦❧s ✇❡ ♥❡❡❞ t♦ ❞❡s❝r✐❜❡ t❤❡ ❜❛s✐❝s ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✳ ❚❤✐s s✉❜s❡❝t✐♦♥ ❞♦❡s ♥♦t s❡❡❦ t♦ ❜❡ ❡①❤❛✉st✐✈❡ ❛♥❞ t♦t❛❧❧② r✐❣♦r♦✉s ✐♥ st♦❝❤❛st✐❝ ❝❛❧❝✉❧✉s ❜✉t ❥✉st t♦ ❣✐✈❡ ❛ ❣❡♥❡r❛❧ ✐❞❡❛ ❛❜♦✉t t❤❡ t♦♦❧s ✇❡ ✇✐❧❧ ❜❡ ✉s✐♥❣ ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧s ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❋♦r ❢✉rt❤❡r ❞❡t❛✐❧s ❛❜♦✉t st♦❝❤❛st✐❝ ❝❛❧❝✉❧✉s ♣❧❡❛s❡ r❡❢❡r t♦ t❤❡ ❡①❝❡❧❧❡♥t ❬✺❪✳ ◆✉♠❡r❛✐r❡ ❆ ◆✉♠❡r❛✐r❡ ✐s ❛ ♣r✐❝❡ ♣r♦❝❡ss (A(t))T ✭❛ ♣r♦❝❡ss ✐s ❛ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✮✱ ✇❤✐❝❤ ✐s str✐❝t❧② ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ t ∈ [O, T ]✳ ◆✉♠❡r❛✐r❡s ❛r❡ ✉s❡❞ t♦ ❡①♣r❡ss ♣r✐❝❡s ✐♥ ♦r❞❡r t♦ ❤❛✈❡ r❡❧❛t✐✈❡ ♣r✐❝❡s✳ ❚❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤✐s r❛t❤❡r ❛❜str❛❝t ❝♦♥❝❡♣t ❝❛♥ ❜❡ s❡❡♥ ✐♥ ✇❤❛t ❢♦❧❧♦✇s✳ ❈❤❛♥❣❡ ♦❢ ♥✉♠❡r❛✐r❡ ▲❡t P ❛♥❞ Q ❜❡ ❡q✉✐✈❛❧❡♥t ♠❡❛s✉r❡s✻ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥✉♠❡r❛✐r❡s A(T ) ❛♥❞ B(t)✳ ❚❤❡ ❘❛❞♦♥✲◆✐❦♦❞②♠ ❞❡r✐✈❛t✐✈❡ t❤❛t ❝❤❛♥❣❡s t❤❡ ❡q✉✐✈❛❧❡♥t ♠❡❛✲ ✺ ❆♠❡r✐❝❛♥ ❛♥❞ ❇❡r♠✉❞❡❛♥ s✇❛♣t✐♦♥ ❛❧s♦ ❡①✐st ❜✉t ❛r❡ ♥♦t ❛s ❧✐q✉✐❞ ❛♥❞ ❛s ✈❛♥✐❧❧❛ t❤❛♥ ❊✉r♦♣❡❛♥ ✻ P ❛♥❞ Q ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✿ P(M ) = 0 ↔ Q(M ) = 0, ∀M ∈ F ✶✵ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s s✉r❡ P ✐♥ Q ✐s ❣✐✈❡♥ ❜②✿ R= dP A(T )B(t) = dQ A(t)B(T ) ✭✶✳✼✮ ❚❤✐s ❞❡r✐✈❛t✐✈❡ ✐s ✈❡r② ✉s❡❢✉❧✿ ❞✉❡ t♦ t❤❡ ♥♦ ❛r❜✐tr❛❣❡ r✉❧❡ t❤❡ ♣r✐❝❡ ♦❢ ❛♥ ❛ss❡t X s❤♦✉❧❞ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ♠❡❛s✉r❡ ❛♥❞ ♥✉♠❡r❛✐r❡✿ A(t)EP ■❢ ♦♥❡ ✐♥tr♦❞✉❝❡s✿ G(T ) = X(T ) X(T ) |Ft = B(t)EQ |Ft A(T ) B(T ) X(T ) A(T ) ❛♥❞ ❞♦✐♥❣ s♦♠❡ s✐♠♣❧❡ ♠❛♥✐♣✉❧❛t✐♦♥ ♦♥ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥✿ EP (G(T )|Ft ) = EQ G(T ) A(T )B(t) |Ft A(t)B(T ) = EQ (G(T )R|Ft ) ❲❡ ❝❛♥ s❡❡ t❤❛t ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ❥✉st ❜② ♠✉❧t✐♣❧②✐♥❣ t❤❡ ♠❛rt✐♥❣❛❧❡ ❜② ✐ts ❘❛❞♦♥✲◆✐❦♦❞②♠ ❞❡r✐✈❛t✐✈❡✳ ●✐rs❛♥♦✈ t❤❡♦r❡♠ ❋♦r ❛♥② ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss k(t) ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✲ ❞✐t✐♦♥✿ 1 E e2 t 0 k2 (s)ds < +∞, ❈♦♥s✐❞❡r t❤❡ ❘❛❞♦♥✲◆✐❦♦❞②♠ ❞❡r✐✈❛t✐✈❡ R = t k(s)dW (s) − R = exp 0 1 2 dP dQ ❣✐✈❡♥ ❜②✿ t k 2 (s)ds , 0 ✇❤❡r❡ W ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✉♥❞❡r t❤❡ ♠❡❛s✉r❡ Q✳ ❯♥❞❡r t❤❡ ♠❡❛s✉r❡ P t❤❡ ♣r♦❝❡ss t W P (t) = W (t) − k(s)ds, 0 ✶✶ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡ ♠❛✐♥ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ●✐rs❛♥♦✈ t❤❡♦r❡♠ ✐s t❤❛t ✇❤❡♥ ♦♥❡ ❝❤❛♥❣❡s ♠❡❛s✉r❡s t❤❡ ❞r✐❢t ❝♦♠♣♦♥❡♥t ✐s ✐♠♣❛❝t❡❞ ❜✉t t❤❡ ✈♦❧❛t✐❧✐t② ❝♦♠♣♦♥❡♥t r❡✲ ♠❛✐♥s ✉♥❛✛❡❝t❡❞✳ ❖♥❡ ❝❛♥ s❛② t❤❛t s✇✐t❝❤✐♥❣ ❢r♦♠ ♦♥❡ ♠❡❛s✉r❡ t♦ ❛♥♦t❤❡r ❥✉st ❝❤❛♥❣❡s t❤❡ r❡❧❛t✐✈❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ♣❛t❤ ❜❡✐♥❣ ❝❤♦s❡♥✳ ❋♦r ❡①❛♠♣❧❡ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ W (t) ❛❜♦✈❡ ♠✐❣❤t ❢♦❧❧♦✇ ❛ ♣❛t❤ ✇❤✐❝❤ ❞r✐❢ts ❞♦✇♥✇❛r❞ ❛t ❛ r❛t❡ ♦❢ ❛❜♦✉t −k ❜✉t ✉♥❞❡r t❤❡ ♠❡❛s✉r❡ P ✐t ✐s ♠♦r❡ ❧✐❦❡❧② t♦ ❞r✐❢t t♦ 0✳ ❚❤❡ ❣❡♥❡r❛❧ ♣✉r♣♦s❡ ♦❢ t❤✐s t❤❡♦r❡♠ ✐s t♦ ❣❡t r✐❞ ♦❢ t❤❡ ❞r✐❢t✳ ❋♦r ♣r♦♦❢ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ ♣❧❡❛s❡ ❝♦♥s✐❞❡r ❬✺❪✱ ♣❛❣❡ ✶✺✸✲✶✺✼✳ ❊q✉✐✈❛❧❡♥t ▼❛rt✐♥❣❛❧❡ ▼❡❛s✉r❡ ❆♥ ❊q✉✐✈❛❧❡♥t ▼❛rt✐♥❣❛❧❡ ▼❡❛s✉r❡ ✭❊▼▼✮ Q ✐s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ t❤❡ s♣❛❝❡ (Ω, F) s✉❝❤ t❤❛t✿ ❼ Q ❛♥❞ Q0 ❛r❡ ❡q✉✐✈❛❧❡♥t ❼ ❚❤❡ ❘❛❞♦♥✲◆②❦♦❞②♠ ❞❡r✐✈❛t✐✈❡ R = ❼ ❚❤❡ ♣r♦❝❡ss W Q (t) = W Q0 (t) − dQ0 dQ t 0 k(s)ds ✐s ♣♦s✐t✐✈❡ ✐s ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ Q✳ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ ❆ss❡t Pr✐❝✐♥❣ ❆❧❧ t❤❡s❡ ❞❡✜♥✐t✐♦♥s ❧❡❞ ✉s t♦ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠✳✼ ✿ ❆ ♠❛r❦❡t ❤❛s ♥♦✲❛r❜✐tr❛❣❡ ♦♣♣♦rt✉♥✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❊▼▼✳ ❆ ♠❛r❦❡t ✐s ❝♦♠♣❧❡t❡ ✭❆❧❧ ❝♦♥t✐♥❣❡♥t ❝❧❛✐♠s ❝❛♥ ❜❡ r❡♣❧✐❝❛t❡❞ ✉s✐♥❣ ❛❞♠✐ss✐❜❧❡ ♣♦rt❢♦❧✐♦✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❊▼▼✳ ❋♦r✇❛r❞ ♠❡❛s✉r❡ ❲❡ ♥❛♠❡ ❋♦r✇❛r❞ ♠❡❛s✉r❡✱ Pi ✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ✇✐t❤ ❛s ♥✉♠❡r❛✐r❡ ✼ ❚❤✐s t❤❡♦r❡♠ ✐s ✈❡r② ✇❡❧❧ ♣r♦✈❡❞ ❛♥❞ ❞❡s❝r✐❜❡❞ ✐♥ ❬✺❪ ✶✷ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s t❤❡ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ♠❛t✉r✐♥❣ ❛t Ti ✱ ♥❛♠❡❧② B(t, Ti )✳ ❯♥❞❡r t❤✐s ♠❡❛s✉r❡✱ X(t) B(t, Ti ) ✐s ❛ ♠❛rt✐♥❣❛❧❡ ❢♦r ❛❧❧ ❝♦♥t✐♥❣❡♥t ❝❧❛✐♠ X(t) ❛♥❞ ✇❡ ❝❛♥ ♣r✐❝❡ ✐t s❛②✐♥❣✿ X(t) = B(t, Ti )Ei [X(Ti )|Ft ] ❙♣♦t ♠❡❛s✉r❡ ❯s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❏❛♠s❤✐❞✐❛♥ ✐♥ ❬✶✸❪ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ s♣♦t ♠❡❛s✉r❡✳ ❈♦♥s✐❞❡r ❛ ♣♦rt❢♦❧✐♦ ♦❢ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ❝r❡❛t❡❞ ❜② t❤❡ ✐♥✈❡st♠❡♥t str❛t❡❣② ❢♦❧❧♦✇✐♥❣✿ ❼ ❆t t = 0✱ ✇❡ ✐♥✈❡st ✶ ❜✉②✐♥❣ ❼ ❆t t = T1 ✱ ✇❡ r❡❝❡✐✈❡ 1 B(0,T1 ) ❩❡r♦ 1 B(0,T1 ) ❝♦✉♣♦♥ ♠❛t✉r✐♥❣ ❛t T1 ❛♥❞ ✇❡ ❜✉② 1 1 B(0,T1 ) B(0,T2 ) ❩❡r♦ ❝♦✉♣♦♥ ♠❛t✉r✐♥❣ ❛t T2 ❼ ❆t t = T2 ✱ ✇❡ r❡❝❡✐✈❡ 1 1 B(0,T1 ) B(0,T2 ) ❛♥❞ ✇❡ ❜✉② 1 1 1 B(0,T1 ) B(0,T2 ) B(0,T3 ) ❩❡r♦ ❝♦✉♣♦♥ ♠❛t✉r✐♥❣ ❛t T3 ❼ ... ❍❡♥❝❡✱ ❛t ❡✈❡r② t✱ ♦♥❡ ❤♦❧❞ ❛ ♣♦rt❢♦❧✐♦ ♦❢ t ) j=1 1 B(Tj−1 ,Tj ) ✭✇❤❡r❡ t ✐s t❤❡ ♥❡①t ❞❛t❡ ✐♥ t❤❡ t❡♥♦r✮✳ ❚❤✐s ♣♦rt❢♦❧✐♦ ❝❛♥ ❜❡ ❝❤♦s❡♥ ❛s ❛ ♥✉♠❡r❛✐r❡ ❢♦r ❛ ❝❡rt❛✐♥ ♠❡❛s✉r❡ t❤❛t ✇❡ ✇✐❧❧ ❝❛❧❧ t❤❡ s♣♦t ♠❡❛s✉r❡ ♥♦t❡❞ P∗ ✳ ✶✳✷ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❙✐♥❝❡ t❤❡② ❤❛✈❡ ❜❡❡♥ ♠♦r❡ ❛♥❞ ♠♦r❡ ✉s❡❞ s❡✈❡r❛❧ ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ ❞❡s❝r✐❜❡ ✐♥t❡r❡st r❛t❡s ✉s✐♥❣ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s✳ ❚❤✐s ♣❛rt ✇✐❧❧ ❞❡s❝r✐❜❡ t❤❡ t✇♦ ♠♦❞❡❧s✱ t❤❡ ♠♦st ✉s❡❞ ✐♥❝❧✉❞✐♥❣ ❛t t❤❡ ❘♦②❛❧ ❇❛♥❦ ♦❢ ❙❝♦t❧❛♥❞✳ ❋♦r ❢✉rt❤❡r ❞❡t❛✐❧s ♦♥❡ ❝❛♥ r❡❢❡r t♦ ❬✹❪ ❛ ✈❡r② ❞❡t❛✐❧❡❞ r❡✈✐❡✇ ❜② ❘❡❜♦♥❛t♦ ♦❢ ❤♦✇ t❤❡s❡ ♠♦❞❡❧s ✇❡r❡ ❜✉✐❧t ❛♥❞ ❤♦✇ ❞✐❞ ✇❡ ❣❡t t❤❡r❡✳ ✶✸ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✶✳✷✳✶ ❙❤♦rt t❡r♠ ✐♥t❡r❡st r❛t❡s ❚❤❡ ✜rst ❣❡♥❡r❛t✐♦♥ ♦❢ ♠♦❞❡❧s t♦ ♣r✐❝❡ ■♥t❡r❡st ❘❛t❡s str✉❝t✉r❡❞ ♣r♦❞✉❝ts ✇❡r❡ ❞❡✈❡❧♦♣❡❞ ✐♥ t❤❡ ❡❛r❧② ✽✵✬s✳ ❙✐♥❝❡✱ ♥✉♠❡r♦✉s ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ ❝r❡❛t❡❞ ❛♥❞ ✇❡ ✇✐❧❧ ♥♦t ❞❡s❝r✐❜❡ ❛❧❧ ♦❢ t❤❡♠ ❛s t❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣❛rt ✐s t♦ s❤♦✇ ❤♦✇ ✐s ❜✉✐❧t t❤❡ ♥❡①t ❣❡♥❡r❛t✐♦♥ ♦❢ ♠♦❞❡❧s✳ ❋♦r ❡♥r✐❝❤♠❡♥t ♣✉r♣♦s❡ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r ♦t❤❡r ✐♠♣♦rt❛♥t s❤♦rt t❡r♠ str✉❝✲ t✉r❡ ♠♦❞❡❧s✱ ✐♥❝❧✉❞✐♥❣ ❈♦①✱ ■♥❣❡rs♦❧❧ ❛♥❞ ❘♦ss ▼♦❞❡❧ ❬✻❪✱ ❍♦✲▲❡❡ ❬✼❪✱ ❇❧❛❝❦✲ ❑❛r❛s✐♥s❦✐ ❬✽❪✱ ❱❛s✐❝❡❦ ❬✾❪✱ ❘❡♥❞❧❡♠❛♥ ❛♥❞ ❇❛rtt❡r❬✶✵❪✳ ❚❤❡ ♠♦st ✉s❡❞ s❤♦rt✲t❡r♠ ✐♥t❡r❡st r❛t❡s ♠♦❞❡❧ ✐♥ t❤❡ ✜♥❛♥❝✐❛❧ ✐♥❞✉str② ✐s t❤❡ ♦♥❡ ❜② ❍✉❧❧ ❛♥❞ ❲❤✐t❡ ✭✇✐t❤ ♦♥❡ ♦r t✇♦ ❢❛❝t♦rs✮✳ ❆❝t✉❛❧❧②✱ t❤✐s ♠♦❞❡❧ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛♥t❡r✐♦r ❱❛s✐❝❡❦ ♠♦❞❡❧ ✭❙❡❡ ❬✾❪✮✳ ❍✉❧❧ ❛♥❞ ❲❤✐t❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ❛ ❱❛s✐❝❡❦ ♠♦❞❡❧ ✇❤✐❝❤ ♠♦❞❡❧s t❤❡ ✐♥st❛♥t❛♥❡♦✉s s❤♦rt✲t❡r♠ ✐♥t❡r❡st r❛t❡ ❛s✿ dr = a(b − r)dt + σdz, a, b, σ constant ✭✶✳✽✮ ▼❡❛♥ ❘❡✈❡rs✐♦♥ ❚❤✐s ♠♦❞❡❧ ✐s ❞❡s❝r✐❜✐♥❣ t❤❡ ♠❡❛♥✲r❡✈❡rs✐♦♥ ♣❤❡♥♦♠❡♥♦♥✿ ✉♥❧✐❦❡ ❛ st♦❝❦✱ ✐♥t❡r❡st r❛t❡s ❛♣♣❡❛r t♦ ❜❡ ♣✉❧❧❡❞ ❜❛❝❦ t♦ s♦♠❡ ❧♦♥❣✲r✉♥ ❛✈❡r❛❣❡ ❧❡✈❡❧ ♦✈❡r t✐♠❡✳ Pr❛❝t✐❝❛❧❧②✱ ✐t ♠❡❛♥s t❤❛t ✇❤❡♥ rt ✐s ❤✐❣❤✱ ♠❡❛♥ r❡✈❡rs✐♦♥ t❡♥❞s t♦ ❝❛✉s❡ ✐t t♦ ❤❛✈❡ ❛ ♥❡❣❛t✐✈❡ ❞r✐❢t❀ ✇❤❡♥ rt ✐s ❧♦✇✱ ♠❡❛♥ r❡✈❡rs✐♦♥ t❡♥❞s t♦ ❝❛✉s❡ ✐t t♦ ❤❛✈❡ ❛ ♣♦s✐t✐✈❡ ❞r✐❢t✳ ❚❤✐s ❢❡❛t✉r❡ ❝❛♥ ❜❡ ❥✉st✐✜❡❞ ❡❝♦♥♦♠✐❝❛❧❧②❀ ❜❛s✐❝❛❧❧②✱ ✇❤❡♥ r❛t❡s ❛r❡ ❤✐❣❤✱ t❤❡ ❡❝♦♥♦♠② t❡♥❞s t♦ s❧♦✇ ❞♦✇♥ ❛♥❞ t❤❡ ❞❡♠❛♥❞ ❢♦r ❢✉♥❞ ❢r♦♠ ❜♦rr♦✇❡r ❞❡❝r❡❛s❡✳ ❍❡♥❝❡✱ r❛t❡s t❡♥❞ t♦ ❣♦ ❞♦✇♥✱ s♦ t❤❡ ❞❡♠❛♥❞ ❢♦r ❢✉♥❞ ❢r♦♠ ❜♦r✲ r♦✇❡rs ✐♥❝r❡❛s❡ ❛♥❞ r❛t❡s t❡♥❞ t♦ ✐♥❝r❡❛s❡✳ ■♥ ❱❛s✐❝❡❦ ♠♦❞❡❧✱ t❤❡ s❤♦rt r❛t❡ t❡♥❞s t♦ ❣♦ t♦ b ❛t ❛ r❛t❡ a✳ ❚❤❡ ✐❞❡❛ ♦❢ ❍✉❧❧ ❛♥❞ ❲❤✐t❡ ✐s t♦ ✉s❡ t❤❡ s❛♠❡ r❛t❡ a ❛♥❞ t❤❡ s❛♠❡ ❝♦♥st❛♥t ✈♦❧❛t✐❧✐t② ❜✉t t♦ ❛❞❞ ❛ t✐♠❡ ❞❡♣❡♥❞❡♥t ❢❡❛t✉r❡ t♦ t❤❡ ♠❡❛♥ ✈❛❧✉❡✿ θ(t) a ✳ ✶✹ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❍✉❧❧✲❲❤✐t❡ ▼♦❞❡❧ ❯s✐♥❣ t❤❡s❡ ❝♦♥s✐❞❡r❛t✐♦♥s✱ t❤❡ ❍✉❧❧✲❲❤✐t❡ ♠♦❞❡❧ ❝♦♥s✐❞❡r t❤❡ ✐♥st❛♥t❛♥❡♦✉s s❤♦rt t❡r♠ ❞②♥❛♠✐❝s ❛s✿ dr = [θ(t) − ar]dt + σdt ✭✶✳✾✮ ✇❤❡r❡ t❤❡ ♣❛r❛♠❡t❡rs ❛r❡ ❛s ❡①♣❧❛✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳ ❚❤❡ θ(t) ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ t❡r♠ str✉❝t✉r❡ ❜② ✉s✐♥❣ ❛ ❝❤❛♥❣❡ ♦❢ ♥✉♠❡r❛✐r❡✳ ❲❡ ❣❡t✿ θ(t) = ∂f (0, t) σ2 + af (0, t) + (1 − e−2at ) ∂t 2a ❆ss✉♠✐♥❣ t❤❛t t❤❡ ❧❛st t❡r♠ ✐s ✈❡r② s♠❛❧❧ ✭✇❤✐❝❤ ✐s tr✉❡ ✐♥ ♣r❛❝t✐❝❡✮✱ t❤✐s ❡q✉❛t✐♦♥ ✐♠♣❧✐❡s t❤❛t t❤❡ s❤♦rt t❡r♠ ✐♥t❡r❡st r❛t❡ rt ❢♦❧❧♦✇s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ✐♥st❛♥t❛♥❡♦✉s ❢♦r✇❛r❞ r❛t❡ ❝✉r✈❡✳ ❲❤❡♥ ✐t ❞❡✈✐❛t❡s ❢r♦♠ t❤✐s ❝✉r✈❡✱ ✐t r❡✈❡rts ❜❛❝❦ t♦ a✱ ❢♦❧❧♦✇✐♥❣ t❤❡ ♠❡❛♥✲r❡✈❡rs✐♦♥ ❢❡❛t✉r❡✳ ❇♦♥❞ ♣r✐❝❡s ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ✉s✐♥❣ ❱❛s✐❝❡❦ ❬✾❪ ✐❞❡❛✳ ❋✐rst✱ ♦♥❡ ❝❛♥ ✇r✐t❡ t❤❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✈❡r✐✜❡❞ ❜② ❛♥② ❝♦♥t✐♥❣❡♥t ❝❧❛✐♠ ❛♥❞ t❤❡♥ ❛♣♣❧② t❤❡ ❜♦✉♥❞❛r✐❡s ❝♦♥❞✐t✐♦♥s t♦ ♦❜t❛✐♥ t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞✳ ❍❡♥❝❡✱ t❤❡ ♣r✐❝❡ B(t, T ) ❛t t✐♠❡ t ♦❢ ❛ ❩✳❈✳ ❜♦♥❞ ♠❛t✉r✐♥❣ ❛t T ❝❛♥ ❜❡ ❣✐✈❡♥ ✉s✐♥❣ ✭✶✳✶✵✮ ✐♥ t❡r♠s ♦❢ t❤❡ s❤♦rt r❛t❡ ❛t t✐♠❡ t ❛♥❞ t❤❡ ♣r✐❝❡s ♦❢ t❤❡ ❩✳❈✳ ❜♦♥❞ t♦❞❛② B(0, T ) ❛♥❞ B(0, t)✳ B(t, T ) = C(t, T ) exp−D(t,T )r(t) ✇❤❡r❡✱ D(t, T ) = ✭✶✳✶✵✮ 1 − e−a(T −t) a ❛♥❞✱ ln C(t, T ) = ln B(0, T ) 1 + B(t, T )F (0, t) − 3 σ 2 (e−aT − eaT )2 (e2at − 1)) B(0, t) 4a ✶✺ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❲✐t❤ t❤❡s❡ ❡q✉❛t✐♦♥s ✇❡ ❤❛✈❡ ❞❡✜♥❡❞ ❡✈❡r②t❤✐♥❣ ✐♥ ♦✉r ♠♦❞❡❧ t♦ ♣r✐❝❡ ❛♥② ❝♦♥t✐♥❣❡♥t ❝❧❛✐♠✳ ❚❤❡ ✐ss✉❡ ❛❜♦✉t t❤✐s ♠♦❞❡❧ ✐s t❤❛t t❤❡ ✉♥❞❡r❧②✐♥❣✱ ♥❛♠❡❧② t❤❡ s❤♦rt✲t❡r♠ ✐♥t❡r❡st r❛t❡ ✐s ♥♦t ❛♥ ♦❜s❡r✈❛❜❧❡ ♦❢ t❤❡ ♠❛r❦❡t✳ ❖♥ t❤❡ ❝♦♥tr❛r②✱ s♦♠❡ ③❡r♦ ❝♦✉♣♦♥ ❜♦♥❞s ❛r❡ tr❛❞❡❞ ✐♥ ❛ ❧✐q✉✐❞ ✇❛② ✐♥ t❤❡ ♠❛r❦❡t ❛♥❞ ❤❡♥❝❡ ❛r❡ ♦❜s❡r✈❛❜❧❡ ♦❢ t❤❡ ♠❛r❦❡t✳ ■t ✇♦✉❧❞ ❜❡ ❡❛s✐❡r t♦ ❤❛✈❡ ❛ ♠♦❞❡❧ t❤❛t ❞❡s❝r✐❜❡s ♦❜s❡r✈❛❜❧❡ ♣r♦❞✉❝ts ❧✐❦❡ ❋♦r✇❛r❞ r❛t❡s✳ ❚❤✐s ✐s t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✳ ✶✳✷✳✷ ❍❡❛t❤ ❏❛rr♦✇ ❛♥❞ ▼♦rt♦♥ ❋r❛♠❡✇♦r❦ ❚❤❡ ♣r❡✈✐♦✉s ❢r❛♠❡✇♦r❦s ✇❡ ❥✉st ❞✐s❝✉ss❡❞ ❛r❡ ❡❛s② t♦ ✐♠♣❧❡♠❡♥t ❛♥❞ ❣✐✈❡✱ ✇❤❡♥ ✉s❡❞ ✇✐t❤ ❝❛✉t✐♦♥✱ ❣♦♦❞ ♣r✐❝❡s ✇✐t❤ r❡s♣❡❝t t♦ ❛❝t✐✈❡❧② tr❛❞❡❞ ✐♥str✉✲ ♠❡♥ts ❧✐❦❡ ❝❛♣s ❛♥❞ ✢♦♦rs✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ❧✐♠✐t❛t✐♦♥s t♦ t❤✐s ❛♣♣r♦❛❝❤✿ t❤❡ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ t✐♠❡ ❛♥❞ ♦♥❡ ❝❛♥ ♥♦t ❛❞❛♣t t❤✐s str✉❝t✉r❡ ✐♥ t❤❡ t✐♠❡ ❛s t❤❡ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ✐♥ t❤❡ ❢✉t✉r❡ ✇✐❧❧ ♣r♦❜❛❜❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦♥❡ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ♠❛r❦❡t ❛t t✳ ■♥ ✶✾✾✷✱ ❍❡❛t❤✱ ❏❛rr♦✇ ❛♥❞ ▼♦rt♦♥ ♣✉❜❧✐s❤❡❞ ❛♥ ✐♠♣♦rt❛♥t ♣❛♣❡r ❬✶✶❪ t♦ ❞❡s❝r✐❜❡ t❤❡ ♥♦✲❛r❜✐tr❛❣❡ ❝♦♥❞✐t✐♦♥ t❤❛t ♠✉st ❜❡ s❛t✐s✜❡❞ ❜② ❡✈❡r② ♠♦❞❡❧ ♦❢ ②✐❡❧❞ ❝✉r✈❡✳ ❚❤❡ ♠❛✐♥ ✐❞❡❛ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❞②♥❛♠✐❝s ♦❢ ✐♥st❛♥t❛♥❡♦✉s✱ ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞❡❞ ❢♦r✇❛r❞ r❛t❡s f (t, T ) ✐♥st❡❛❞ ♦❢ t❤❡ s❤♦rt✲t❡r♠ r❛t❡ r✳ ❆t t✐♠❡ t✱ ❢♦r ❛ ♠❛t✉r✐t② T + dt✿ df (t, T ) = a(t, T )dt + γ(t, T ) · dWt , ✭✶✳✶✶✮ ✇❤❡r❡ a(t, T ) ❛♥❞ γ(t, T ) ❛r❡ ❛❞❛♣t❡❞ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛♥❞ Wt ✐s ❛ d✲ ❞✐♠❡♥s✐♦♥❛❧ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛❝t✉❛❧ ♣r♦❜❛❜✐❧✐t② P✳ ❚❤✐s r❛t❡ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ r❛t❡ t❤❛t ♦♥❡ ❝♦♥tr❛❝t ❢♦r ❛t t✐♠❡ t ♦♥ ❛ r✐s❦ ✶✻ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❧❡ss ❧♦❛♥ t❤❛t ❜❡❣✐♥s ❛t ❞❛t❡ T ❛♥❞ ✐s r❡t✉r♥❡❞ ❛♥ ✐♥st❛♥t ❧❛t❡r✳✽ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ♥♦ ❛r❜✐tr❛❣❡ ✐♥ t❤✐s ♠❛r❦❡t ✐♠♣❧✐❡s ❛ ✉♥✐q✉❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞r✐❢t a ❛♥❞ t❤❡ ✈♦❧❛t✐❧✐t② γ ✳ ❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ ✜♥❞ ♦✉t ✇❤❛t ✐s t❤✐s r❡❧❛t✐♦♥✳ ❚❤❡ ♥♦ ✐♥st❛♥t❛♥❡♦✉s ❢♦r✇❛r❞ r❛t❡ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s❧② ❝♦♠♣♦✉♥❞ ✇❛② ✭s❛♠❡ ♣r♦❝❡ss t❤❛t ❢♦r ❞❡t❡r♠✐♥✐♥❣ ✭✶✳✸✮✮ ✐s r❡❧❛t❡❞ t♦ t❤❡ ❩❡r♦ ❈♦✉♣♦♥ ❜♦♥❞❀ ❜② ❛r❜✐tr❛❣❡ ✇❡ ❤❛✈❡✿ Ft (T, T + δ) = 1 ln δ B(t, T ) B(t, T + δ) ❍❡♥❝❡ ✇❤❡♥ δ ❣♦❡s t♦ 0✱ ✇❡ ❝❛♥ ✜♥❞ f (t, T )✿ f (t, T ) = − ∂ln(B(t, T )) ∂T ✭✶✳✶✷✮ ❚❤❡♥ ❜② ❛♣♣❧②✐♥❣ t❤❡ ■tô ❧❡♠♠❛ t♦ ✭✶✳✶✷✮ ✇✐t❤ t❤❡ ❞②♥❛♠✐❝ ❣✐✈❡♥ ✐♥ ✭✶✳✶✮ ♦♥❡ ❝❛♥ ❣❡t✿ df (t, T ) = σ B (t, T ) ∂σ B (t, T ) ∂σ B (t, T ) dt − dWt ∂T ∂T ✭✶✳✶✸✮ ❚❤✐s ❡q✉❛t✐♦♥ ❣✐✈❡s t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ t❤❡ ❞r✐❢t ❛♥❞ t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ✐♥✲ st❛♥t❛♥❡♦✉s ❢♦r✇❛r❞ r❛t❡ f (t, T )✳ ❚❤❡r❡❢♦r❡✱ ✐♥t❡❣r❛t✐♥❣ ❜❡t✇❡❡♥ t ❛♥❞ T ✱ ♦♥❡ ❝❛♥ ♦❜t❛✐♥✿ T σ B (t, T ) − σ B (t, t) = t ∂σ B (t, τ ) dτ ∂τ ❲❡ s❡t σ B (t, t) = 0 ❛s ✐t s❡❡♠s ♦❜✈✐♦✉s t❤❛t t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ ❛ ❩❡r♦ ❈♦✉♣♦♥ ❜♦♥❞ ❛t ♠❛t✉r✐t② ✐s ♥✐❧✱ ❛♥❞✿ T σ B (t, T ) = t ✽ ∂σ B (t, τ ) dτ ∂τ ✭✶✳✶✹✮ ❖♥❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t t❤✐s ✐s ❥✉st t❤❡ ❢♦r✇❛r❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ rt = f (t, t) ✶✼ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❯s✐♥❣ t❤❡ ♥♦t❛t✐♦♥ ♦❢ t❤❡ ♣r❡❧✐♠✐♥❛r✐❡s ♦❢ t❤✐s s❡❝t✐♦♥ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ ❢✉♥✲ ❞❛♠❡♥t❛❧ ❍❏▼ r❡s✉❧t✿ T γ(t, τ )dτ a(t, T ) = γ(t, T ) t ✭✶✳✶✺✮ ❘❡♠❛r❦✿ ❚❤✐s r❡s✉❧t ✇❛s ♣r♦✈❡❞ ✐♥ ❛ ♦♥❡ ❢❛❝t♦r ❝❛s❡✳ ■t ✐s q✉✐❡t str❛✐❣❤t ❢♦r✇❛r❞ t♦ s❤♦✇ ✐t ✇✐t❤ s❡✈❡r❛❧ ✐♥❞❡♣❡♥❞❡♥t ❢❛❝t♦rs✱ s❡❡ ❬✶✶❪✳ ■❢ ✇❡ s✉♣♣♦s❡ ✐♥ ❛ r✐s❦ ♥❡✉tr❛❧ ✇♦r❧❞ ❛ ❞②♥❛♠✐❝ ❢♦r t❤❡ ✐♥st❛♥t❛♥❡♦✉s ❢♦r✇❛r❞ r❛t❡ s✉❝❤ t❤❛t✿ d γ k (t, T )dWk df (t, T ) = a(t, T )dt + ✭✶✳✶✻✮ k=1 ✇✐t❤ t❤❡ γ k (t, T ) ❛r❡ ❛ ❢❛♠✐❧② ♦❢ ✈♦❧❛t✐❧✐t② ❝♦❡✣❝✐❡♥ts ❢♦r ❡❛❝❤ ❢❛❝t♦r Wk ✭■♥❞❡♣❡♥❞❡♥t ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s✮ ❧❡❢t ✉♥s♣❡❝✐✜❡❞ ❡①❝❡♣t ♦♥ ✐♥t❡❣r❛❜✐❧✐t② ❛♥❞ ♠❡❛s✉r❛❜✐❧✐t② ✭q✉✐❡t ✇❡❛❦ ❝♦♥❞✐t✐♦♥s✮ t❤❡♥ ♦♥❡ ❝❛♥ ❣❡t✿ d T γ k (t, T ) a(t, T ) = k=1 t ∂γ k (t, τ ) dτ ∂τ ✭✶✳✶✼✮ ❚❤✐s ♥❡✇ ❝♦♥❞✐t✐♦♥ ✐s ❛♣♣❧✐❝❛❜❧❡ t♦ ❡✈❡r② ✐♥t❡r❡st r❛t❡s ♠♦❞❡❧s✱ ✐♥❝❧✉❞✐♥❣ s❤♦rt✲t❡r♠ ✐♥t❡r❡st r❛t❡s ♠♦❞❡❧s ❧✐❦❡ t❤❡ ❍✉❧❧✲❲❤✐t❡ ♦♥❡ ✇❡ r❡✈✐❡✇❡❞ ❜❡❢♦r❡✳ ❇✉t ✐t st✐❧❧ ❣✐✈❡s ❝♦♥❞✐t✐♦♥ ♦♥ ❛♥ ✉♥♦❜s❡r✈❛❜❧❡ ♦❢ t❤❡ ♠❛r❦❡t✱ t❤❡ ✐♥st❛♥t❛✲ ♥❡♦✉s ❢♦r✇❛r❞ r❛t❡✳ ❍♦✇❡✈❡r✱ t❤✐s ♥❡✇ ✐♠♣❧✐❡❞ ❝♦♥❞✐t✐♦♥ ❣❛✈❡ ❛ ♥❡✇ ❛♥❣❧❡ ♦❢ st✉❞② ❛♥❞ ❇r❛❝❡✱ ●❛t❛r❡❦ ❛♥❞ ▼✉s✐❡❧❛ ✐♥ ❬✶✷❪ ❤❛✈❡ ❛♣♣❧✐❡❞ ✐t t♦ ❋♦r✇❛r❞ ▲✐❜♦r r❛t❡✱ ✇❤✐❝❤ ❛r❡ ❞✐r❡❝t❧② ♦❜s❡r✈❛❜❧❡ ♦♥ t❤❡ ♠❛r❦❡t✱ ❞❡✈❡❧♦♣✐♥❣ t❤❡ s♦✲❝❛❧❧❡❞ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✶✳✷✳✸ ❚❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❚❤✐s ♠♦❞❡❧ ✐s ✈❡r② ✐♠♣♦rt❛♥t ♥♦✇❛❞❛②s ✐♥ t❤❡ ✜♥❛♥❝✐❛❧ ✐♥❞✉str② ❛♥❞ ✐s s✉❜✲ ❥❡❝t t♦ ❛ ❧♦t ♦❢ r❡s❡❛r❝❤ ✐♥ t❤❡ ❜❛♥❦s ✐♥❝❧✉❞✐♥❣ t❤❡ ❘♦②❛❧ ❇❛♥❦ ♦❢ ❙❝♦t❧❛♥❞ ❛s ✐t ✐s ❤❛r❞❡r t♦ ✐♠♣❧❡♠❡♥t t❤❛♥ t❤❡ s❤♦rt r❛t❡ ♠♦❞❡❧ ✐♥ t❡r♠ ♦❢ ❝❛❧✐❜r❛t✐♦♥✳ ✶✽ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ●❡♥❡r❛❧ Pr✐♥❝✐♣❧❡ ❆s t♦❧❞ ♣r❡✈✐♦✉s❧② t❤✐s ♠♦❞❡❧ ✐s ✉s✐♥❣ ❛s ✐♥♣✉ts t❤❡ ❢♦r✇❛r❞ r❛t❡s ❛♥❞ ❢r♦♠ t❤❡♠ ❜✉✐❧❞ t❤❡ ❩❡r♦ ❈♦✉♣♦♥s ❝✉r✈❡✳ ❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t ❢♦r✇❛r❞ r❛t❡s ❢♦❧❧♦✇ ❛ ❧♦❣✲♥♦r♠❛❧ ❞②♥❛♠✐❝✳ ❖♥❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t t❤✐s ♣r❛❝t✐❝❡ ✐s ❞✐r❡❝t❧② t❛❦❡♥ ❢r♦♠ ❡q✉✐t② ♠❛r❦❡ts✿ ♦♣❡r❛t♦rs ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛ ♠♦❞❡❧ s♦ t❤❛t ▲✐❜♦r r❛t❡s ❛♥❞ s✇❛♣s r❛t❡s ❢♦❧❧♦✇ ❛ ❧♦❣✲♥♦r♠❛❧ ♣r♦❝❡ss✳ ❖♥❡ s❤♦✉❧❞ ❤✐❣❤❧✐❣❤t t❤❡ ❢❛❝t t❤❛t t❤✐s ❛ss✉♠♣t✐♦♥ ✐s ♥♦t r❡❧❛t❡❞ t♦ t❤❡ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❛s ✐t ✐s ❢♦r ❡q✉✐t② ♣r✐❝❡s ❜✉t ❜❡❝❛✉s❡ ❤✐st♦r✐❝❛❧❧② t❤❡ ♠❛r❦❡t q✉♦t❡s ▲✐❜♦r r❛t❡s ❛♥❞ s✇❛♣s r❛t❡s ✉s✐♥❣ ❇❧❛❝❦ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧ ✐♥ ❬✷❪✳ ❍❡♥❝❡✱ t❤❡ ❧♦❣ ♥♦r♠❛❧ ❛ss✉♠♣t✐♦♥ ❢♦r t❤♦s❡ r❛t❡s ❛r✐s❡s ♥❛t✉r❛❧❧②✳ ❆ss✉♠♣t✐♦♥ ♦♥ t❤❡ ❞②♥❛♠✐❝s ♦❢ t❤❡ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡s ■♥ ✶✾✾✼✱ ❇r❛❝❡ ❡t ❛❧✳ ♣r♦♣♦s❡s ❛ ♠♦❞❡❧ ✇❤❡r❡ t❤❡ ▲✐❜♦r r❛t❡s ❢♦❧❧♦✇ ❛ ❧♦❣ ♥♦r♠❛❧ ♣r♦❝❡ss ✐♥ t❤❡ ❢♦r✇❛r❞ ♠❡❛s✉r❡ ❛ss♦❝✐❛t❡❞✳ ◆❛♠❡❧②✱ ❢♦r ❛ ❣✐✈❡♥ ♠❛✲ t✉r✐t② δ ✱ ✭t❤❡ t②♣✐❝❛❧ ♠❛t✉r✐t② ❛r❡ ✸✱ ✻✱ ✾ ❛♥❞ ✶✷ ♠♦♥t❤s✮✱ t❤❡ ❛ss♦❝✐❛t❡❞ ❢♦r✇❛r❞ ▲✐❜♦r r❛t❡ ♣r♦❝❡ss {L(t, T ); t ≥ 0} ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❜② T +δ 1 + δL(t, T ) = exp f (t, ν)dν T ✭✶✳✶✽✮ ❢♦❧❧♦✇s ❛ ❧♦❣ ♥♦r♠❛❧ ♣r♦❝❡ss ✐♥ t❤❡ s♣♦t ♠❛rt✐♥❣❛❧❡ ♠❡❛s✉r❡ P∗ ✭❛♥❞ ❛ ♠❛r✲ t✐♥❣❛❧❡ ♣r♦❝❡ss ✐♥ ✐ts ❋♦r✇❛r❞ ♠❡❛s✉r❡ Pi ✮✿ dL(t, T ) = (. . .)dt + L(t, T )γ(t, T )dWt∗ ✭✶✳✶✾✮ ✇✐t❤ γ(t, T ) ❛ ❞❡t❡r♠✐♥✐st✐❝ ❢✉♥❝t✐♦♥ ❜♦✉♥❞❡❞ ❛♥❞ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢♦❧✲ ❧♦✇✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s t♦ ❛♣♣❧② t❤❡ ●✐rs❛♥♦✈ t❤❡♦r❡♠✳ ✶✾ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ▼❛✐♥ r❡s✉❧ts ✐✳ ❙❡t✉♣ ♦❢ ❛ ✉♥✐q✉❡ ②✐❡❧❞ ❝✉r✈❡ ❢♦r♠ t❤❡ ❋♦r✇❛r❞ ▲■❇❖❘ ❘❛t❡ ❲❡ ✇✐❧❧ ✉s❡ ❏❛♠s❤✐❞✐❛♥ ❛♣♣r♦❛❝❤ ❬✶✸❪ t♦ ❡①♣❧❛✐♥ ❤♦✇ t❤✐s ♠♦❞❡❧ ✐s ❜✉✐❧t ❛♥❞ ❤♦✇ ✐t ✐s r❡❧❛t❡❞ t♦ t❤❡ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞✳ ❲❡ ❛♣♣❧② t❤❡ ■t♦ ▲❡♠♠❛ t♦ t❤❡ ❡q✉❛❧✐t② s❤♦✇♥ ❜❡❢♦r❡ ❛♥❞ ✉s✐♥❣ ✶✳✶✿ B(t, T ) B(t, T + δ) B(t, T )B(t, T + δ)(m(t, T ) − m(t, T + δ)) δdL(t, T ) = dt B 2 (t, T + δ) B(t, T )B(t, T + δ)(σ B (t, T ) − σ B (t, T + δ)) dWt + B 2 (t, T + δ) B(t, T )B(t, T + δ)(σ B (t, T + δ))2 − B(t, T )B(t, T + δ)σ B (t, T + δ)σ B (t, T ) + dWt2 B 2 (t, T + δ) 1 + δL(t, T ) = B(t, T ) (m(t, T ) − m(t, T + δ)) − σ B (t, T + δ)(σ B (t, T ) − σ B (t, T + δ)) dL(t, T ) = + dt δB(t, T + δ) B(t, T )(σ B (t, T ) − σ B (t, T + δ)) dWt δB(t, T + δ) ❘❡✲♦r❣❛♥✐③✐♥❣ t❤✐s ❡q✉❛t✐♦♥✱ ✇❡ ❝❛♥ ✜♥❞ t❤❛t✿ dL(t, T ) = µ(t, T )dt + γ(t, T )L(t, T )dWt ✭✶✳✷✵✮ ✇❤❡r❡✿ µ(t, T ) = B(t, T ) m(t, T ) − m(t, T + δ) − γ(t, T )L(t, T )σ B (t, T + δ) δB(t, T + δ) ❛♥❞ γ(t, T )L(t, T ) = B(t, T ) (σ B (t, T ) − σ B (t, T + δ)) σ B (t, T + δ) ✇❤✐❝❤ ❣✐✈❡s t❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡❧❛t✐♦♥ (σ B (t, T ) − σ B (t, T + δ)) = δL(t, T )γ(t, T ) 1 + δL(t, T ) ✭✶✳✷✶✮ ✷✵ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❇r❛❝❡✱ ❡t ❛❧✳ ✭✶✾✾✼✮ ❤❛✈❡ ♥♦t✐❝❡❞ t❤❛t t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ❡q✉❛t✐♦♥ ✭✶✳✷✶✮ ✐s ❛❝t✉❛❧❧② ❛ r❡❝✉rr❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ σ B (t, T )✿ j σ B (t, t ))−σ B (t, T +(j+1)δ) = k= δ −1 t (δL(t, t + kδ)) γ(t, t+kδ) ✭✶✳✷✷✮ 1 + δL(t, t + kδ) ✇❤❡r❡ δ −1 t ✐s t❤❡ t❤❡ ♥❡①t ✐♥t❡❣❡r✳ ■❢ ✇❡ ❛ss✉♠❡ t❤❡ ❙♣♦t ▲✐❜♦r ▼❡❛s✉r❡ P∗ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♠❛r❦❡t ♠❡❛s✉r❡ P✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ht ✱ s♦♠❡ ❛❞❛♣t❡❞ ♣r♦❝❡ss✱ t❤❡ ❘❛❞♦♥ ◆②❦♦❞②♠ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ t✇♦ ♠❡❛s✉r❡s s✉❝❤ t❤❛t✿ dWt = dWt∗ + ht dt ❯s✐♥❣ t❤❡ ❝❤❛♥❣❡ ♦❢ ♥✉♠❡r❛✐r❡ t❡❝❤♥✐q✉❡s ❛♥❞ t❤❡ ■t♦ ▲❡♠♠❛✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t✿ m(t, T ) − m(t, t ) = σ B (t, t ) − ht (σ B (t, T ) − σ B (t, t )) ❈♦♠❜✐♥✐♥❣ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥ ✇✐t❤ ✶✳✷✶ ✇❡ ♦❜t❛✐♥✿ B(t, T ) m(t, T ) − m(t, T + δ) = γ(t, T )L(t, T ) · σ B (t, t ) − ht δB(t, T + δ) ❙♦ ✇❡ ✜♥❛❧❧② ❣❡t t♦✿ σ B (t, t ) − σ B (t, T + δ) − ht dt + dWt dL(t, T ) = γ(t, T )L(t, T ) ▼♦r❡ ❡①❤❛✉st✐✈❡❧②✿ j dL(t, T ) = γ(t, T )L(t, T ) k= δ −1 t (δL(t, t + kδ)) γ(t, t+kδ) dt+L(t, T )γ(t, T )dWt∗ 1 + δL(t, t + kδ) ✭✶✳✷✸✮ ❚❤✐s ♣r♦❝❡ss ✜♥✐s❤❡s t❤❡ s❡t✉♣ ♦❢ t❤❡ ②✐❡❧❞ ❝✉r✈❡ ❞②♥❛♠✐❝s ❛s ✇❡ ❛r❡ ❣✐✈❡♥ t❤❡ δ−▲✐❜♦r r❛t❡ ♣r♦❝❡ss✱ t❤❡ ③❡r♦ ❝♦✉♣♦♥ ✈♦❧❛t✐❧✐t② ✐♥ ✭✶✳✷✷✮ ❛♥❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢♦r✇❛r❞ ❝✉r✈❡ t♦❞❛②✳ ❲❤❛t s❤♦✉❧❞ ❜❡ ❡♠♣❤❛s✐③❡ ✐s t❤❛t ✇❡ ❤❛✈❡ ✇♦r❦❡❞ t❤❡ ✷✶ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ♦t❤❡r ✇❛② t❤❛t t❤❡ ♦t❤❡r s❤♦rt t❡r♠ ✐♥t❡r❡st r❛t❡s ♠♦❞❡❧✿ ❢r♦♠ t❤❡ ❋♦r✇❛r❞ r❛t❡s ❦♥♦✇♥ ❛t t✐♠❡ 0 ✭t❤❡ ♦❜s❡r✈❛❜❧❡s ✮ ✇❡ ❤❛✈❡ ❞❡✜♥❡❞ ❛ ✉♥✐q✉❡ ②✐❡❧❞ ❝✉r✈❡ ❞②♥❛♠✐❝ ✉s✐♥❣ t❤❡ ❛r❜✐tr❛❣❡✲❢r❡❡ ❛ss✉♠♣t✐♦♥ ❛♥❞ ❍❏▼ r❡s✉❧t ❞❡s❝r✐❜❡❞ ✐♥ ✶✳✷✳✷✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ③❡r♦✲❝♦✉♣♦♥ ✐s ❛ ♣r✐♦r✐ st♦❝❤❛st✐❝✳ ❘❡♠❛r❦ ✶✿ ❇r❛❝❡ ❡t ❛❧✳ ✭✶✾✾✼✮ ❤❛✈❡ s❤♦✇♥ ✇✐t❤ ❞❡t❛✐❧s t❤❛t t❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s ♣r♦❜❧❡♠ ❡①✐sts ❛♥❞ ✐s ✉♥✐q✉❡✳ ❘❡♠❛r❦ ✷✿ ❚❤✐s ♠♦❞❡❧ r❡s♣❡❝ts t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ t❤❡ ♠❡❛♥ r❡✈❡rs✐♦♥ ❜❡❤❛✈✲ ✐♦r ♦❢ ✐♥t❡r❡st r❛t❡s ✐♥ t❤❡ ♠❛r❦❡t ❛s ✐t ❝❛♥ ❜❡ ✇❡❧❧ ♦❜s❡r✈❡❞ ♦♥ ❡♠♣✐r✐❝❛❧ st✉❞✐❡s ❢♦r ✐♥st❛♥❝❡ ✐♥ ❬✶✷❪✳ ❘❡♠❛r❦ ✸✿ ❚❤✐s ❡①♣r❡ss✐♦♥ ✐s ✈❡r② ❝♦♥✈❡♥✐❡♥t ❛♥❞ ✇❛s ♣r♦♣♦s❡❞ ❜② ❏❛♠s❤✐❞✲ ✐❛♥ ✐♥ ❬✶✸❪ ❛s ✐t ♣❡r♠✐ts t♦ ✐♠♣❧❡♠❡♥t ♥✉♠❡r✐❝❛❧❧② t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✇✐t❤ ♦♥❧② ♦♥❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❋♦r✇❛r❞ ♠❡❛s✉r❡ ♦♥❡s✳ ❚❤✐s ✐s t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✳ ✐✐✳ ❊①♣r❡ss✐♦♥ ♦❢ t❤❡ ▲■❇❖❘ ❋♦r✇❛r❞ ❘❛t❡s ✉♥❞❡r ❞✐✛❡r❡♥t ♥✉✲ ♠❡r❛✐r❡s ✭❋♦r✇❛r❞ ♠❡❛s✉r❡s✮ ❊✈❡♥ ✐❢ t❤❡② ❛r❡ ❧❡ss ❝♦♥✈❡♥✐❡♥t ❢♦r ❝♦♠✲ ♣✉t❛t✐♦♥ t❤❡s❡ ❡①♣r❡ss✐♦♥s ❣✐✈❡ s❡♥s❡ t♦ ✇❤❛t ✐s ❜❡❤✐♥❞ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✳ ❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② ❛♥❞ ❢♦r s✐♠♣❧✐✜❝❛t✐♦♥ ♣✉r♣♦s❡✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❝♦♥s✐❞❡r ❢r♦♠ ♥♦✇ ❛ ❢❛♠✐❧② ♦❢ δ ▲✐❜♦r ❢♦r✇❛r❞ r❛t❡s {L(t, Tk ), t ≤ 0}n ✇❤✐❝❤ ♠❛t✉r❡s ❛t {Tk }n ✳ ❍❡♥❝❡✱ ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② Lk (t) t❤❡ ▲✐❜♦r r❛t❡ s✉❝❤ t❤❛t✿ Lk (t) = L(t, Tk − δ) ✭✶✳✷✹✮ ❲✐t❤ t❤❡ ♥❡✇ ♥♦t❛t✐♦♥s ❢♦r t❤❡ ❋♦r✇❛r❞ r❛t❡s Li (t) t❤❡ ♣r❡✈✐♦✉s ❡①♣r❡ss✐♦♥ ✐♥ t❤❡ s♣♦t ♠❛rt✐♥❣❛❧❡ ♠❡❛s✉r❡ ❜❡❝♦♠❡s✿ Tk Lk (t) j=1 δLj (t)(γk (t) γj (t)) dt + Lk (t)γk (t)dWt∗ 1 + δLj (t) ✭✶✳✷✺✮ ✷✷ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❈♦♥s✐❞❡r t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ Pk ✱ t❤❡ ❢♦r✇❛r❞ ♠❡❛s✉r❡ ✇✐t❤ ♠❛t✉r✐t② Tk ✱ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ♥✉♠❡r❛✐r❡ B( , Tk )✱ t❤❡ ❩❡r♦ ❝♦✉♣♦♥ ❜♦♥❞ ♠❛t✉r✐♥❣ ❛t Tk ✳ ❲❡ ❤❛✈❡ s❡❡♥ ♣r❡✈✐♦✉s❧② t❤❛t✿ Lk (t)B(t, Tk ) = B(t, Tk−1 ) − B(t, Tk ) δ ✭✶✳✷✻✮ ❖♥❡ ❝❛♥ ♦❜s❡r✈❡ t❤❛t ✇❡ ❝❛♥ r❡♣❧✐❝❛t❡ Lk (t)B(t, Tk ) ❜② ❜✉②✐♥❣ ❛♥❞ s❡❧❧✲ ✐♥❣ t❤❡ ❜♦♥❞s B(t, Tk−1 ) ❛♥❞ B(t, Tk )✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❛ss❡t Lk (t)B(t, Tk ) ❞✐✈✐❞❡❞ ❜② t❤❡ ♥✉♠❡r❛✐r❡ B( , Tk ) ✐s ❛ ♠❛rt✐♥❣❛❧❡ ✉♥❞❡r Pk ❛♥❞ ✐s ❛s ❛ ♠❛tt❡r ♦❢ ❢❛❝t Lk (t)✳ ❙♦ ♦♥❡ ❝❛♥ ✇r✐t❡✿ dLk (t) = Lk (t)γk (t)dWtk , t ≤ Tk−1 ✭✶✳✷✼✮ ❋♦r t❤❡ ♦t❤❡r ❝❛s❡s ✐♥ ♦r❞❡r t♦ ❡①♣r❡ss Lk (t) ✐♥ t❤❡ ❢♦r✇❛r❞ ♠❡❛s✉r❡ Pi ✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ✉s❡ ●✐rs❛♥♦✈ tr❛♥s❢♦r♠❛t✐♦♥ ❢♦r Pk t♦ Pi ✳ ❲❡ ❝❛♥ s❤♦✇ t❤❛t ❝❛s❡ i < k ❛s t❤❡ ❝❛s❡ i > k ✐s ❛♥❛❧♦❣♦✉s✳ ❲❡ ♣r♦❝❡❡❞ ❜② r❡❝✉rr❡♥❝❡✳ ❚❤❡ ❘❛❞♦♥ ◆✐❦♦❞②♠ ❞❡r✐✈❛t✐✈❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❝❤❛♥❣❡ ♦❢ ♥✉♠❡r❛✐r❡ ❢r♦♠ Pk t♦ Pk−1 ✐s✿ R= ∂Pk−1 B(t, Tk−1 ) B(Tk−1 , Tk ) = B(t, Tk ) B(Tk−1 , Tk−1 ) ∂Pk ✭✶✳✷✽✮ ❆❝❝♦r❞✐♥❣ t♦ ●✐rs❛♥♦✈ t❤❡♦r❡♠ ✇❡ ❦♥♦✇ t❤❛t R ✐s ❛♥ ❡①♣♦♥❡♥t✐❛❧ ♠❛rt✐♥❣❛❧❡ ✉♥❞❡r Pk s✉❝❤ t❤❛t ✐t ❡①✐sts φ ❛ r❡❣✉❧❛r ♣r♦❝❡ss✾ s♦✿ dR = φ dWtk R ✇❤❡r❡ dWtk = dWtk−1 + φ dt ✾ ❘❡❣✉❧❛r ❤❡r❡ ♠❡❛♥s s❡✈❡r❛❧ ❝♦♥❞✐t✐♦♥s ✐♥❝❧✉❞✐♥❣ ✐♥t❡❣r❛❜❧❡ ✐♥ L2 ✷✸ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ❞❡t❡r♠✐♥❡ t❤✐s ♣r♦❝❡ss φ B(t,T ) d( B(t,Tk−1) ) dR d(1 + δLk (t)) = B(t,T k ) = k−1 R 1 + δLk (t) B(t,Tk ) = δdLk (t) γk (t)Lk (t) = dWtk 1 + δLk (t) 1 + δLk (t) ❚❤❡r❡❢♦r❡ ✇❤❡♥ ❛ss❡♠❜❧✐♥❣ t❤❡ t✇♦ s✐❞❡s✱ dWtk = dWtk−1 + γk (t)Lk (t) dt 1 + δLk (t) ✭✶✳✷✾✮ ❆♥ ✐♠♣♦rt❛♥t t❤✐♥❣ t♦ r❡♠✐♥❞ ✐s t❤❛t ✐♥ ❛ ♠♦❞❡❧ ✇✐t❤ d ❢❛❝t♦rs✱ dWt ✐s ❛ d✲❞✐♠❡♥s✐♦♥ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❛♥❞ γk (t) ✐s ❛ d✲❞✐♠❡♥s✐♦♥ ✈❡❝t♦r✳ ❇② r❡❝✉rr❡♥❝❡✱ ✇❡ ❝❛♥ ❡①♦❣❡♥♦✉s❧② ❣✐✈❡ t❤❡ ❞②♥❛♠✐❝ ♦❢ t❤❡ k✲t❤ ❢♦r✇❛r❞ r❛t❡ ✉♥❞❡r ♠❡❛s✉r❡ i✳ ❋✐♥❛❧❧②✱ s✉♠♠✐♥❣ ✉♣ t❤❡ ❞✐✛❡r❡♥t ❡①♣r❡ss✐♦♥s ♦❢ Lk (t) ✉♥❞❡r t❤❡ ❋♦r✇❛r❞ ♠❡❛s✉r❡ Pi ✿  δL (t)(γk (t) γj (t))   dt + Lk (t)γk (t)dWtk , i < k, t ≤ Ti ; Lk (t) kj=i+1 j 1+δL   j (t)  dLk (t) = i = k, t ≤ Tk−1 ; Lk (t)γk (t)dWti ,     δLj (t)(γk (t) γj (t))  −Lk (t) k dt + Lk (t)γk (t)dWtk , i < k, t ≤ Tk−1 ; j=i+1 1+δLj (t) ✭✶✳✸✵✮ ✇✐t❤ W i t❤❡ st❛♥❞❛r❞ d✲❞✐♠❡♥s✐♦♥❛❧ ❲✐❡♥❡r ♣r♦❝❡ss ✉♥❞❡r Pi ✳ ❆❧❧ t❤❡ ♣♦✐♥t ✇✐t❤ t❤✐s ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ❋♦r✇❛r❞ ❘❛t❡s ✐s t❤❛t ✇❡ ❝❛♥ s❡❡ ❛r✐s❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤♦s❡ ❋♦r✇❛r❞ r❛t❡s✿ d γi (t) γj (t) = (γi )k (γj )k k=1 γi (t) γj (t) = ρij γi ✇❤❛t ✇❡ ♥♦t❡ = ρij σi σj γj ✷✹ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❲✐t❤ ρij t❤❡ ✐♥st❛♥t❛♥❡♦✉s ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ i✲t❤ ❛♥❞ j ✲t❤ ❋♦r✇❛r❞ r❛t❡✳ ❲❡ ✇✐❧❧ st✉❞② ✐♥ ❝❤❛♣t❡r ✷ t❤♦s❡ t✇♦ ❝♦♠♣♦♥❡♥ts σi ❛♥❞ ρi ✳ ✶✳✷✳✹ ▲✐❜♦r ▼❛r❦❡t ♠♦❞❡❧ s✉♠♠❛r② ❚❤❡ ▲✐❜♦r ♠❛r❦❡t ♠♦❞❡❧ ✐s ❛♥ ✐♥t❡r❡st r❛t❡s ♠♦❞❡❧ ✇❤♦s❡ ✐♥♣✉t ❛r❡✿ ❼ ❆ s❡t ♦❢ ❜♦♥❞ ♠❛t✉r✐t✐❡s {Ti }n ❼ ❚❤❡ ▲✐❜♦r ❋♦r✇❛r❞ r❛t❡s ❛t t✐♠❡ ③❡r♦ L1 (0), . . . , Ln (0) ❼ ❚❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t✐❡s ♦❢ t❤❡ ❢♦r✇❛r❞ r❛t❡s γi ( ) ❢♦r i − 1, . . . , n ❚❤❡ γi ( ) ❛r❡ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❇●▼ ♠♦❞❡❧ ❛♥❞ t❤♦s❡ ♥❡❡❞ t♦ ❜❡ ❝❛❧✐✲ ❜r❛t❡❞ s♦ t❤❛t ♦✉r ♠♦❞❡❧ r❡✢❡❝ts ❝♦rr❡❝t❧② t❤❡ ♣r✐❝❡s ♦❢ ❛ss❡ts tr❛❞❡❞ ❛❝t✐✈❡❧② ✐♥ t❤❡ ♠❛r❦❡ts✳ ❚❤✐s ❝❛❧✐❜r❛t✐♦♥ ♣r♦❝❡❞✉r❡ ✇✐❧❧ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ ❈❤❛♣t❡r ✷✳ ✶✳✸ Pr✐❝✐♥❣ ❱❛♥✐❧❧❛ ❉❡r✐✈❛t✐✈❡s ❱❛♥✐❧❧❛ ❞❡r✐✈❛t✐✈❡s ❛r❡ t❤❡ ♠♦st ❧✐q✉✐❞ ✇❤✐❝❤ ♠❛❦❡s t❤❡♠ ✈❡r② ❡✣❝✐❡♥t t♦ tr❛❝❦ ✈♦❧❛t✐❧✐t② ✐♥❢♦r♠❛t✐♦♥ ✐♥ ✐♥t❡r❡st r❛t❡ ♠❛r❦❡ts✳ ❖♥ t❤❡ ❝♦♥tr❛r② ♦❢ t❤❡ ❙✇❛♣s ❛♥❞ t❤❡ ❋♦r✇❛r❞ ❘❛t❡s ✐♥ ✶✳✶✳✺ ✇❡ ♥❡❡❞ ✐♥ ♦r❞❡r t♦ ♣r✐❝❡ t❤❡♠ t♦ ✉s❡ t❤❡ ♣r❡✈✐♦✉s ♠♦❞❡❧s ❛♥❞ ❛ss✉♠♣t✐♦♥s ✇❡ ❞❡s❝r✐❜❡❞ ❜❡❢♦r❡✳ ✶✳✸✳✶ ■♥t❡r❡st r❛t❡ ♦♣t✐♦♥s✿ ❝❛♣ ❛♥❞ ✢♦♦r ▲❡t ❝♦♥s✐❞❡r ❛ ✢♦❛t✐♥❣ r❛t❡ ♥♦t❡ ✇❤❡r❡ t❤❡ ✐♥t❡r❡st r❛t❡ ✐s r❡s❡t ❡q✉❛❧ t♦ ▲■❇❖❘ ♣❡r✐♦❞✐❝❛❧❧② ✭✉s✉❛❧❧② ✉s✐♥❣ ❛ t❡♥♦r ♦❢ ✸ ♠♦♥t❤s✮✳ ❚♦ ♣r♦t❡❝t ❤✐♠s❡❧❢ ❛❣❛✐♥st t❤❡ r✐s❡ ♦❢ ▲■❇❖❘✱ t❤❡ ✐♥✈❡st♦r ❝❛♥ ❜✉② ❛♥ ✐♥t❡r❡st r❛t❡ ❝❛♣ s♦ t❤❛t t❤❡ ✢♦❛t✐♥❣✲r❛t❡ ✇✐❧❧ ♥♦t r❛✐s❡ ❛❜♦✈❡ ❛ ❝❡rt❛✐♥ ❧❡✈❡❧✿ t❤❡ ❝❛♣ r❛t❡✳ ■♥ ❛ ❢♦r✇❛r❞ ❝❛♣✱ s❡tt❧❡❞ ✐♥ ❛rr❡❛rs ❛t t✐♠❡ Tj , j = 1 . . . n✱ t❤❡ ❝❛s❤✲✢♦✇s ❛r❡ (Lj (Tj ) − κ)+ δ ♣❛✐❞ ❛t t✐♠❡ Tj+1 ✇✐t❤ ❛ ♥♦t✐♦♥❛❧ 1✳ ❚❤❡ r✉❧❡ ♦❢ ♥♦ ✷✺ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❛r❜✐tr❛❣❡ ❛♥❞ t❤❡ ❞✐s❝♦✉♥t ❢❛❝t♦rs B(t, Tj+1 ) ❣✐✈❡s t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❝❛♣✿ n−1 B(t, Tj+1 )Ej+1 [(Lj (Tj ) − κ)+ δ] capt = ✭✶✳✸✶✮ j=0 ✇❤❡r❡ ❤❡r❡ Ej+1 ✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ✉♥❞❡r t❤❡ ❢♦r✇❛r❞ ♠❡❛s✉r❡ Pj+1 ❛s ✇❡ ❞❡✜♥❡❞ ✐t ✐♥ s❡❝t✐♦♥ ✶✳✶✳✻✳ ❚❤❡ ❢♦r♠✉❧❛ ✭✶✳✸✶✮ ♣❡r♠✐ts t♦ ❝♦♥s✐❞❡r ❛ ❝❛♣ ❛s ❛ ♣♦rt❢♦❧✐♦ ♦❢ n ✐♥t❡r❡st r❛t❡ ♦♣t✐♦♥s ❛❧s♦ ❦♥♦✇♥ ❛s ❝❛♣❧❡ts✿ t❤❡ ❡❧❡♠❡♥t❛r② ❝❛s❤✲✢♦✇ (Lj (Tj ) − κ)+ δ ✐s t❤❡ ♣❛② ♦✛ ♦❢ ❛ ❝❛❧❧ ♦♣t✐♦♥ ♦♥ t❤❡ ▲■❇❖❘ r❛t❡ ♦❜s❡r✈❡❞ ✐♥ ❛rr❡❛rs ❛t t✐♠❡ Tj ❛♥❞ s❡tt❧❡❞ ❛t t✐♠❡ Tj+1 ✳ ❙✐♠✐❧❛r❧②✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡ ❛ ✢♦♦r ✇❤✐❝❤ ♣r♦✈✐❞❡s ❛♥ ✐♥s✉r❛♥❝❡ t❤❛t t❤❡ ✢♦❛t✐♥❣ r❛t❡ ✇✐❧❧ ♥♦t ❢❛❧❧ ✉♥❞❡r t❤❡ ✢♦♦r r❛t❡ t♦ ❜❡ ❞❡✜♥❡❞✳ ❚❤❡ ✢♦♦r❧❡t ✐s ❛ ♣✉t ♦♣t✐♦♥ ♦♥ t❤❡ ▲■❇❖❘ r❛t❡ ♦❜s❡r✈❡❞ ❛t t✐♠❡ Tj ❛♥❞ s❡tt❧❡❞ ❛t t✐♠❡ Tj+1 ✳ Pr✐❝✐♥❣ ❝❛♣❧❡ts ✇✐t❤ ❇❧❛❝❦ ❋♦r♠✉❧❛ ❯s✐♥❣ ❇❧❛❝❦ ✐♥ ❬✷❪ ❛ ❝❧♦s❡❞ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♣r✐❝❡ ♦❢ ❛ ❝❛♣❧❡t ❝❛♥ ❜❡ ❞❡r✐✈❡❞✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦r✇❛r❞ r❛t❡s ❛r❡ ❧♦❣✲♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ✉♥❞❡r s♦♠❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ Q ✶✵ ❛♥❞ ❤❛✈❡ ❛ ❝♦♥st❛♥t ✈♦❧❛t✐❧✐t② σ > 0✳ ✭✶✳✸✷✮ dLi (t) = Li (t)σdWt ❚❤✐s st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❝❛♥ ❡❛s✐❧② ❜❡ s♦❧✈❡❞✿ 1 Li (t) = Li (0) exp(σWt − 2 σ 2 t2 ) , ∀t ∈ [0, Ti ], ✭✶✳✸✸✮ ❛♥❞ ✇❡ ❦♥♦✇ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❜②✿ Li (0) = ✶✵ 1 δ B(0, T ) −1 B(t, T ) ◆♦ ❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ✐s ❛✈❛✐❧❛❜❧❡ ❢♦r t❤✐s ♣r♦❜❛❜✐❧✐t②✱ ✇❡ ✇✐❧❧ r❡❢❡r t♦ Q ❛s t❤❡ ♠❛r❦❡t ♣r♦❜❛❜✐❧✐t② ✷✻ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❚❤❡ ♣❛②♦✛ ♦❢ t❤❡ ❝❛♣❧❡t ✇✐t❤ str✐❦❡ κ ❛t t✐♠❡ Ti ♦✈❡r t❤❡ ▲■❇❖❘ r❛t❡ Li (Ti ) ♦♥ ❛ ♥♦t✐♦♥❛❧ ❛♠♦✉♥t 1 ✐s✿ 1δ max (Li (Ti ) − κ, 0), ❚❤❡♥✱ t❤❡ ♣r✐❝❡ ♦❢ t❤✐s ❝❛♣❧❡t ❛t t✐♠❡ t ✐s✿ CapletBl (t) = δB(t, Ti+1 )EQ (Li (Ti ) − κ)+ |Ft ❯s✐♥❣ ❇❧❛❝❦✲❙❝❤♦❧❡s ❢♦r♠✉❧❛ ✇❡ ❣❡t✱ ∀t ∈ [0, Ti ]✱ CapletBl (t) = 1δB(t, Ti+1 )[Li (t)N (d1 (t, Ti )) − κN (d2 (t, Ti ))], ✇✐t❤✱ d1 = ln(Li (t)/κ) + σ 2 (Ti2−t) √ σ Ti − t d2 = d1 − σ ✭✶✳✸✺✮ ✭✶✳✸✻✮ Ti − t ✇❤❡r❡ N : R → [0, 1] ✐s t❤❡ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✿ N (x) = ✭✶✳✸✹✮ √1 2 2 x − z2 −∞ e ❋♦r ❛ ❝❛♣✱ ♦♥❡ ❝❛♥ ❣❡t✿ n−1 CapBl (t) = δB(t, Tj+1 ) Lj (t)N (d1 (t, Tj )) − κN (d2 (t, Tj )) ✭✶✳✸✼✮ j=0 ❚❤❡ ♣❛r❛♠❡t❡r σ ✐s ✉s✉❛❧❧② r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❋♦r✇❛r❞ ✈♦❧❛t✐❧✐t② ♦❢ Li ✳ ❈❛♣s ❛r❡ q✉♦t❡❞ ❢♦r ✐♥❞✐❝❛t✐✈❡ ♣r✐❝❡s ❜② t❤❡ ✈♦❧❛t✐❧✐t② ❢♦r ❛ str✐❦❡ ❡q✉❛❧ t♦ t❤❡ ❢♦r✇❛r❞ r❛t❡✱ t❤❡② ❛r❡ t❤❡ ❢❛♠♦✉s ❛t t❤❡ ▼♦♥❡② ❇❧❛❝❦ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✳ ■♥ ♦r❞❡r t♦ ❣❡t t❤❡ ✢♦♦r ♣r✐❝❡ ♦♥❡ ❝❛♥ ✉s❡ t❤❡ ❝❛♣✲✢♦♦r ♣❛r✐t② ✇❤✐❝❤ ❝❛♥ ❜❡ s❤♦✇♥ str❛✐❣❤t❢♦r✇❛r❞ ✇r✐t✐♥❣ t❤❡ ❝❛♣ ❛♥❞ ✢♦♦r ❞❡✜♥✐t✐♦♥s ❛♥❞ ✉s✐♥❣ t❤❡ dz ✳ ✷✼ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ♥♦✲❛r❜✐tr❛❣❡ ♣r♦♣❡rt②✿ n Cap(t) − F loor(t) = (B(t, Ti )[Li (t) − κ]) ✭✶✳✸✽✮ i=0 Pr✐❝✐♥❣ ❝❛♣❧❡ts ✐♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❲❡ ❢♦❧❧♦✇ t❤❡ ❣❡♥❡r❛❧ ✐❞❡❛ ♦❢ ▼✐❧t❡rs❡♥ ❡t ❛❧✳ ✐♥ ❬✶✺❪✳ ❆s s❡❡♥ ❜❡❢♦r❡ ✇❡ ♣❧❛❝❡ ♦✉rs❡❧✈❡s ✐♥ t❤❡ ❢♦r✇❛r❞ ♠❡❛s✉r❡ Pi ✳ ❯♥❞❡r t❤✐s ♠❡❛s✉r❡ t❤❡ i✲t❤ ▲✐❜♦r ❋♦r✇❛r❞ r❛t❡ ✐s ❛ ♠❛rt✐♥❣❛❧❡✿ dLi (t) = Li (t)γi (t) dWti , ✭✶✳✸✾✮ t ≤ Ti ❲❡ r❡❝♦❣♥✐③❡ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ♠❛rt✐♥❣❛❧❡ ✐♥ t❤✐s st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛✲ t✐♦♥ ❛♥❞ ✇❡ ❝❛♥ ❝❤❡❝❦ ❜② t❤❡ ■t♦ ▲❡♠♠❛ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s s♦❧✉t✐♦♥ ♦❢ t❤✐s ❡q✉❛t✐♦♥✿ Li (t) = Li (0)e Ti t γi (s) dWsi − 21 Ti t γi (s) 2 ds , t ≤ Ti , ✭✶✳✹✵✮ ❍❡♥❝❡ Li (Ti ) ✐s ❛ ♠❛rt✐♥❣❛❧❡ ✉♥❞❡r ✐ts ♠❡❛s✉r❡ ❛♥❞ ✇❡ ❝❛♥ ✉s❡ t❤❡ ♥♦ ❛r❜✐✲ tr❛❣❡ r✉❧❡✿ CapletLM M (t) = δB(t, Ti+1 )Ei+1 (Li (Ti ) − κ)+ |Ft = δB(t, Ti+1 )Ei+1 Li (Ti )1D |Ft − κδB(t, Ti+1 )P rob(D|Ft ) = δB(t, Ti+1 )(I1 − I2 ), ✇❤❡r❡ D = {Li (Ti ) > κ} ✐s t❤❡ ❡①❡r❝✐s❡ s❡t✳ ❋✉rt❤❡r♠♦r❡✱ γi ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❢✉♥❝t✐♦♥✱ ❤❡♥❝❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧❛✇ ✉♥❞❡r Ei ♦❢ ✐ts ■t♦ ✐♥t❡❣r❛❧ ✐s ●❛✉ss✐❛♥ ✇✐t❤ ♠❡❛♥ 0 ❛♥❞ ❛ ✈❛r✐❛♥❝❡ ζi (t)✿ Ti ζi (t) = t γi (s) 2 ds ✷✽ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ❙♦ ✇❡ ❣❡t✿ ln(Li (t)) − ln κ − 12 ζi2 (t) ζi (t) I2 = κN ✭✶✳✹✶✮ ❚❤❡ ❞❡r✐✈❛t✐♦♥ ✐s s✐♠✐❧❛r ❢♦r I2 ❛♥❞ ✇❡ ✇✐❧❧ ♥♦t r❡♣r♦❞✉❝❡ ✐t✿ I1 = Li (t)N ln(Li (t)) − ln κ + 12 ζi2 (t) ζi (t) ✭✶✳✹✷✮ ❋✐♥❛❧❧② s✉♠♠✐♥❣ ❡✈❡r②t❤✐♥❣ ❢♦r t❤❡ ❝❛♣❧❡ts ❛♥❞ ❡✈❡r② ❝❛♣❧❡ts t♦ ❣❡t t❤❡ ❝❛♣ ♣r✐❝❡✿ n−1 CapLM M (t) = δB(t, Ti+1 ) Li (t)N (d1(t) − κN (d2(t)) i=0 ✇✐t❤ d1,2 (t) = ln(Li (t)) − ln κ ± 21 ζi2 (t) ζi (t) ❛♥❞ Ti ζi2 (t) = γi (s) 2 ds t ❘❡♠✐♥❞✐♥❣ ✭✶✳✸✹✮ ✇❡ ❝❛♥ ❞❡✜♥❡ σnBlack,LM M t❤❡ ❇❧❛❝❦ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ♦❢ ❝❛♣❧❡t ♣r✐❝❡❞ ❜② ▲▼▼✳ σiBlack,LM M = 1 Tn Ti γi (s) 2 ds 0 ✭✶✳✹✸✮ ❍❡♥❝❡✱ t❤❡ ❇●▼ ❝❛♣❧❡t ❝❛♥ ❛❧s♦ ❜❡ q✉♦t❡❞ ✐♥ t❡r♠s ♦❢ ✐ts ❇❧❛❝❦ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✳ ❚❤❛t ✐s t❤❡ ✇❛② ❝❛♣❧❡ts ❛r❡ ❣❡♥❡r❛❧❧② q✉♦t❡❞ ✉s✐♥❣ ❛t t❤❡ ♠♦♥❡② r❛t❡✳ ❯s✐♥❣ t❤✐s ❢♦r♠✉❧❛ ✇❡ s❡❡ ✇❡❧❧ ✇❤② t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✐s ❛✉t♦ ❝❛❧✐❜r❛t❡❞ ♦♥ t❤❡ ❝❛♣❧❡ts ✈♦❧❛t✐❧✐t✐❡s ❛s ✇❡ ❤❛✈❡ ♥♦t ❞♦♥❡ ❛♥② ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❤✐s ❞❡r✐✈❛t✐♦♥✳ ❋❧♦♦r ♣r✐❝❡s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ ❝❛♣✲✢♦♦r ♣❛r✐t② ❡q✉❛t✐♦♥ s❤♦✇♥ ♣r❡✈✐♦✉s❧② ✐♥ ✶✳✸✽✳ ✷✾ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✶✳✸✳✷ ❙✇❛♣t✐♦♥s ❆s ♣r❡✈✐♦✉s❧② ❞❡s❝r✐❜❡❞✱ ✇❡ ❛r❡ ♥♦✇ ❣♦✐♥❣ t♦ ❞❡r✐✈❡ ❛♥❛❧②t✐❝❛❧ ❢♦r♠✉❧❛ t♦ ♣r✐❝❡ ❛ s✇❛♣t✐♦♥✱ ✐✳❡✳ ❛ ❝♦♥tr❛❝t ✇❤❡r❡ ②♦✉ ♣❛② ❛ ♣r❡♠✐✉♠ t♦ ❣❡t t❤❡ ♦♣t✐♦♥ t♦ ❡♥t❡r ❛ s✇❛♣ ♦❢ ❛ ❝❡rt❛✐♥ t❡♥♦r ❛t ♠❛t✉r✐t② ✇❤❡r❡ ②♦✉ ♣❛② ❛ ♣r❡✲♥❡❣♦t✐❛t❡❞ ✜①❡❞ r❛t❡ ✭t❤❡ str✐❦❡✮ ❛❣❛✐♥st ❛ ✢♦❛t✐♥❣ ♦♥❡✳ ❇❧❛❝❦ ❋♦r♠✉❧❛ ❲❡ ❤❛✈❡ s❡❡♥ ♣r❡✈✐♦✉s❧② ❤♦✇ t♦ ❡①♣r❡ss t❤❡ s✇❛♣ r❛t❡ Sp,q ✳ ❚❤✉s✱ ✇❡ ❛r❡ ❣♦✲ ✐♥❣ t♦ ❞❡❞✉❝❡ t❤❡ s✇❛♣t✐♦♥ ♣r✐❝❡ t❤❡ s❛♠❡ ✇❛② ❛s ❢♦r ❝❛♣❧❡ts✿ t❤❛t ♠❡❛♥s ✇❡ ❛ss✉♠❡ ❧♦❣✲♥♦r♠❛❧✐t② ♦❢ t❤❡ ❢♦r✇❛r❞ s✇❛♣ r❛t❡ ❛♥❞ ❝♦♥st❛♥t ♣♦s✐t✐✈❡ ✈♦❧❛t✐❧✲ ✐t② σ ✳ ❈♦♠♣❛r✐♥❣ t❤❡ ❢✉t✉r❡ ❝❛s❤✲✢♦✇s ♦♥ ❛ s✇❛♣ r❛t❡ st❛rt✐♥❣ ❛t Tp ✇✐t❤ ✜①❡❞ r❛t❡ Sp,q (Tp ) t♦ t❤♦s❡ ♦❢ ❛ s✇❛♣ st❛rt✐♥❣ ❛t Tp ✇✐t❤ ✜①❡❞ r❛t❡ κ✱ ✇❡ ❝❛♥ s❤♦✇ t❤❡ ♣❛②♦✛ ♦❢ ❛ ♣❛②❡r s✇❛♣t✐♦♥ ♦♥ ❛ ✉♥✐t❛r② ♥♦t✐♦♥❛❧ ❛s ❛ s❡r✐❡s ♦❢ ❝❛♣❧❡t ♣❛②♦✛s ♣❛✐❞ ❧❛t❡r ✿ q [max(Sp,q (Tp ) − κ) ✭✶✳✹✹✮ i=p+1 ❍❡♥❝❡ ✉s✐♥❣ t❤❡ ♥♦✲❛r❜✐tr❛❣❡ ❛ss✉♠♣t✐♦♥ ❛♥❞ ✐♥ t❤❡ ♠❛r❦❡t ♣r♦❜❛❜✐❧✐t② ♠❡❛✲ s✉r❡ ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞ Q ✇❡ ❤❛✈❡✿ q SwaptionBl p,q (t) B(0, Ti )EQ (Sp,q (Tp ) − κ)+ |Ft = ✭✶✳✹✺✮ i=p+1 ❍❡♥❝❡ ✇❡ ❝❛♥ ✉s❡ ❇❧❛❝❦ ❋♦r♠✉❧❛ ✶✳✸✼ ❛❞❛♣t❡❞ t♦ ❛ ❞❡❧❛②❡❞ ♣❛②♦✛ ✭❢r♦♠ Tp t♦ T − i✮ ❛♥❞ ♦♥❡ ❝❛♥ ❣❡t✿ q SwaptionBl p,q (t) B(0, Ti )[(Sp,q (t)N (d1 ) − κN (d2 )] = i=p+1 ✭✶✳✹✻✮ ✸✵ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✇✐t❤ d1 = ln((Sp,q (t)/κ) + σ 2 (Ti2−t) (Tp − t) σ d2 = d1 − σ Tp − t ❋✐♥❛❧❧② ✇❡ ❛❧s♦ ♦❜t❛✐♥ ❤❡r❡ ❛ ❇❧❛❝❦ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ✇❤✐❝❤ ✇✐❧❧ ❜❡ ✉s❡❞ ❧❛t❡r t♦ ❣✐✈❡ ❛ ♣r✐❝❡ t♦ t❤♦s❡ s✇❛♣t✐♦♥s✳ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡♠♣❤❛s✐③❡ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ t❤❡ ❧♦❣✲♥♦r♠❛❧✐t② ♦❢ t❤❡ ❢♦r✇❛r❞ s✇❛♣ r❛t❡ ✇❤✐❝❤ ✐s ♥♦t t❤❡ ❝❛s❡ ✐♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✳ Pr✐❝✐♥❣ ✐♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✲ ❙✇❛♣ ▼❛r❦❡t ▼♦❞❡❧ ■♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ t❤❡ ♣r✐❝✐♥❣ ❝❛♥♥♦t ❜❡ ❞♦♥❡ ✉s✐♥❣ ❛♥ ❡①❛❝t ❝❧♦s❡❞ ❢♦r♠✉❧❛ ❛♥❞ t❤✐s ✐s t❤❡ ♣✉r♣♦s❡ ♦❢ ❝❤❛♣t❡r ✷✳ ❍♦✇❡✈❡r✱ ♦♥❡ ❝❛♥ ❞❡✈❡❧♦♣ t❤❡ s❛♠❡ ♠♦❞❡❧ ❛s t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❜✉t ✉s✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❋♦r✇❛r❞ s✇❛♣ r❛t❡s ❛r❡ ❧♦❣✲♥♦r♠❛❧✿ t❤✐s ♠♦❞❡❧ ✐s ❝❛❧❧❡❞ t❤❡ ❙✇❛♣ ▼❛r❦❡t ▼♦❞❡❧✳ ❙❡❡ ❬✶✸❪ ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧s ❛❜♦✉t t❤✐s ♠♦❞❡❧✳ ❍❡♥❝❡✱ ❛♥ ❡①❛❝t ♣r✐❝❡ ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❛s ❢♦r t❤❡ ❝❛♣❧❡ts ✐♥ ▲▼▼✳ ❲✐t❤ str❛✐❣❤t❢♦r✇❛r❞ ♥♦t❛t✐♦♥s✿ q M SwaptionSM (t) p,q B(t, Ti )[Sp,q (t)N (d1 ) − κN (d2 )] = i=p+1 ✇✐t❤ d1,2 (t, Ti ) = ln((Sp,q (t)/κ) ± 21 ζ 2 (t, Ti ) ζ(t, Ti ) ❛♥❞ t❤❡ ❇❧❛❝❦ ✈♦❧❛t✐❧✐t② ζ ❝♦♠♣✉t❡❞ ❛s✿ Ti ζ 2 (t, Ti ) = t νi (s) 2 ds ✭✶✳✹✼✮ ✸✶ ■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s ✇❤❡r❡ νi ✐s t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ✈♦❧❛t✐❧✐t② ✭✇❡❧❧ ❛❞❛♣t❡❞✮ ♦❢ t❤❡ ❢♦r✇❛r❞ s✇❛♣ r❛t❡ ✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢♦r✇❛r❞ s✇❛♣ ♠❡❛s✉r❡✳ ❈❤❛♣t❡r ✷ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✷✳✶ ❚❤❡ s❡tt✐♥❣s✿ ▼❛✐♥ ♣✉r♣♦s❡ ♦❢ t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❇❡❢♦r❡ st❛rt✐♥❣ ❛ ❝❛❧✐❜r❛t✐♦♥✱ ❛ ❧✐st ♦❢ ❝❛❧✐❜r❛t✐♦♥ ♦❜❥❡❝ts s❤♦✉❧❞ ❜❡ ❣✐✈❡♥✳ ❆ ❝❛❧✐❜r❛t✐♦♥ ♦❜❥❡❝t ❝❛♥ ❜❡ ❡✐t❤❡r ❛ ❝❛♣❧❡t ♣r✐❝❡✱ ❛ ❢♦r✇❛r❞ r❛t❡ ❝♦rr❡❧❛t✐♦♥ ♦r ❛ s✇❛♣t✐♦♥ ♣r✐❝❡✳ ❊❛❝❤ ♦❢ t❤❡ ❡♥tr✐❡s ✐♥ t❤✐s ❧✐st r❡q✉✐r❡s ❛ ♣r❡❝✐s❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♦❜❥❡❝t ✐ts❡❧❢ ✲ ❢♦r ✐♥st❛♥❝❡✱ ❢♦r ❛ s✇❛♣t✐♦♥✿ ✇❤✐❝❤ t❡♥♦r ♣❡r✐♦❞ t❤❡ s✇❛♣t✐♦♥ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛♥❞ ✇❤❛t t❤❡ ❡①♣✐r② ❞❛t❡ ✐s ✲ ❛♥❞ ♦❢ ❝♦✉rs❡ ♠❛r❦❡t ✈❛❧✉❡ ♦❢ t❤❡ ❧✐q✉✐❞ tr❛❞❡❞ s❡❝✉r✐t✐❡s ✇❡ ❝♦♥s✐❞❡r✳ ◆♦t❡ t❤❛t ❝❛♣❧❡t ❛♥❞ s✇❛♣t✐♦♥ ♣r✐❝❡s ❛r❡ q✉♦t❡❞ ❤❡r❡ ✐♥ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐✲ t✐❡s✳ ❙❛② ❛ ❝❛❧✐❜r❛t✐♦♥ ❤❛s ▼ ❝❛❧✐❜r❛t✐♦♥ ♦❜❥❡❝ts✱ ✇✐t❤ ♠❛r❦❡t ✈❛❧✉❡s xTk raded , k = 1, . . . , M ✳ ●✐✈❡♥ ❛ s❡t ♦❢ ♣❛r❛♠❡t❡rs✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝♦♠♣✉t❡ t❤❡ ♠♦❞❡❧ ✈❛❧✲ ✉❡s ♦❢ t❤❡ M ❝❛❧✐❜r❛t✐♦♥ ♦❜❥❡❝ts ✇✐t❤ t❤❡ ❢♦r♠✉❧❛s ❞❡r✐✈❡❞ ✐♥ t❤❡ ✜rst ♣❛rt✳ odel , k = 1, . . . , M ✳ ❚❤✐s ✇✐❧❧ ❧❡❛❞ ✉s t♦ ❚❤✐s ✇✐❧❧ ②✐❡❧❞ ▼ ♠♦❞❡❧ ✈❛❧✉❡s xM k odel ❛♥❞ t❤❡ ❤✐❣❤❧✐❣❤ts M ❞✐✛❡r❡♥t ❡rr♦rs ❜❡t✇❡❡♥ t❤❡ k✲t❤ ♠♦❞❡❧ ✈❛❧✉❡ xM k ♠❛r❦❡t ✈❛❧✉❡ xTk raded ✳ ❆s ❛ ❜♦tt♦♠ ❧✐♥❡✱ ✇❡ ❛❞❞ ❡✈❡r② ❡rr♦rs t♦ ♦❜t❛✐♥ ❤♦✇ ❢❛r ♦✉r ♣❛r❛♠❡t❡rs ❢♦r t❤❡ ♠♦❞❡❧s ❛r❡ ❢r♦♠ t❤❡ ♠❛r❦❡t ✈❛❧✉❡✳ ❚❤❡ ❝❛❧✐❜r❛t✐♦♥ ♣r♦❝❡ss ❝♦♥s✐sts ✸✸ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✐♥ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤✐s ❡rr♦r ♦✈❡r t❤❡ ♣❛r❛♠❡t❡rs s♦ ❛s t♦ ❣❡t t❤❡ ♠♦❞❡❧ t♦ r❡s❡♠❜❧❡ t❤❡ ♠❛r❦❡t ❛s ❝❧♦s❡ ❛s ♣♦ss✐❜❧❡✳ ❲❤❛t ✇❡ ❝♦✉❧❞ s✉♠ ✉♣ ❜②✿ M odel Errork (xM (param); xTk raded ) k min param ✭✷✳✶✮ k=1 ■♥ t❤✐s ♣❛rt✱ ✇❡ ❞✐s❝✉ss t❤❡ ♠❛✐♥ ♠❡t❤♦❞s ♦❢ ❝❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✳ ❇② ❝❛❧✐❜r❛t✐♦♥ ✇❡ ♠❡❛♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ✭t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t✐❡s ❛♥❞ ❝♦rr❡❧❛t✐♦♥s✮ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ s♦ ❛s t♦ ♠❛t❝❤ ❛s ❝❧♦s❡❧② ❛s ♣♦ss✐❜❧❡ ❞❡r✐✈❛t✐✈❡ ♣r✐❝❡s ❝♦♠♣✉t❡❞ ❛♥❞ ♦❜s❡r✈❡❞ ♣r✐❝❡s ♦❢ ❛❝t✐✈❡❧② tr❛❞❡❞ s❡❝✉r✐t✐❡s✿ ❝❛♣❧❡ts ❛♥❞ s✇❛♣t✐♦♥s✳ ■t ✐s ✈❡r② ❡❛s② t♦ ❝❛❧✐❜r❛t❡ t❤❡ ❇●▼ ♠♦❞❡❧ t♦ ❝❛♣❧❡t ✈♦❧❛t✐❧✐t✐❡s ❛s ✐t ✐s ❛❧♠♦st str❛✐❣❤t ❢♦r✇❛r❞ ❜❡❝❛✉s❡ ✇❡ ❛ss✉♠❡❞ t❤❡ ❧♦❣ ♥♦r♠❛❧✐t② ♦❢ t❤❡ ✉♥❞❡r✲ ❧②✐♥❣ ✭❚❤❡ ❢♦r✇❛r❞ r❛t❡s✮✳ ❇✉t ✐♥ ♦r❞❡r t♦ ♣r✐❝❡ ♣r♦❞✉❝ts ✐♥✈♦❧✈✐♥❣ s✇❛♣ ♣r✐❝❡s✱ ✇❡ ♥❡❡❞ t♦ ❝❛❧✐❜r❛t❡ ✐t ❛❧s♦ ♦♥ t❤❡ s✇❛♣t✐♦♥ ♠❛r❦❡t ❛♥❞ t❤❡ s✇❛♣ r❛t❡s ❛r❡ ♥♦t ❧♦❣✲♥♦r♠❛❧ ✐❢ t❤❡ ❢♦r✇❛r❞ r❛t❡s ❛r❡✳ ❋✐rst ✇❡ ❤❛✈❡ t♦ t❛❦❡ ❝❛r❡ ♦❢ t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ❋♦r✇❛r❞ r❛t❡s t❤❛t ✇❡ ❞❡✜♥❡❞ ♣r❡✈✐♦✉s❧②✿ ❛ss✉♠✐♥❣ t❤❡✐r ❧♦❣✲♥♦r♠❛❧✐t② ❝r❡❛t❡❞ t❤✐s ✈♦❧❛t✐❧✐t②✳ ❉✐✛❡r❡♥t ♣❛r❛♠❡t❡r✐③❛t✐♦♥s ❛r❡ ♣♦ss✐❜❧❡ ❢♦r t❤✐s✳ ❚♦ ♣r✐❝❡ ❝♦rr❡❝t❧② ✇❡ ❤❛✈❡ t♦ ✇♦r❦ ✉s✐♥❣ ❛ s♦❧❡ ♥✉♠❡r❛✐r❡ ✭❚❤❡ s♣♦t ♠❡❛s✉r❡✮ ✇❤✐❝❤ ✐♠♣❧✐❡s ❛ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞✐✛❡r❡♥t ❢♦r✇❛r❞ r❛t❡s ❛♥❞ ❝❤❛♥❣❡s t❤❡ ❞r✐❢t ✭✇❤✐❝❤ ❞♦❡s ♥♦t ✐♠♣❛❝t ♦✉r st✉❞②✮ ❚❤✐s ✇✐❧❧ ❧❡❛❞ ✉s t♦ t❤❡ ❞❡❜❛t❡ ❜❡t✇❡❡♥ ❤✐st♦r✐❝❛❧ ❛♥❞ ✐♠♣❧✐❡❞ ❞❛t❛✳ ❲❡ ✇✐❧❧ s❤♦✇ ❞✐✛❡r❡♥t s♦❧✉t✐♦♥s ❢♦r t❤❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡ ❛♥❞ ✐♥ ❧❛st s❡❝t✐♦♥ ✐❢ ✇❡ s❤♦✉❧❞ ❝❤♦♦s❡ ❤✐st♦r✐❝❛❧ ❞❛t❛ ♦r ✐♠♣❧✐❡❞ ❞❛t❛ ❛s ✐♥♣✉ts✳ ■♥ t❤✐s ❝❤❛♣t❡r ✇❡ ❝♦♥s✐❞❡r ❛ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✇✐t❤ d✲❢❛❝t♦rs ❞❡s❝r✐❜❡❞ ❜②✿ d γik (t)dWtk dLi (t) = µi (t)dt + k=1 ✭✷✳✷✮ ✸✹ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✇❤❡r❡ ❛❧❧ t❤❡ Wtk ❛r❡ ♦rt❤♦❣♦♥❛❧ ❛♥❞ t❤❡ γik ❛r❡ t❤❡ ❧♦❛❞✐♥❣s ♦❢ ❡❛❝❤ ❢❛❝t♦rs✳ ❲❡ ❦♥♦✇ t❤❛t ✇❡ ❤❛✈❡ t❤❡ r❡❧❛t✐♦♥✿ d bik dWtk γi dWt = σi ✭✷✳✸✮ k=1 ❙♦ ✇❡ ❝❛♥ s❡❡ t❤❡ r❡❧❛t✐♦♥ ✇✐t❤ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❛r✐s❡s✿ γi γj = σi σj ρij d = σi σj bik bkj k=1 ✇❤❡r❡ bi ❛r❡ ❝♦rr❡❧❛t✐♦♥ ✈❡❝t♦rs ✐♥ (R+ )d ❛♥❞ γk : [0, Tk−1 ] → (R+ )d ✳ ❖♥ t♦♣ ♦❢ t❤✐s✱ ✇❡ ❤❛✈❡ ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ ❛ ❣♦♦❞ ♣r✐❝✐♥❣ ♦❢ t❤❡ ❝❛♣❧❡ts d b2ik = 1 ✭✷✳✹✮ k=1 ❚❤✐s ❞❡s❝r✐♣t✐♦♥ ❤❛s t❤❡ ❤✉❣❡ ❛❞✈❛♥t❛❣❡ t♦ ❞✐st✐♥❣✉✐s❤ t❤❡ ✈♦❧❛t✐❧✐t② ❛♥❞ t❤❡ ❝♦rr❡❧❛t✐♦♥ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡♥ ❛ s❡♣❛r❛t❡ ❝❛❧✐❜r❛t✐♦♥ ✐s ♣♦ss✐❜❧❡ ✇❤❡r❡ σi ✇✐❧❧ ✐♥✢✉❡♥❝❡ ♣r✐❝❡ ♦❢ t❤❡ ❝❛♣❧❡ts ✭❙❡❡ ❬✶✷❪✮ ❛♥❞ t❤❡ ❝❤♦✐❝❡ ♦❢ (bik ✇✐❧❧ ✐♥✢✉❡♥❝❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡✳ ✷✳✷ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t② ❆s ♣r❡✈✐♦✉s❧② ❡①♣❧❛✐♥❡❞ ✇❡ ❤❛✈❡ t♦ ❣✐✈❡ ❛ s❤❛♣❡ t♦ t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✲ ✐t② ♦❢ t❤❡ ❢♦r✇❛r❞ r❛t❡s✳ ❚♦ ❝❧❛r✐❢②✱ ✇❡ ❤❛✈❡ t♦ ✜❧❧ ✐♥ t❤❡ ♠❛tr✐① ❣✐✈❡♥ ✐♥ ✷✳✶✳ ❲❡ ❛r❡ ❣✐✈❡♥ t❤❡ ❝❤♦✐❝❡ ❜❡t✇❡❡♥ s❡✈❡r❛❧ ♣❛r❛♠❡t❡r✐③❛t✐♦♥s ❢♦r t❤❡ str✉❝t✉r❡ ✇✐t❤ ❞✐✛❡r❡♥t ❛❞✈❛♥t❛❣❡s✳ ✷✳✷✳✶ ❚♦t❛❧ ♣❛r❛♠❡t❡r✐③❡❞ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ❆ ✜rst s✐♠♣❧❡ ✐❞❡❛ ✇♦✉❧❞ ❜❡ t♦ ❝❤♦♦s❡ ❛ t♦t❛❧ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ❝♦♥s✐❞❡r✐♥❣ t❤❛t ❡❛❝❤ σij ✐s ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✜t t❤❡ ♠❛tr✐① t♦ ❜♦t❤ ❝❛♣❧❡ts ❛♥❞ s✇❛♣✲ ✸✺ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ■♥st❛♥t✳ ❱♦❧ t ∈ (T0 , T1 ] (T1 , T2 ] (T2 , T3 ] · · · L1 (t) σ1,1 ❞❡❛❞ ··· ··· L2 (t) σ2,1 σ2,2 ❞❡❛❞ ··· L3 (t) σ3,1 σ3,2 σ3,3 ··· ✳✳ ✳ Li (t) ✳✳ ✳ LM (t) ✳✳ ✳ σi,1 ✳✳ ✳ σM,1 ✳✳ ✳ σi,2 ✳✳ ✳ σM,2 ✳✳ ✳ ··· σi,3 ··· σM,3 ··· ··· ✳✳ ✳ (TM −1 , TM ] ❞❡❛❞ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ❞❡❛❞ σM,M ❚❛❜❧❡ ✷✳✶✿ ●❡♥❡r❛❧ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ t✐♦♥s✳ ❍♦✇❡✈❡r ❛s ✐t ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❬✶✼❪ t❤✐s ♣r♦❝❡ss ✐♥✈♦❧✈❡s ♥✉♠❡r♦✉s ✐ss✉❡s ✐♥❝❧✉❞✐♥❣ ♦✈❡r✲♣❛r❛♠❡t❡r✐③❛t✐♦♥✳ ❚❤♦✉❣❤✱ t❤❡ s②st❡♠ ♦♥❧② ❤❛✈❡ ❛ ✜♥✐t❡ ♥✉♠✲ ❜❡r ♦❢ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ❛♥❞ ❝❛♥♥♦t ❜❡ ❝♦♥str❛✐♥❡❞ ❡✈❡r②✇❤❡r❡✳ ❚❤❛t ✐s ✇❤② ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ❛ s❡♠✐ ♣❛r❛♠❡t❡r✐③❡❞ str✉❝t✉r❡✳ ✷✳✷✳✷ ●❡♥❡r❛❧ P✐❡❝❡✇✐s❡✲❈♦♥st❛♥t P❛r❛♠❡t❡r✐③❛t✐♦♥ ❆ ✈❡r② ✉s❡❞ str✉❝t✉r❡ ✐s t❤❡ ♦♥❡ t❤❛t ♠❛❦❡s t❤❡ ✈♦❧❛t✐❧✐t② ❞❡♣❡♥❞s ♦♥❧② ♦♥ t❤❡ ❞✐st❛♥❝❡ t♦ ♠❛t✉r✐t②✳ ❋♦r ♣r❛❝t✐❝❛❧ ♣✉r♣♦s❡s✱ ✐❢ ✇❡ ❢♦r❝❡ t❤❡ ✈♦❧❛t✐❧✐t② t♦ ❜❡ ❝♦♥st❛♥t ♦♥ ❡❛❝❤ t✐♠❡ ❜✉❝❦❡t✱ ✇❡ ❝❛♥ ✇r✐t❡✿ σi (t) = σ(Ti − t) = ηi−k , t = [Tk ; Tk+1 ] ❋✐♥❛❧❧② ✇❡ ❝❛♥ ♦r❣❛♥✐③❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t✐❡s ✐♥ ❛ ♠❛tr✐① ❛s ❢♦❧❧♦✇s✿ ❲❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t ❞✉❡ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs✱ t❤❡ ♠❛✐♥ ✐ss✉❡ ✇✐t❤ t❤✐s str✉❝t✉r❡ ✐s t❤❛t ✐t ❞♦❡s ♥♦t ❛❧❧♦✇ ❛ s✐♠✉❧t❛♥❡♦✉s ❝❛❧✐❜r❛t✐♦♥ ♦❢ ❜♦t❤ ❝❛♣❧❡ts ❛♥❞ s✇❛♣t✐♦♥s ✈♦❧❛t✐❧✐t✐❡s ❜✉t ♦♥❧② ❢♦r ♦♥❡ ♦❢ t❤❡♠ ✭✐♥ ▲▼▼✱ ✐t ✐s ♦♥ ❝❛♣❧❡ts✮✳ ❙❡❡ ❬✶✾❪ ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧s ❛❜♦✉t ✐t✳ ✸✻ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ■♥st❛♥t✳ ❱♦❧ t ∈ (T0 , T1 ] (T1 , T2 ] (T2 , T3 ] · · · L1 (t) η1 ❞❡❛❞ ··· ··· L2 (t) η2 η1 ❞❡❛❞ ··· L3 (t) η3 η2 η1 ··· ✳✳ ✳ Li (t) ✳✳ ✳ LM (t) ✳✳ ✳ ηi ✳✳ ✳ ηM ✳✳ ✳ ηi−1 ✳✳ ✳ ηM −1 ✳✳ ✳ ❞❡❛❞ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ··· ηi−2 ··· ηM −2 ··· ··· ✳✳ ✳ (TM −1 , TM ] ❞❡❛❞ η1 ❚❛❜❧❡ ✷✳✷✿ P✐❡❝❡✇✐s❡✲❝♦♥st❛♥t ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ✷✳✷✳✸ ▲❛❣✉❡rr❡ ❢✉♥❝t✐♦♥ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ t②♣❡ ✈♦❧❛t✐❧✐t② ❘❡❜♦♥❛t♦ ❤❛s ♣r♦♣♦s❡❞ ❛ ♠♦r❡ ❛❝❝✉r❛t❡ str✉❝t✉r❡ ❛❞❞✐♥❣ ♦♥❡ ♠♦r❡ ♣❛r❛♠❡✲ t❡r t♦ t❤❡ ❢♦r✇❛r❞ r❛t❡s ❛♥❞ ❦❡❡♣✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ✈♦❧❛t✐❧✐t② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ t♦ ♠❛t✉r✐t②✳ ❆s ❛ ♠❛tt❡r ♦❢ ❢❛❝t ❞♦✐♥❣ t❤✐s ✇❡ ❡♥r✐❝❤ t❤❡ str✉❝t✉r❡ ❛♥❞ ♣❡r♠✐ts ❛ ❜❡tt❡r ✜t ✇✐t❤ ♠❛r❦❡t ♣r✐❝❡s ✭♦♥ ❜♦t❤ ❝❛♣❧❡ts ❛♥❞ s✇❛♣t✐♦♥s✮ t❤❛♥ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ❜② ❛❞❞✐♥❣ ❛ st❛t✐♦♥❛r② ♣❛rt ηi−k ✿ σi (t) = ci ηi−k , t = [Tk ; Tk+1 ] ❖♥❝❡ ❛❣❛✐♥ ✇❡ ❝❛♥ s✉♠ ✉♣ t❤✐s str✉❝t✉r❡ ✐♥ ❛ ♥❡✇ ♠❛tr✐①✿ ■♥st❛♥t✳ ❱♦❧ t ∈ (T0 , T1 ] (T1 , T2 ] ··· ··· L1 (t) c1 η1 ❞❡❛❞ (T2 , T3 ] ··· L2 (t) c2 η2 c2 η1 ❞❡❛❞ ··· L3 (t) c3 η3 c3 η2 c3 η1 ··· (TM −1 , TM ] Li (t) ci ηi ci ηi−1 ci ηi−2 ··· ❞❡❛❞ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ LM (t) cM ηM cM ηM −1 cM ηM −2 ··· ··· cM η1 ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ··· ❞❡❛❞ ❚❛❜❧❡ ✷✳✸✿ ▲❛❣✉❡rr❡ t②♣❡ ✈♦❧❛t✐❧✐t② str✉❝t✉r❡ ❖❢ ❝♦✉rs❡✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡ t❤❛t ✇❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ 2N ♣❛r❛♠❡t❡rs ✐♥✲ st❡❛❞ ♦❢ N ✐♥ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✳ ❚♦ ❡❛s❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ✸✼ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✉s❡ ❘❡❜♦♥❛t♦ ✐❞❡❛ ❛❜♦✉t t❤❡ st❛t✐♦♥❛r② ♣❛rt ♦❢ t❤❡ ✈♦❧❛t✐❧✐t② ηi ✳ ▼♦st ♦❢ t❤❡ t✐♠❡ t❤✐s ♣❛rt ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ✇✐t❤ ❛ s♠❛❧❧ ❤✉♠♣ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❝✉r✈❡✳ ❋✐♥❛♥❝✐❛❧ ❥✉st✐✜❝❛t✐♦♥ ❢♦r t❤✐s ❤✉♠♣ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✻❪✳ ❚❤❡ ✐❞❡❛ ✐s t♦ r❡♣r❡s❡♥t ✐t ✉s✐♥❣ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ▲❛❣✉❡rr❡ ❢✉♥❝t✐♦♥s✱ ❡s♣❡❝✐❛❧❧② t❤❡ t✇♦ ✜rst✳ τ ζ1 : τ → e− 2 τ ζ2 : τ → τ e− 2 ❙♦ ✇❡ ♦❜t❛✐♥ ❢♦r η ✿ η(τ ) = ae−βτ + bτ e−βτ + c η(τ ) = e−βτ (a + bτ ) + c ❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② ✇❡ ❢♦r❝❡✿ η(0) = 1 = a + c ❛♥❞ ✇❡ ❣❡t ✇✐t❤ ❛ s❧✐❣❤t ❝❤❛♥❣❡ ♦❢ ♥♦t❛t✐♦♥ t♦ r❡✢❡❝t ✇❤❛t t❤❡s❡ ❝♦♥st❛♥ts r❡♣r❡s❡♥t ✿ η(τ ) = η∞ + (1 − η∞ + bτ )e−βτ ❆♥❞ ✜♥❛❧❧② ✇❡ ❣❡t ✿ ||γi (t)|| = σi (t) = ci η(τ ) ✭✷✳✺✮ ❚❤✐s str✉❝t✉r❡ ❢♦r ✈♦❧❛t✐❧✐t② ✐s ❛ ❣♦♦❞ ❝❤♦✐❝❡ ❜❡t✇❡❡♥ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ q✉❛❧✐t② ♦❢ t❤❡ ✜t✿ ❝♦♠♣❛r❡ t♦ t❤❡ ♣r❡✈✐♦✉s str✉❝t✉r❡✱ ✇❡ ❤❛✈❡ t♦ ♣r♦✲ ♣♦s❡ ✈❛❧✉❡s ❢♦r η∞ , β, b ♦♥ t❤❡ t♦♣ ♦❢ t❤❡ ci ✭t❤❡② ❛r❡ ❤❡r❡ ❛s ♥♦r♠❛❧✐③❛t✐♦♥ ❢❛❝t♦rs ❛❢t❡r t❤❡ ✜rst ❝♦❡✣❝✐❡♥ts ❤❛✈❡ ✇❡❧❧ r❡♣r♦❞✉❝❡❞ t❤❡ s❤❛♣❡ ♦❢ t❤❡ t❡r♠✲ str✉❝t✉r❡ ✈♦❧❛t✐❧✐t②✮ ❛♥❞ t❤✐s ❣✐✈❡s t❤❡ ❜❡st ✜t t♦ t❤❡ ♠❛r❦❡t ❛s ✇❡ ❝❛♥ ✉s❡ ❛❧s♦ ❞❛t❛ ❢r♦♠ t❤❡ s✇❛♣t✐♦♥ ♠❛r❦❡t ✭❚❤❡ ♣✐❡❝❡✲✇✐s❡ str✉❝t✉r❡ ♦♥❧② ♣❡r♠✐ts ✸✽ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ t♦ ✜t t❤❡ ❝❛♣❧❡t ✈♦❧❛t✐❧✐t✐❡s ✐♥ ❡❛❝❤ ❜✉❝❦❡t✮✳ ❲❡ ❝❛♥ ❢♦r ✐♥st❛♥❝❡ s❡t t❤❡ ci ✉s✐♥❣ t❤❡ ❇❧❛❝❦ ✈♦❧❛t✐❧✐t② ❞❡✜♥✐t✐♦♥ ❢♦r ❛ ❝❛♣❧❡t ❛♥❞ ✜t ♣❡r❢❡❝t❧② t❤❡ ❝❛♣❧❡t ♠❛r❦❡t ❛♥❞ ♦♣t✐♠✐③❡ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ η(τ ) ♦♥ t❤❡ s✇❛♣t✐♦♥ ✈♦❧❛t✐❧✐t✐❡s ✿ ci = √ σiBS Ti Ti 0 ✭✷✳✻✮ η(Ti − s)ds ❍❡♥❝❡✱ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤✐s ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t② t❡r♠ str✉❝t✉r❡ ❢♦r t❤❡ ♥❡①t ♣❛rts✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ str✉❝t✉r❡ ✐s ❣✐✈❡♥ ✐♥ ✷✳✶ ✸✾ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✶✿ ❊①❛♠♣❧❡ ♦❢ ❛ ❤✉♠♣❡❞ ▲❛❣✉❡rr❡✲t②♣❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t② ❢♦r b = 5.60, β = 1.75, ❛♥❞ η∞ = 0.96 ❜❡❢♦r❡ ♥♦r♠❛❧✐③❛t✐♦♥ ❜② t❤❡ ci ❢❛❝t♦r ✹✵ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✷✳✸ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❛♠♦♥❣ t❤❡ ❋♦r✇❛r❞ ❘❛t❡s ❚♦ ♣r✐❝❡ ❛♥ ✐♥t❡r❡st r❛t❡ ❞❡r✐✈❛t✐✈❡✱ ✐t s❡❡♠s ♣r❡tt② ❝❧❡❛r t❤❛t ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❢❛❝❡ ❝♦rr❡❧❛t✐♦♥ ✐ss✉❡s ❛♠♦♥❣ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s✳ ❍❡♥❝❡✱ ✇❡ ❤❛✈❡ t♦ ❝♦♥s✐❞❡r t❤❛t t❤❡ ❢♦r✇❛r❞ r❛t❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ t♦ ❡st✐♠❛t❡ t❤✐s✳ ▲❡t ❝♦♥s✐❞❡r t❤❡ ❢❛♠✐❧② ♦❢ t❤❡ ❢♦r✇❛r❞ r❛t❡s {Li (t)} ✇❡ ❝❛♥ ✇r✐t❡✿ dLi (t) = µi ({Li (t)}, t)dt + γi (t) · dWt Li (t) ✇❤❡r❡ ✇❡ ❝❛♥ r❡❝♦❣♥✐③❡ t❤❡ ✈♦❧❛t✐❧✐t② t❡r♠ ✇❡ ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r ❛♥❞ ✇❤❡r❡ Wt ✐s t❤❡ ✉s✉❛❧ ❞✲❞✐♠❡♥s✐♦♥♥❛❧ ♦rt❤♦❣♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❚❤❡ ❝♦rr❡❧❛t✐♦♥ ✈❡r② s✐♠♣❧② ❛♣♣❡❛rs ✇❤❡♥ t❛❦✐♥❣ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦❢ t❤❡ ✈♦❧❛t✐❧✐t② t❡r♠s✿ ❉❡✜♥✐t✐♦♥✿ ❚❤❡ ✐♥st❛♥t❛♥❡♦✉s ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ❢♦r✇❛r❞ r❛t❡s Li (t) ❛♥❞ Lj (t) ✐s ❞❡✜♥❡❞ ❜②✿ ρij = cov(Li (t), Lj (t)) V ar(Li (t))V ar(Lj (t)) ✭✷✳✼✮ ■♥ t❤❡ ❇●▼ ❝❛s❡✱ t❤✐s ❞❡✜♥✐t✐♦♥ ❜❡❝♦♠❡s✿ ρij = γi (t) · γj (t) = bi bj |γi (t)||γj (t)| ❋✐♥❛❧❧②✱ t❤❡ ❝❛❧✐❜r❛t✐♦♥ ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ ❛ ♠❛tr✐① B ∈ M(M, d) ✇✐t❤ M t❤❡ ♥✉♠❜❡r ♦❢ ❢♦r✇❛r❞ r❛t❡s ♥❡❝❡ss❛r② t♦ ❜✉✐❧❞ t❤❡ ♣r✐❝❡ ♦❢ ♦✉r ❞❡r✐✈❛t✐✈❡ ❛♥❞ d t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝t♦rs ♦❢ ♦✉r ♠♦❞❡❧ ✇❤✐❝❤ ♣❡r♠✐ts t❤❡ ❜❡st t♦ ❛♣♣r♦❛❝❤ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ✉s✐♥❣ ❛ ♥♦r♠ ✇❡ ❤❛✈❡ t♦ ❞❡✜♥❡✳ ❖♥❡ s✉❝❤ ❞✐st❛♥❝❡ ❝♦✉❧❞ ❜❡ t❤❡ ❋r♦❜❡♥✐✉s ♥♦r♠ ❛s ✇❡ ✇✐❧❧ s❡❡ ✐♥ s❡❝t✐♦♥ ✷✳✸✳✷✳ ✹✶ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✷✳✸✳✶ ❍✐st♦r✐❝ ❝♦rr❡❧❛t✐♦♥ ✈s ♣❛r❛♠❡tr✐❝ ❝♦rr❡❧❛t✐♦♥ ❚❤❡ ❝❤♦✐❝❡ ♦❢ t❤✐s str✉❝t✉r❡ ✐s ♦♥❡ ♦❢ t❤❡ ❦❡② ♦❢ ❛ ❣♦♦❞ ❇●▼ ❝❛❧✐❜r❛t✐♦♥✳ ❲❡ ✇✐❧❧ s❡❡ ✇❤❛t ❛r❡ t❤❡ ❞✐✛❡r❡♥t ♣♦ss✐❜✐❧✐t✐❡s ❛♥❞ ✇❤❛t ✐s t❤❡ ❜❡st ✇❛② t♦ ❝❛❧✐❜r❛t❡ t❤❡ ❝♦rr❡❧❛t✐♦♥✳ ❍✐st♦r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ❆ r❛t❤❡r ♥❛t✉r❛❧ ❝❤♦✐❝❡ ✇♦✉❧❞ ❜❡ t♦ ❝♦♥s✐❞❡r t❤❡ ❤✐st♦r✐❝ ❝♦rr❡❧❛t✐♦♥ ❜❡✲ t✇❡❡♥ ❢♦r✇❛r❞ r❛t❡s ❛s ❛ ❣♦♦❞ ❡st✐♠❛t✐♦♥ ❢♦r t❤❡ ♣r❡s❡♥t ♦♥❡✳ ■♥ ♣r❛❝t✐❝❡✱ ②♦✉ ♥❡❡❞ t♦ ❝♦❧❧❡❝t ❞✉r✐♥❣ t❤❡ ❧❛r❣❡st ♣❡r✐♦❞ ♦❢ t✐♠❡ t❤❡ ❞❛✐❧② ❝❤❛♥❣❡s ✐♥ t❤❡ ❞✐✛❡r❡♥t ❢♦r✇❛r❞ r❛t❡s ❛♥❞ ❝♦♠♣✉t❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ✭❍❡r❡ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ✐s ❝♦♥st❛♥t ♦✈❡r t✐♠❡ ❛s ✇❡ ❝♦♥s✐❞❡r ❛ ❧❛r❣❡ ♣❡r✐♦❞ ♦❢ t✐♠❡ ✭✶✾✾✹✲✷✵✵✻✮ ❜✉t s♦♠❡ ♦♣❡r❛t♦rs ♦❢ t❤❡ ♠❛r❦❡t ❤❛✈❡ ♦❜s❡r✈❡❞ t❤❛t ❞✉♦ t♦ ♠❛r❦❡t ❥✉♠♣s t❤✐s ✐♥❢♦r♠❛t✐♦♥ ✐s ♥♦t ❛❝❝✉r❛t❡ ❛♥❞ ♣r♦♣♦s❡ t♦ ✉s❡ ❛ s❧✐❞✐♥❣ ✇✐♥❞♦✇ ♦❢ N ❞❛②s t❤❛t ❡①❝❧✉❞❡ s♣❡❝✐❛❧ ❞❛②s ❧✐❦❡ ❋❊❉ ♠❡❡t✐♥❣s✱ ❈P■ ❛♥♥♦✉♥❝❡♠❡♥ts✳✳✳✳ ❲❡ r❡♠✐♥❞ t❤❡ ❢♦r♠✉❧❛ t♦ ❡st✐♠❛t❡ t❤❡ ❤✐st♦r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ρij ❜❡t✇❡❡♥ t❤❡ ❋♦r✇❛r❞ ❘❛t❡s L(Ti ) ❛♥❞ L(Tj ) ✐s ❣✐✈❡♥ ❜② ✭✷✳✼✮ ❚❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ s❤♦✇ ❛ ❝❧❡❛r❧② ✈✐s✐❜❧❡ ❞❡✲❝♦rr❡❧❛t✐♦♥ ❛❧♦♥❣ t❤❡ ❝♦❧✉♠♥s ✇❤❡♥ ♠♦✈✐♥❣ ❛✇❛② ❢r♦♠ t❤❡ ❞✐❛❣♦♥❛❧✳ ❋✐♥❛❧❧②✱ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤♦s❡ ❞❛t❛ ❛r❡ ✈❡r② ♦❢t❡♥ ❞✐st✉r❜❡❞ ❛s s❤♦✇♥ ✐♥ ✷✳✷✳ ✹✷ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✷✿ ❍✐st♦r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ❛♠♦♥❣ ❋♦r✇❛r❞ ✶❨✲▲✐❜♦r r❛t❡s ❜❡t✇❡❡♥ ✶✾✾✹ ❛♥❞ ✷✵✵✻ ✇✐t❤ ❞❛✐❧② ♦❜s❡r✈❛t✐♦♥s ✹✸ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋♦r t❤❡s❡ r❡❛s♦♥s✱ s❡✈❡r❛❧ ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ✐♥ ♦r❞❡r t♦ ❣✐✈❡ ❛ ♠♦r❡ r❡❣✉❧❛r s❤❛♣❡ t♦ ❤✐st♦r✐❝❛❧ ❞❛t❛ ❛♥❞ s✐♠♣❧✐❢② t❤❡ ❝♦♠♣✉t❛t✐♦♥✳ ❋✉r✲ t❤❡r♠♦r❡✱ ❚❤✐s ✐s ❜❡tt❡r ✐♥ t❡r♠s ♦❢ ❝♦♥s✐st❡♥t ♣r✐❝✐♥❣ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛s t❤❡ ❣r❡❡❦s ✇✐❧❧ ❣❡t s♠♦♦t❤❡r ✇✐t❤ ❛ s♠♦♦t❤❡r ❝♦rr❡❧❛t✐♦♥ s✉r❢❛❝❡✳ P❛r❛♠❡t❡r✐③❡❞ ❝♦rr❡❧❛t✐♦♥ ♠♦❞❡❧s ❙✐♠♣❧❡ ❡①♣♦♥❡♥t✐❛❧ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ❚❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥❛❧ ❢♦r♠ ❢♦r ❛ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐s ♣♦ss✐❜❧② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ρij = exp[−β|Ti − Tj |], t ≤ min(Ti , Tj ) ✭✷✳✽✮ ✇✐t❤ Ti ❛♥❞ Tj ✱ t❤❡ ❡①♣✐r✐♥❣ ❞❛t❡s ♦❢ t❤❡ i✲t❤ ❛♥❞ j ✲t❤ ❢♦r✇❛r❞ r❛t❡s✱ ❛♥❞ β ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ❚❤✐s ❢♦r♠ r❡s♣❡❝ts s❡✈❡r❛❧ ✜♥❛♥❝✐❛❧ r❡q✉✐r❡♠❡♥ts✿ ✶✳ ❚❤❡ ❢❛rt❤❡r ❛♣❛rt t✇♦ ❢♦r✇❛r❞ r❛t❡s ❛r❡✱ t❤❡ ♠♦r❡ ❞❡✲❝♦rr❡❧❛t❡❞ t❤❡② ❛r❡✳ ✷✳ ❚❤❡ ❝♦♥❞✐t✐♦♥ β ≥ 0 ❛ss✉r❡ t❤❛t t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① [ρij ] ✐s ❛❞♠✐s✲ s✐❜❧❡ ✭❆ r❡❛❧ s②♠♠❡tr✐❝ ♠❛tr✐① ✇✐t❤ ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s✮✳ ✸✳ ❍♦✇❡✈❡r✱ ♦♥❡ ♠❛② ♥♦t✐❝❡ t❤❛t t❤✐s ❢♦r♠ ✐s ♥♦t ♣r❡❝✐s❡ ❡♥♦✉❣❤ ❛s ✐t ❞♦❡s ♥♦t ❣✐✈❡ t❤❡ ♣♦ss✐❜✐❧✐t② t♦ ✐♥❞✐❝❛t❡ ❤♦✇ ❢❛st ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t✐♠❡ ❜❡t✇❡❡♥ t❤❡ ❡①♣✐r✐♥❣ ❞❛t❡s t❤❡ ❢♦r✇❛r❞ r❛t❡s ❞❡✲❝♦rr❡❧❛t❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✸✵❨ ❋♦r✇❛r❞ r❛t❡ ❛♥❞ t❤❡ ✶✵❨ ❋♦r✇❛r❞ r❛t❡ ❤❛✈❡ t❤❡ s❛♠❡ ❝♦rr❡❧❛t✐♦♥ t❤❛t t❤❡ ✷✵❨ ❋♦r✇❛r❞ r❛t❡ ❛♥❞ t❤❡ ✸♠ ❋♦r✇❛r❞ ❘❛t❡✳ ❖♥❡ ❝❛♥ r❡❢❡r t♦ t❤❡ ❝♦rr❡❧❛t✐♦♥ s✉r❢❛❝❡ ❣✐✈❡♥ ✐♥ ✷✳✸✳ ❚❤✐s ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤✐s ❢♦r♠ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t✐♠❡ t ❡①♣❧✐❝✐t❧② ❛s ♦♥❡ ❝❛♥ s❡❡ ✐♥ ❡q✉❛t✐♦♥ ✷✳✽✳ ❖♥❡ ✉♥❞❡rst❛♥❞s t❤❛t t❤✐s ❢❡❛t✉r❡ ✐s ❛❧s♦ ❛♥ ❛❞✈❛♥t❛❣❡ ♦♥ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇ ✭❢♦r t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ρij σi (t)σi (t)dt✮ ❜✉t ✐s t❤✐s s✐♠♣❧✐✜❝❛t✐♦♥ ✇♦rt❤ ✐t❄ ✹✹ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐♥❛❧❧②✱ ✇❡ ❝❛♥ ❣❡♥❡r❛❧✐③❡ t❤✐s ❢✉♥❝t✐♦♥❛❧ ❢♦r♠ ✷✳✽ ❜② ❛❞❞✐♥❣ ❛ t❡r♠ ♦❢ ❛s②♠♣✲ t♦t✐❝ ❞❡✲❝♦rr❡❧❛t✐♦♥ ✇❤✐❝❤ ♠❡❛♥s t❤❛t ✇❤❡♥ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❡①♣✐r✐♥❣ ❞❛t❡s ❣♦❡s t♦ +∞ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❝❛♥♥♦t ❣♦ t♦ ③❡r♦ ❜✉t t♦ ❛ ✜♥✐t❡ ❧❡✈❡❧ ρ∞ ✳ ❚❤❡ ❡q✉❛t✐♦♥ ✷✳✽ ✐s ❝❤❛♥❣❡❞ ✐♥t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♥❡✿ ρij = ρ∞ + (1 − ρ∞ ) exp[−β|Ti − Tj |] ✭✷✳✾✮ ❖♥❡ ❝❛♥ ❝❤❡❝❦ t❤❛t t❤✐s str✉❝t✉r❡ ❣✐✈❡s ❛ ♠❛tr✐① ♦❢ ❝♦✉rs❡ r❡❛❧✱ s②♠♠❡tr✐❝ ❛♥❞ ❤❛s ♣♦s✐t✐✈❡ ❡✐❣❡♥✈❛❧✉❡s✿ ✐t ✐s ❛♥ ❛❞♠✐ss✐❜❧❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✳ ✹✺ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✸✿ ❙✐♠♣❧❡ ❊①♣♦♥❡♥t✐❛❧ P❛r❛♠❡t❡r✐③❡❞ ❝♦rr❡❧❛t✐♦♥ ❛♠♦♥❣ ❋♦r✇❛r❞ r❛t❡s ✇✐t❤ β = 9% ✹✻ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ▼♦❞✐✜❡❞ ❡①♣♦♥❡♥t✐❛❧ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ❘❡❜♦♥❛t♦ ✐♥ ❬✶✻❪ ❤❛s ♣r♦♣♦s❡❞ ❛ s❧✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ✇❤✐❝❤ ❣✐✈❡s ❜❡tt❡r r❡s✉❧ts✿ ρij = exp[−βmin(Ti ,Tj ) |Ti − Tj |] ✭✷✳✶✵✮ ❍❡r❡ βmin(Ti ,Tj ) ✐s ♥♦t ❛ ❝♦♥st❛♥t ❛♥②♠♦r❡ ❜✉t ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡❛r❧✐❡st ❡①✲ ♣✐r✐♥❣ ❢♦r✇❛r❞ ❞❛t❡✳ ◆❡✈❡rt❤❡❧❡ss✱ ❙❝❤♦❡♥♠❛❦❡rs ❛♥❞ ❈♦✛❡② ✐♥ ❬✶✽❪ ❤❛✈❡ s❤♦✇♥ t❤❛t t❤✐s t②♣❡ ♦❢ ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ❛ss✉r❡ ❛♥②♠♦r❡ t❤❛t t❤❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ✇✐❧❧ r❡♠❛✐♥ ♣♦s✐t✐✈❡✱ ❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥ ❢♦r ❛ ♠❛tr✐① t♦ ❜❡ ❝♦rr❡✲ ❧❛t✐♦♥ ❛❞♠✐ss✐❜❧❡✳ ❇✉t✱ ✐❢ ✇❡ ❝❤♦♦s❡✿ βmin(Ti ,Tj ) = β0 exp(−γ min(Ti , Tj )) ✭✷✳✶✶✮ t❤❡♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ρij ❛r❡ ❛❧❧ ♣♦s✐t✐✈❡✳ ❚❤✐s ❢♦r♠ ✜ts t❤❡ r❛t❡ ♦❢ ❞❡✲ ❝♦rr❡❧❛t✐♦♥ ❢❡❛t✉r❡ ❞✐s❝✉ss❡❞ ❜❡❢♦r❡ ✇❤✐❧❡ st✐❧❧ ♥♦t ❞❡♣❡♥❞✐♥❣ ♦❢ t ♣r❡s❡r✈✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❢❡❛t✉r❡✳ ✹✼ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✹✿ ▼♦❞✐✜❡❞ ❊①♣♦♥❡♥t✐❛❧ P❛r❛♠❡t❡r✐③❡❞ ❝♦rr❡❧❛t✐♦♥ ❛♠♦♥❣ ❋♦r✇❛r❞ r❛t❡s ✇✐t❤ β0 = 12% ❛♥❞ γ = 33% ✹✽ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❙❝❤♦❡♥♠❛❦❡rs✲❈♦✛❡② ❛♣♣r♦❛❝❤ ❙❝❤♦❡♥♠❛❦❡rs✲❈♦✛❡② ❤❛✈❡ ♣r♦♣♦s❡❞ ✐♥ ❬✶✽❪ ❛ s❡♠✐✲♣❛r❛♠❡tr✐❝ ❢✉❧❧ r❛♥❦ str✉❝t✉r❡ ❢♦r t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✳ ❚❤✐s s❡♠✐✲♣❛r❛♠❡tr✐❝ str✉❝t✉r❡ ♣r♦✈✐❞❡s ❛ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❜② s✉❜❥❡❝t✐♥❣ ❛ r❛✲ t✐♦ ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡ ✇❤✐❝❤ ♦❜❡②s t♦ s✐♠♣❧❡ ❡❝♦♥♦♠✐❝❛❧ ♣r✐♥❝✐♣❧❡s✳ ❚❤❡② ❞❡s❝r✐❜❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ρi,i+p ✇✐t❤ ❛♥ ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ i ✇❤❡♥ p ✐s ✜①❡❞✳ ❚❤✐s str✉❝t✉r❡ ✐s ♠♦r❡ ✐♥✈♦❧✈❡❞ ❜✉t ✐t ❤❛s t❤❡ ♠♦r❡ r♦❜✉st♥❡ss ❛♥❞ ❣❡♥❡r❛t❡s ❛❞♠✐ss✐❜❧❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐❝❡s✳ ρij = exp − |i − j| ln ρ∞ m−1 i2 + j 2 + ij − 3mi − 3mj + 3i + 3j + 2m2 − m − 4 + η1 + (m − 2)(m − 3) i2 + j 2 + ij − mi − mj − 3i − 3j + 3m + 2 , − η2 (m − 2)(m − 3) (i, j) ∈ [1, m]2 , 3η1 ≤ η2 ≤ 0, 0 ≤ η1 + η2 ≤ −lnρ∞ ❚❤✐s str✉❝t✉r❡ ❡♥❥♦②s s♦♠❡ ✈❡r② ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s✿ ❋✐rst❧② ✿ t❤❡ ♠❛tr✐❝❡s ♣r♦❞✉❝❡❞ ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ♣♦s✐t✐✈❡ s❡♠✐✲❞❡✜♥✐t❡✱ ❛s ❡✈❡r② ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❤❛s t♦ ❜❡✳ ❙❡❝♦♥❞❧②✿ t❤❡ str✉❝t✉r❡ ♣r♦❞✉❝❡s ❝♦rr❡❧❛t✐♦♥ ❞❡❝r❡❛s✐♥❣ ❛s t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ r❛t❡s ✐♥❝r❡❛s❡s✳ ❋✐♥❛❧❧②✿ t❤❡ s✉❜✲❞✐❛❣♦♥❛❧s ♦❢ t❤❡ r❡s✉❧t✐♥❣ ♠❛tr✐① ❛r❡ ✐♥❝r❡❛s✐♥❣ ✇❤✐❧❡ ♠♦✈✲ ✐♥❣ t♦ ❧♦♥❣❡r t❡♥♦rs ✭❙♦✉t❤ ❊❛st ♦❢ t❤❡ ♠❛tr✐①✮✳ ❚❤✐s ♣r♦♣❡rt② ✐s ❛❧s♦ ✈✐s✐❜❧❡ ✐♥ t❤❡ ♠♦❞✐✜❡❞ ❡①♣♦♥❡♥t✐❛❧ ❢♦r♠ ❛♥❞ ♠❡❛♥s t❤❛t ❝❤❛♥❣❡s ✐♥ ❧♦♥❣ t❡♥♦r ❋♦r✇❛r❞ ❘❛t❡s ❛r❡ ♠♦r❡ ❝♦rr❡❧❛t❡❞✳ ❚❤❡r❡❛❢t❡r ✐♥ ✷✳✺ ✐s ❣✐✈❡♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ s✉r❢❛❝❡ ✇✐t❤ ♣❛r❛♠❡t❡rs t❤❛t ❘❇❙ ✐s ✉s✐♥❣ t♦ ❜♦♦❦ tr❛❞❡s✳ ✹✾ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✺✿ ❙❝❤♦❡♥♠❛❦❡rs ❈♦✛❡② ❝♦rr❡❧❛t✐♦♥ ❛♠♦♥❣ ❋♦r✇❛r❞ ▲✐❜♦r r❛t❡s ✇✐t❤ η1 = 19.99%, η2 = 59.99% ❛♥❞ ρ∞ = 45% ✺✵ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✷✳✸✳✷ ❘❛♥❦ ❘❡❞✉❝t✐♦♥ ♠❡t❤♦❞s ◆♦✇ t❤❛t ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ ❛ s♠♦♦t❤❡r ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❢♦r ♦✉r ▲✐❜♦r ▼❛r❦❡t ♠♦❞❡❧ ❣✐✈✐♥❣ t❤❡ ✐♥♣✉ts✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❝❛❧✐❜r❛t❡ ♦✉r ♠♦❞❡❧ ✇✐t❤ ❛ s♠❛❧❧❡r ♥✉♠❜❡r ♦❢ ❢❛❝t♦rs t❤❛♥ t❤❡ ♥✉♠❜❡r ♦❢ ❋♦r✇❛r❞ r❛t❡s t❤❛t ✐s ✐♥♣✉tt❡❞ ♦r✐❣✐♥❛❧❧② ❛s ❛ ❇●▼ ♠♦❞❡❧ ✇✐t❤ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥ ✇✐t❤ ✶✺ ❢❛❝t♦rs ✐s ♥♦t ♣♦ss✐❜❧❡✳ ❘❡❜♦♥❛t♦ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ❘❡❜♦♥❛t♦ ✐♥ ❬✶✻❪ ❣✐✈❡s ❛♥ ✐♥t❡r❡st✐♥❣ ✇❛② t♦ t❛❝❦❧❡ t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ ❝♦rr❡❧❛✲ t✐♦♥ ♠❛tr✐① ❢♦r t❤❡ ▲▼▼ ✇✐t❤ d ❢❛❝t♦rs✳ ●❡♥❡r❛❧✐③✐♥❣ t❤❡ ❇●▼ ♠♦❞❡❧ ❛♥❞ ♠♦r❡ s♣❡❝✐✜❝❛❧❧② ✶✳✸✵ t♦ d ❢❛❝t♦rs ✇❡ ❝❛♥ ✇r✐t❡ ✐♥ ❛♥② ❋♦r✇❛r❞ ♠❡❛s✉r❡✿ d γik (t)dWtk dLi (t) = µi (t)dt + ✭✷✳✶✷✮ k=1 ❲❤❡r❡ ❛❧❧ t❤❡ Wtk ❛r❡ ♦rt❤♦❣♦♥❛❧ ❛♥❞ t❤❡ γik ❛r❡ t❤❡ ❧♦❛❞✐♥❣s ♦❢ ❡❛❝❤ ❢❛❝t♦rs ❛s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤✐s ❝❤❛♣t❡r✳ ❲❡ ❦♥♦✇ t❤❛t ✇❡ ❤❛✈❡ t❤❡ r❡❧❛t✐♦♥✿ d bik dWtk γi dWt = σi ✭✷✳✶✸✮ k=1 ❙♦ ✇❡ ❝❛♥ s❡❡ t❤❡ r❡❧❛t✐♦♥ ✇✐t❤ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❛r✐s❡s✿ γi γj = σi σj ρij d = σi σj bik bkj k=1 ❆♥❞ ✇❡ ❤❛✈❡ ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ ❛ ❣♦♦❞ ♣r✐❝✐♥❣ ♦❢ t❤❡ ❝❛♣❧❡ts✿ d b2ik = 1 ✭✷✳✶✹✮ k=1 ❲❡ ❛r❡ ❣♦✐♥❣ t♦ s❤♦✇ t❤❛t t❤✐s ✈❡r② ❣❡♥❡r❛❧ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❇●▼ ♠♦❞❡❧ ♣❡r♠✐ts ✉s t♦ ♣❛r❛♠❡t❡r✐③❡ t❤❡ γi ✳ ✺✶ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❚✇♦✲❢❛❝t♦r ❈❛s❡ ▲❡t ❛ss✉♠❡ t❤❛t d = 2✱ t❤❡♥ ✐♥ t❤❡✐r ❢♦r✇❛r❞ ♠❡❛s✉r❡ ✭❞r✐❢ts ❛r❡ ✐rr❡❧❡✈❛♥t ✐♥ t❤✐s ❞✐s❝✉ss✐♦♥✮✿ dLi (t) = σi (t)[b1i (t)dWt1 + b2i (t)dWt2 ] Li (t) t❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥ ✷✳✹ ❜❡❝♦♠❡s✿ b21i (t) + b22i (t) = 1 ✭✷✳✶✺✮ ❚❤❡r❡ ✇❡ ❝❛♥ ✐♥tr♦❞✉❝❡ ❛♥② ❝♦❡✣❝✐❡♥t θ ❛♥❞ ✐t ✐s ❛❧✇❛②s ❝♦rr❡❝t t❤❛t cos2 (θ) + sin2 (θ) = 1, ✇❤✐❝❤ s♣❡❝✐✜❡s ❛ s❡t ♦❢ ❝♦❡✣❝✐❡♥ts b1i , b2i ❛♥❞ ❤❡♥❝❡ ❛ ♣♦ss✐❜❧❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❧♦❛❞✐♥❣s ♦♥t♦ t❤❡ t✇♦ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ♦✉r ❇●▼ ♠♦❞❡❧✳ ❍♦✇ ❝❛♥ ✇❡ ❝❤♦♦s❡ ❛♠♦♥❣ ❛❧❧ t❤❡ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s❄ ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ✐♠♣♦s❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦♥❞✐t✐♦♥ t♦ t❤✐s ❝❤♦✐❝❡ ♦❢ θ✳ ❯s✐♥❣ ✭✷✳✼✮✿ E ρik = E dLk (t) dLi (t) Lk (t) Li (t) dLk (t) dLk (t) Lk (t) Lk (t) E ✭✷✳✶✻✮ dLi (t) dLi (t) Li (t) Li (t) ❋✐rst✱ t❤❡ ❞❡♥♦♠✐♥❛t♦r✿ E dLk (t) dLk (t) = σk2 (t)E [b1k (t)dWt1 + b2k (t)dWt2 ][b1k (t)dWt1 + b2k (t)dWt2 ] Lk (t) Lk (t) = σk2 (t)(b1k (t)2 + b2k (t)2 )dt = σk2 (t)dt ❆s ✇❡ ❤❛✈❡ ❝❤♦s❡♥ ❛ 2✲❞✐♠❡♥s✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✇✐t❤ ♦rt❤♦❣♦♥❛❧ ❇r♦✇✲ ♥✐❛♥ ✐♥❝r❡♠❡♥ts✳ ❍❡♥❝❡✱ t❤❡ ❞❡♥♦♠✐♥❛t♦r s✐♠♣❧✐✜❡s t♦✿ E dLk (t) dLk (t) dLi (t) dLi (t) E = σk (t)σi (t)dt Lk (t) Lk (t) Li (t) Li (t) ✭✷✳✶✼✮ ✺✷ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋♦r t❤❡ ♥✉♠❡r❛t♦r ✇❡ ❞❡r✐✈❡ t❤❡ s❛♠❡ ❝❛❧❝✉❧✉s ✉s✐♥❣ t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ❜❡✲ t✇❡❡♥ t❤❡ t✇♦ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥s✿ E dLk (t) dLi (t) = E σk (t)[b1k (t)dWt1 + b2k (t)dWt2 ]σi (t)[b1i (t)dWt1 + b2i (t)dWt2 ] Lk (t) Li (t) = E σk [sin θk dWt1 + cos θk dWt2 ]σi (t)[sin θi dWt1 + cos θi dWt2 ] = σk σj [sin θk sin θi + cos θk cos θi ]dt = σk σj [cos(θk − θi )]dt ❋✐♥❛❧❧②✱ ρik = [cos(θk − θi )] ✭✷✳✶✽✮ ❍❡♥❝❡✱ t❤✐s ❛♣♣❧✐❝❛t✐♦♥ t♦ ❛ 2✲❢❛❝t♦r ❝❛s❡ s❤♦✇ t❤❛t t❤❡ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ✷ ❋♦r✇❛r❞ r❛t❡s ✐s ♣✉r❡❧② ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✧❛♥❣❧❡s✧ ✇❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❧♦❛❞✐♥❣s bik ✳ ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❛ d ❢❛❝t♦r ❝❛s❡ ❚❤✐s ❝❛s❡ ✐s ❣❡♥❡r❛❧✐③❛❜❧❡ t♦ ❛ d ❢❛❝t♦rs ❝❛s❡✳ ❘❡♠✐♥❞✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ d 2 k=1 bik = 1✱ ✇❡ r❡❝♦❣♥✐③❡ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ ❤②♣❡r✲s♣❤❡r❡ ♦❢ r❛❞✐✉s ✶✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♣♦❧❛r ❝♦✲♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ✉♥✐t✲r❛❞✐✉s ❤②♣❡r✲s♣❤❡r❡ ❣✐✈❡s✿ k−1 bik = cos θik sin θjk , k = 1, 2, . . . , d − 1 j=1 k−1 bik = sin θjk , k=d j=1 ❚❤✐s ♣❛r❛♠❡t❡r✐③❛t✐♦♥ {θ} ✐s ✈❡r② ✉s❡❢✉❧ ♦♥ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ s✐❞❡ ❛s ✇❡ ✇✐❧❧ s❡❡ ✐t ❧❛t❡r✳ ✺✸ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❚❤❡ ❋r♦❜❡♥✐✉s ♥♦r♠ ❲❡ ❡①♣❧❛✐♥❡❞ ❜❡❢♦r❡ ✇❡ ✇❡r❡ tr②✐♥❣ t♦ ✜♥❞ B ∈ M(M, d) s♦ t❤❛t BB T ✇❛s ♥❡❛r A = [ρij ]T raded ∈ M(M, M )✳ ❚❤✐s s✉❜s❡❝t✐♦♥ ✇✐❧❧ ❣✐✈❡ ❛ s❡♥s❡ t♦ ✇❤❛t ♥❡❛r ♠❡❛♥✳ ■♥ ♦♣t✐♠✐③❛t✐♦♥ s❡✈❡r❛❧ ✈✐❡✇s ❝❛♥ ❜❡ t❛❦❡♥ ❛❜♦✉t ❞✐st❛♥❝❡ ✉s✐♥❣ s✉❜♦r❞✐♥❛t❡❞ ♥♦r♠s✱ ♣❡♥❛❧t② ❢✉♥❝t✐♦♥✱ ♦❜st❛❝❧❡ ❢✉♥❝t✐♦♥✳ ❲❡ ✇✐❧❧ st✐❝❦ t♦ t❤❡ s✐♠♣❧❡st ❝❛s❡ ♦❢ t❤❡ ❋r♦❜❡♥✐✉s ♥♦r♠✳ ❋♦r♠❛❧❧②✱ ✇❡ ❝♦♥s✐❞❡r ❛ ✇❡✐❣❤t❡❞ ❋r♦❜❡♥✐✉s ✐♥♥❡r ♣r♦❞✉❝t , W ♦♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ ♦❢ r❡❛❧ s②♠♠❡tr✐❝ ♠❛tr✐① M × M ❞❡✜♥❡❞ ❜②✿ X, Y W X, Y ∈ M(M, M ) = trace(XW Y W ), ✭✷✳✶✾✮ ❲❡ ✉s❡ t❤❡ ❡q✉❛❧❧② ✇❡✐❣❤t❡❞ ❋r♦❜❡♥✐✉s ♥♦r♠✱ ❤❡♥❝❡ W = I ❛♥❞ ✇❡ ❣❡t t❤❡ ♥♦r♠ ✐♥❞✉❝❡❞ ❜② , X 2 W✿ = X, X W = trace(X 2 ), X ∈ M(M, M ) ✭✷✳✷✵✮ ❆♣♣❧②✐♥❣ t❤✐s ♥♦r♠ t♦ ♦✉r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✿ ✇❡ ❛r❡ tr②✐♥❣ t♦ r❡❞✉❝❡ t❤❡ ❞✐st❛♥❝❡ [ρij ]model − [ρij ]traded ✇❤✐❝❤ ❝❛♥ ❜❡ tr❛❞✉❝❡❞ ✐♥✿ χ2 = [ρij ]model − [ρij ]traded 2 |[ρij ]model − [ρij ]traded |2 = d = (bjr brk ) − [ρij ]traded |2 | r=1 ❚❤✐s ♥♦r♠ ❞❡✜♥❡s ❤♦✇ ♥❡❛r ✐s ♦✉r ♠♦❞❡❧❡❞ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❢r♦♠ t❤❡ ♠❛r❦❡t✳ Pr✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s ✲ P❈❆ ❇❛❝❦ t♦ t❤❡ ❘❡❜♦♥❛t♦ ❛♥❣❧❡ ♣❛r❛♠❡tr✐③❛t✐♦♥✱ ✇❡ ❝❛♥ ♦❜❥❡❝t t❤❛t t❤✐s ❤❛s ♦♥❧② ♠❛❞❡ ✉s ❣♦ ❢r♦♠ ❝❛❧✐❜r❛t✐♥❣ M × d ❢❛❝t♦rs t♦ M × (d − 1) ❢❛❝t♦rs t❤❛t ✐♥t❡❣r❛t❡ t❤❡ ❝♦♥str❛✐♥ts ♦❢ ✷✳✹✳ ✺✹ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ✉s❡ ❛ ✸ ❢❛❝t♦rs ♠♦❞❡❧ t♦ s✐♠✉❧❛t❡ t❤❡ ✶✵❨ ❯❙❉ ▲✐❜♦r r❛t❡ ✭q✉♦t❡❞ ✐♥ ❛♥♥✉❛❧❧② ❝♦♠♣♦✉♥❞✮ ✇❡ st✐❧❧ ❤❛✈❡ ❛ ♣r♦❜❧❡♠ ♦❢ 10 × (3 − 1) = 20 ✈❛r✐❛❜❧❡s✳ ❍❡♥❝❡✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ❣♦♦❞ st❛rt t♦ ✜♥❞ ♦✉t t❤❡ s♦❧✉t✐♦♥✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ♣r✐♥❝✐♣❧❡ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s✳ ❚❤✐s t❡❝❤♥✐q✉❡ ✐s t❤❡ ♦♣t✐♠❛❧ ❧✐♥❡❛r tr❛♥s❢♦r♠ t❤❛t tr❛♥s❢♦r♠s t❤❡ ❝♦r✲ r❡❧❛t✐♦♥ ♠❛tr✐① t♦ ❛ ♥❡✇ ✈❡❝t♦r ❜❛s✐s✳ ❚❤✐s ✈❡❝t♦r s②st❡♠ ✭❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✮ ✐s s✉❝❤ t❤❛t t❤❡ ❣r❡❛t❡st ✈❛r✐❛♥❝❡ ❜② ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❞❛t❛ ❝♦♠❡s t♦ ❧✐❡ ♦♥ t❤❡ ✜rst ❝♦♦r❞✐♥❛t❡✱ t❤❡ s❡❝♦♥❞ ❣r❡❛t❡st ✈❛r✐❛♥❝❡ ♦♥ t❤❡ s❡❝♦♥❞ ❝♦♦r❞✐♥❛t❡✱ ❛♥❞ s♦ ♦♥✳ Pr❛❝t✐❝❛❧❧②✱ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① [ρij ] t❤❛t ✇❡ ❝❛♥ ❛❧✇❛②s ❞✐❛❣♦♥❛❧✐③❡ t♦ ✜♥❞ ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐① Λ = [λi ] ❛♥❞ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❞✐❛❣♦♥❛❧ ♠❛tr✐① V s✉❝❤ t❤❛t [ρ] = V ΛV −1 ✳ ❚❤❡s❡ ♠❛tr✐❝❡s ❛r❡ ❡❛s✐❧② ❢♦✉♥❞ ✉s✐♥❣ ❛ ◗❘ ❛❧❣♦r✐t❤♠ ✇✐t❤ ●r❛♠✲❙❝❤♠✐❞t ♠❡t❤♦❞✳ ❚❤❡♥✱ ②♦✉ ❝❛♥ ❢♦r♠ ❛ ♠❛tr✐① B ∈ M(M, d) ❞❡✜♥❡❞ ✇✐t❤✿ √ B= ΛP = λ1 V1 , . . . , λi Vi , . . . , λd Vd ❖♥❡ ❦❡❡♣s t❤❡ d ♠♦st ✐♠♣♦rt❛♥t ❡✐❣❡♥✈❛❧✉❡s {λi } ❛♥❞ t❤❡✐r ❡✐❣❡♥✈❡❝t♦rs {Vi }✳ ❲✐t❤ t❤✐s ❝❤♦✐❝❡ ✇❡ ❤❛✈❡ BB T ∈ M(M, M ) ❝❧♦s❡ ✐♥ ♥♦r♠ t♦ t❤❡ ♠❛r❦❡t ✐♥♣✉t [ρij ]✳ ❖♥ t♦♣ ♦❢ t❤✐s✱ ✇❡ ✇✐❧❧ ✉s❡ t❤✐s B t♦ ❞❡s❝r✐❜❡ t❤❡ ❢❛❝t♦rs bik ❛s ❞❡✜♥❡❞ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ♦✉r ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✐♥ ✷✳✶✸✳ ❲❡ ❛❧s♦ ❤❛✈❡ ❛♥ ✐♥❞✐❝❛t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝t♦rs ✐♠♣♦rt❛♥t t♦ ❝r❡❛t❡ ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ r❛♥❦ M ♠❛tr✐①✳ ■♥ ♦✉r ❡①❛♠♣❧❡✱ ✇❡ ✜♥❞ t❤❛t t❤❡ ✜rst t❤r❡❡ ❡✐❣❡♥✈❛❧✉❡s ❛❝❝♦✉♥t ❢♦r 93.6% ♦❢ t❤❡ s✉♠ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ❛s s❤♦✇♥ ✐♥ ✷✳✹✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❝❛♥ ❡①♣❧❛✐♥ 93.6% ♦❢ t❤❡ ✈❛r✐❛♥❝❡ ✇✐t❤ t❤❡ ✜rst t❤r❡❡ ❢❛❝t♦rs✳ ❆ P❈❆ ■♥t❡r♣r❡t❛t✐♦♥ ❲❡ ❝❛♥ ❡❛s✐❧② ❞r❛✇ ❛ ♣❛r❛❧❧❡❧ ❜❡t✇❡❡♥ t❤♦s❡ ❡✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡ ♠♦✈❡s ♦❢ t❤❡ ❝✉r✈❡✳ ❚❤❡ ✜rst ❢❛❝t♦r✱ t❤❡ ♠♦st ✐♠♣♦rt❛♥t✱ ❡①♣❧❛✐♥s t❤❡ ♣❛r❛❧❧❡❧ s❤✐❢ts ✺✺ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❊✐❣❡♥✈❛❧✉❡s ❱❛❧✉❡ Pr♦♣♦rt✐♦♥ ✶st 7.86 78.6% ✷♥❞ 1.07 10.7% ✸r❞ 0.427 4.27% ❙✉♠ ♦❢ t❤❡ ♦t❤❡rs 0.64 6.40% ❚❛❜❧❡ ✷✳✹✿ ▼❛✐♥ ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✿ t❤❡ P❈❆ ❛r✐s❡s ♥❛t✲ ✉r❛❧❧② t♦ ❡①♣❧❛✐♥ t❤❡ ♠♦✈❡s ♦❢ t❤❡ ❝✉r✈❡ ♠♦✈❡♠❡♥ts ♦❢ t❤❡ ②✐❡❧❞ ❝✉r✈❡✳ ❚❤❡ s❡❝♦♥❞ ♦♥❡ ❡①♣❧❛✐♥s t❤❡ ✐♥✈❡rs✐♦♥ ♠♦✈❡s ♦❢ t❤❡ ❝✉r✈❡✿ ✇❤❡♥ t❤❡ s❤♦rt ❞❛t❡❞ ✐♥❝r❡❛s❡ ✇❤✐❧❡ t❤❡ ❧♦♥❣ ❞❛t❡❞ ❞❡❝r❡❛s❡ ♦r t❤❡ ♦♣♣♦s✐t❡✳ ❋✐♥❛❧❧②✱ t❤❡ t❤✐r❞ ❢❛❝t♦r ❡①♣❧❛✐♥s t❤❡ t♦rs✐♦♥ ♠♦✈❡s ♦❢ t❤❡ ❝✉r✈❡✿ ✇❤❡♥ ❧♦♥❣ ❛♥❞ s❤♦rt r❛t❡s ❞❛t❡❞ ✐♥❝r❡❛s❡ ❛♥❞ ♠✐❞❞❧❡ ❞❛t❡❞ ❞❡❝r❡❛s❡s ♦r t❤❡ ♦♣♣♦s✐t❡✳ ❍❡♥❝❡✱ t❤❛♥❦s t♦ t❤❡ P❈❆✱ ✇❡ ❤❛✈❡ ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡①♦❣❡✲ ♥❡♦✉s❧② ❣✐✈❡♥ ❢✉❧❧ r❛♥❦ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❛t ❛ r❡❧❛t✐✈❡❧② ❧♦✇ ❝♦♠♣✉t❛t✐♦♥ ❝♦st✳ ▼♦r❡♦✈❡r✱ ✇❡ ❦♥♦✇ ❤♦✇ ♠✉❝❤ ♦❢ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ✇❡ ❛❝❝♦✉♥t ❢♦r ✇❤❡♥ ✉s✐♥❣ ❛ 3 ❢❛❝t♦rs ♠♦❞❡❧✳ ❘❡❜♦♥❛t♦ ❛♥❣❧❡s ♦♣t✐♠✐③❡❞ ♠❡t❤♦❞ ●♦✐♥❣ t♦ ❛ ❢✉❧❧ ♦♣t✐♠✐③❛t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ χ2 ✉♥❞❡r ❛ 3 ❢❛❝t♦rs ♠♦❞❡❧✱ ✇❡ ♦❜t❛✐♥ ✈❡r② ❝❧♦s❡ r❡s✉❧ts t♦ t❤❡ P❈❆✳ ❚❤❡r❡❛❢t❡r ✐s ❣✐✈❡♥ ❛ ✜❣✉r❡ ❝♦♠♣❛r✐♥❣ t❤❡ t✇♦ ♠❡t❤♦❞s✳ ❚❤❡ ♦♣t✐♠✐③❛t✐♦♥ ❝❛♥ ❜❡ ❞♦♥❡ ✉s✐♥❣ ❇r♦②❞❡♥✲❋❧❡t❝❤❡r✲ ●♦❧❞❢❛r❜✲❙❤❛♥♥♦ ❛❧❣♦r✐t❤♠ ✭❛s ❞❡t❛✐❧❡❞ ✐♥ ❬✶✹❪✮ ✉s✐♥❣ ♣❛r❛♠❡t❡rs ❢♦r t❤❡ ❙❝❤♦❡♥♠❛❦❡rs✲❈♦✛❡② str✉❝t✉r❡ ✉s❡❞ ❜② ❘❇❙ ❢♦r t❤❡ ❜♦♦❦✐♥❣ ❢♦r ■♥t❡r❡st r❛t❡ ❞❡r✐✈❛t✐✈❡s ❛♥❞ ❜❛s❡❞ ♦♥ ❛ s❧✐❞✐♥❣ ✇✐♥❞♦✇ ♦❢ t❤❡ ❧❛st ✶✷ ②❡❛rs✶ ♦♥ ❯❙❉ ✶✷♠ ▲✐❜♦r✳ ❚❤❡ ❘♦②❛❧ ❇❛♥❦ ♦❢ ❙❝♦t❧❛♥❞ ✐s ✉s✐♥❣ ❛ s❧✐❣❤t❧② ♠♦❞✐✜❡❞ ✈❡rs✐♦♥ ♦❢ t❤✐s ❛❧❣♦r✐t❤♠ t❤❛t ❣✐✈❡s ❜❡tt❡r r❡s✉❧ts✳ ❖❜✈✐♦✉s❧②✱ t❤❡ ♥♦r♠ ♦♣t✐♠✐③❛t✐♦♥ ❧♦♦❦s ❜❡tt❡r ❛♥❞ t❤❡ ❋♦r✇❛r❞ r❛t❡s ❛r❡ ❝❧♦s❡ t♦ t❤❡ ✐♥♣✉t ♠❛tr✐①✳ ❲❡ ❝❛♥ s❡❡ ✐♥ t❤❡ ♥❡①t ✜❣✉r❡ t❤❛t t❤❡ ❡✐❣❡♥✈❡❝t♦rs ❢♦r ❜♦t❤ ♠❡t❤♦❞s ❛r❡ q✉✐❡t s✐♠✐❧❛r ❛❧t❤♦✉❣❤ t❤❡r❡ ✐s ♥♦ ♦rt❤♦❣♦♥❛❧✐③❛t✐♦♥ ♣r♦❝❡ss ✐♥ t❤❡ ❘❡❜♦♥❛t♦ ✶ ❚❤✐s ✇✐♥❞♦✇ ❝❛♥ ❝❤❛♥❣❡ ❛s ♦♥❡ ❝❛♥ ❛r❣✉❡ t❤❛t ❛ s❤♦rt❡r ✇✐♥❞♦✇ ❣✐✈❡s ❛ ❜❡tt❡r tr❡♥❞❀ ❤♦✇❡✈❡r t❤✐s ❝❤♦✐❝❡ ✐s ✈❡r② ❝♦♥❞✐t✐♦♥❛❧ t♦ tr❛❞❡r ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t ♦♣✐♥✐♦♥ ✺✻ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❛♥❣❧❡s ♦♣t✐♠✐③❡❞ ♠❡t❤♦❞ ✭❋✉❧❧② ♦♣t✐♠✐③❡❞ ♠❡t❤♦❞✮✳ ❲❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡ ✐♥ t❤❡s❡ ♣r♦❝❡ss ✐s ❥✉st ❛ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✳ ✺✼ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✻✿ ❊✐❣❡♥✈❡❝t♦rs ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ P❈❆ ❛♥❞ ❘❡❜♦♥❛t♦ ❛♥❣❧❡s ♦♣t✐♠✐③❡❞ ♠❡t❤♦❞ ✺✽ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ r❛♥❦ r❡❞✉❝❡❞ ❝♦rr❡❧❛t✐♦♥s ❚❤❡ ♥❡①t ✜❣✉r❡s ✷✳✼✱ ✷✳✽ ❛♥❞ ✷✳✾ ♣❧♦t ❛ ❝♦❧✉♠♥ ✭❙❡❝♦♥❞✱ ❋✐❢t❤ ❛♥❞ ❚❡♥t❤ ❝♦❧✉♠♥✮ ♦❢ ❡❛❝❤ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❢♦r♠❡❞ ✭▼❛r❦❡t✱ P❈❆ ❛♥❞ ❋✉❧❧② ♦♣t✐✲ ♠✐③❡❞✮ ❛♥❞ ❝♦♠♣❛r❡ t❤❡♠✳ ❲❤❛t ✐s ♣❧♦tt❡❞ ✐s t❤❡ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛ ❋♦r✇❛r❞ ▲✐❜♦r r❛t❡ ✭2nd : ρ2, j, 5th : ρ5, j, 10th : ρ10, j ✮ ✇✐t❤ t❤❡ ♦t❤❡r ❋♦r✲ ✇❛r❞ ▲✐❜♦r r❛t❡s ❢♦r ❡❛❝❤ ♠❛tr✐① ❢♦r♠❡❞✳ ▲♦♦❦✐♥❣ ❛t t❤❡s❡ ✜❣✉r❡s s❡✈❡r❛❧ r❡♠❛r❦s ❝❛♥ ❜❡ ❞♦♥❡✳ ■♥ ❣❡♥❡r❛❧✱ ✇❡ ♦❜s❡r✈❡ t❤❛t t❤❡s❡ r❛♥❦ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞s t❡♥❞ t♦ ♦✈❡r❡st✐♠❛t❡ t❤❡ ❝♦rr❡❧❛✲ t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❛❞❥❛❝❡♥t ❋♦r✇❛r❞ r❛t❡s ✭t❤✉s t❤❡ t❡r♠s ρi,i−1 . . .✮ ❛♥❞ ❧♦✇❡r t❤❡ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞✐st❛♥t ♦♥❡ ✭t❤❡ t❡r♠s ρi, . . .✮✳ ❍❡♥❝❡✱ t❤✐s ❧❡❛❞ t♦ s②st❡♠❛t✐❝❛❧ ♠✐s♣r✐❝❡ ♦♥ t❤❡ s✇❛♣t✐♦♥s✿ s❤♦rt ♠❛t✉r✐t✐❡s s✇❛♣t✐♦♥ ✇✐❧❧ ❛❧✇❛②s ❜❡ t♦♦ ❡①♣❡♥s✐✈❡ ❜❡❝❛✉s❡ ♠♦❞❡❧ ❝♦rr❡❧❛t✐♦♥ ✇✐❧❧ ❜❡ t♦♦ ❤✐❣❤ ❛♥❞ ❧♦♥❣ ♠❛t✉r✐t✐❡s s✇❛♣t✐♦♥ ✇✐❧❧ ❜❡ t♦♦ ❝❤❡❛♣ ❜❡❝❛✉s❡ ♠♦❞❡❧ ❝♦rr❡❧❛t✐♦♥ ✇✐❧❧ ❜❡ t♦♦ ❧♦✇✳ ❲✐t❤ t❤♦s❡ r❡s❡r✈❡s ✐♥ ♠✐♥❞ r❡s✉❧ts r❡♠❛✐♥ ❛❝❝❡♣t❛❜❧❡ ❢♦r ❛t✲t❤❡✲♠♦♥❡② s✇❛♣t✐♦♥s✳ ◆❡✈❡rt❤❡❧❡ss✱ ✇❡ ❝❛♥ s❡❡ t❤❛t ✐♥ ♦✉r ❝❛s❡ t❤❡ ❧♦✇ ❝♦rr❡❧❛t✐♦♥ ❡❢✲ ❢❡❝t ✐s ♥♦t ✈❡r② ✇❡❧❧ ♦❜s❡r✈❡❞✱ t❤✐s ✐s ❞✉❡ t♦ t❤❡ r❛t❤❡r s♠❛❧❧ s✐③❡ ✭✶✵ ❨❡❛rs✮ ♦❢ ♦✉r ♠❛tr✐①✳ ■♥❝r❡❛s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝t♦rs t♦ 4 ❞♦❡s ♥♦t ✐♠♣r♦✈❡ ❛s ♠✉❝❤ ❛s ❢r♦♠ 2 t♦ 3 ❛s t❤❡ 4✲t❤ ❡✐❣❡♥✈❛❧✉❡ ✐s s♠❛❧❧❡r t❤❛♥ t❤❡ ✜rst 3 ✭✐♥ ♦✉r ❝❛s❡ ✇❡ ✇♦✉❧❞ ❤❛✈❡ t❛❦❡♥ ❛❝❝♦✉♥t ♦❢ 95.7% ✭✈s 93.6% ✇✐t❤ 3 ❢❛❝t♦rs✮ ♦❢ t❤❡ ✈❛r✐❛♥❝❡ ✇✐t❤ ✹ ❢❛❝t♦rs✮❀ ❤❡♥❝❡ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ❛♥❞ t❤❡ ❛❝❝✉r❛❝② ♥❡❡❞❡❞ ✇❡ ❝❛♥ ✐♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝t♦rs ❜✉t ♥❡✈❡r ❧❡t ✐t ❣♦ ❜❡❧♦✇ ✸✳ ✺✾ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✼✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ✷❨ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡s ❝♦rr❡❧❛t✐♦♥ s✐♠✉❧❛t❡❞ ❜② P❈❆ ❛♥❞ ❝♦♠♣❧❡t❡ ♦♣t✐♠✐③❛t✐♦♥ ✻✵ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✽✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ✺❨ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡s ❝♦rr❡❧❛t✐♦♥ s✐♠✉❧❛t❡❞ ❜② P❈❆ ❛♥❞ ❝♦♠♣❧❡t❡ ♦♣t✐♠✐③❛t✐♦♥ ✻✶ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐❣✉r❡ ✷✳✾✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ✶✵❨ ❋♦r✇❛r❞ ▲✐❜♦r ❘❛t❡s ❝♦rr❡❧❛t✐♦♥ s✐♠✉✲ ❧❛t❡❞ ❜② P❈❆ ❛♥❞ ❝♦♠♣❧❡t❡ ♦♣t✐♠✐③❛t✐♦♥ ✻✷ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✷✳✹ ❙✇❛♣t✐♦♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛s ❲❡ ❤❛✈❡ s❡❡♥ t❤❛t ❛ ✜♥❛♥❝✐❛❧ ♠♦❞❡❧ ✐s ✉s❛❜❧❡ ❜② ♦♣❡r❛t♦rs ♦♥❧② ✐❢ ✐t r❡✢❡❝ts ♣r✐❝❡s ♦❢ t❤❡ ♠❛r❦❡t✳ ❚❤✐s ❝❛❧✐❜r❛t✐♦♥ ✐s ✉s✉❛❧❧② ❛ ✈❡r② t✐♠❡ ❝♦♥s✉♠✐♥❣ ♦♣❡r❛t✐♦♥✳ ❚❤❡♦r❡t✐❝❛❧❧②✱ ✐♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ ♦✉r ♣r♦❜❧❡♠✱ ✇❡ s❤♦✉❧❞ ♣r♦♣♦s❡ ❛ s❡t ♦❢ ♣❛r❛♠❡t❡rs ❢♦r t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t② ❛♥❞ ❝♦rr❡❧❛t✐♦♥✱ r✉♥ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥ ♦♥ t❤❡ ❋♦r✇❛r❞ ❘❛t❡s ❛♥❞ ❢r♦♠ t❤♦s❡✱ ❞❡r✐✈❡ t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ s✇❛♣t✐♦♥s✳ ❚❤✐s ♣r♦❝❡ss ♥❡❡❞s ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ❛t ❡❛❝❤ st❡♣ ✇❤✐❝❤ ✐s t♦♦ ♠✉❝❤ t✐♠❡ ❝♦♥s✉♠✐♥❣✳ ❍❡♥❝❡✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛♥ ❛♣♣r♦①✐♠❛t❡ ❝❧♦s❡❞ ❢♦r♠✉❧❛ ❢♦r t❤✐s ♣r✐❝❡\✈♦❧❛t✐❧✐t②✳ ■♥ t❤❡ ♠❛r❦❡t✱ s✇❛♣t✐♦♥s ❛t t❤❡ ♠♦♥❡② ❛r❡ q✉♦t❡❞ ✉s✐♥❣ t❤❡✐r ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✿ t❤❡ ♠❛r❦❡t ✉s❡s ❇❧❛❝❦ ❋♦r♠✉❧❛ t♦ ❝r❡❛t❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♣r✐❝❡s ♦❢ t❤❡ s✇❛♣t✐♦♥s ❛♥❞ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ✉s❡❞ ❢♦r t❤❡ q✉♦t❛t✐♦♥✳ ❚❤❡ ✉s❡ ♦❢ t❤✐s ❇❧❛❝❦ ❢♦r♠✉❧❛ r❡q✉❡st t❤❛t ♦♥❡ ❛ss✉♠❡ t❤❡ ❧♦❣ ♥♦r♠❛❧✐t② ♦❢ t❤❡ ❋♦r✇❛r❞ r❛t❡s ✐♥ t❤❡✐r ❋♦r✇❛r❞ ▼❡❛s✉r❡ ❛♥❞ ❛s ❛ ♠❛tt❡r ♦❢ ❢❛❝t ♥♦ ❧♦❣✲♥♦r♠❛❧✐t② ❢♦r t❤❡ s✇❛♣ r❛t❡s✳ ✷✳✹✳✶ ❘❡❜♦♥❛t♦ ❋♦r♠✉❧❛ ❘❡❜♦♥❛t♦ ✐♥ ❬✶✻❪ ♣r♦♣♦s❡❞ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ s✇❛♣✲ t✐♦♥ ♣r✐❝❡s✳ ❆ s✇❛♣ r❛t❡ Sp,q (t) ❛s ✇❡ s❛✇ ✐t ❜❡❢♦r❡ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❋♦r✇❛r❞ r❛t❡s✿ q−1 k wp,q Lk (t) Sp,q (t) = ✭✷✳✷✶✮ k=p ✇❤❡r❡ t❤❡ ✇❡✐❣❤ts {w} ❛r❡ ❣✐✈❡♥ ❜②✿ k wp,q = δB(t, Tk + δ) q−p i=1 δB(t, Ti + iδ) ❍❡r❡✱ ✇❡ ❛ss✉♠❡ t❤❛t ✐♥ t❤❡ ❞②♥❛♠✐❝ ♦❢ t❤❡ s✇❛♣ r❛t❡ dSp,q t❤❡ ✇❡✐❣❤✐♥❣s {w} ✐♥ t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ❛r❡ ❝♦♥st❛♥t ❛♥❞ ❡q✉❛❧ t♦ t❤❡✐r ✈❛❧✉❡ ✐♥ ✵✱ ✻✸ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ wp,q (0)✳ ❍❡♥❝❡✱ q−1 ✭✷✳✷✷✮ wp,q (0)k dLk (t) dSp,q ≈ k=p Black ✉s✐♥❣ t❤❡ r❡❧❛t✐♦♥ s❤♦✇❡❞ ❚❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② σp,q ✐♥ t❤❡ ❝❤❛♣t❡r ✶✳ Tp Black 2 (σp,q ) Tp = γp,q 0 2 Tp dt = 0 2 dSp,q Sp,q ✭✷✳✷✸✮ ❚❤❡r❡❢♦r❡✱ dSp,q Sp,q q−1 2 = j,k=p q−1 = j,k=p q−1 = j,k=p k (0)w j (0)dL (t)dL (t) wp,q p,q j k 2 Sp,q k (0)w j (0)(γ wp,q p,q k γj )Lk (t)Lj (t) dt 2 Sp,q k (0)w j (0)ρ σ σ L (t)L (t) wp,q p,q j kj k j k dt 2 Sp,q ❆♥❞ ✜♥❛❧❧②✱ Tp q−1 Black 2 (σp,q ) Tp ≈ 0 j,k=p q−1 ≈ j,k=p k (0)w j (0)ρ σ σ L (t)L (t) wp,q p,q j kj k j k dt 2 Sp,q k (0)w j (0)L (t)L (t) wp,q p,q j k 2 Sp,q Tp ρkj σk σj dt 0 ❍❡r❡ ✇❡ ❛ss✉♠❡ Lk (t) = Lk (0) q−1 ≈ j,k=p k (0)w j (0)L (t)L (t) wp,q p,q j k ρkj (0) 2 Sp,q Tp σk σj dt 0 ❍❡r❡ ✇❡ ❛❧s♦ ❛ss✉♠❡ ρjk (t) ≈ ρjk (0) ❙♦ ♣✉tt✐♥❣ t❤✐♥❣s t♦❣❡t❤❡r ❘❡❜♦♥❛t♦ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛ ✐s✿ Black σp,q = 1 Tp q−1 j,k=p k (0)w j (0)L (t)L (t) wp,q p,q j k ρkj (0) 2 Sp,q Tp σk (t)σj (t)dt 0 ✭✷✳✷✹✮ ✻✹ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❚❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✇♦r❦s q✉✐❡t ✇❡❧❧ ❜✉t ✐t ❝❛♥ ❜❡ ✜♥❡ t✉♥❡❞ ✉s✐♥❣ ❍✉❧❧ ❛♥❞ ❲❤✐t❡ ✐❞❡❛ ✐♥ ❬✶✾❪✳ ✷✳✹✳✷ ❍✉❧❧ ❛♥❞ ❲❤✐t❡ ❋♦r♠✉❧❛ ■♥ ❬✶✾❪✱ ❍✉❧❧ ❛♥❞ ❲❤✐t❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦❢ t❤❡ ♣r❡✈✐♦✉s ❢♦r✲ ♠✉❧❛ ✉s✐♥❣ t❤❡ ✜rst ♦r❞❡r ❢♦r t❤❡ ❝♦❡✣❝✐❡♥t {w}✳ ❲❡ ✇✐❧❧ ♦♠✐t t❤❡ s✉❜s❝r✐♣ts p ❛♥❞ q t♦ ❧✐❣❤t t❤❡ ♥♦t❛t✐♦♥✳ ❚❤❡ ❞❡r✐✈❛t✐♦♥ ✐s ❢♦r t❤✐s ♦♥❡✿ q−1 d(wk (t)Lk (t)) dSp,q = k=p q−1 wk (t)dLk (t) + Lk (t)dwk (t) = ❛s t❤❡ ✇❡✐❣❤t✐♥❣s ❛r❡ ❞❡t❡r♠✐♥✐st✐❝ ❢✉♥❝t✐♦♥s ♦❢Lk : k=p q−1 q−1 k = w (t)dLk (t) + k=p q−1 Lk (t) i=p k=p q−1 q−1 wk (t)dLk (t) + Lk (t) = i=p k=p ∂wk (t) dLi (t) ∂Li ∂wk (t) dLi (t) ∂Li ❚❤❡ ✜rst ♦r❞❡r ❞❡r✐✈❛t✐✈❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ✇r✐t✐♥❣✿ k wp,q = = B(t, Tk + δ) d i=1 δB(t, Ti + δ) k 1 i=0 1+δLi (t) q−p p+k−1 1 i=1 δ i=0 1+δLi (t) ❚❤❡ ❞❡r✐✈❛t✐♦♥ ✐s str❛✐❣❤t❢♦r✇❛r❞ ❛♥❞ ✇❡ ✇✐❧❧ ♥♦t r❡♣r♦❞✉❝❡ ✐t✳ ❚❤❡ r❡❛❞❡r ❝❛♥ r❡❢❡r t♦ ❬✶✾❪ ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧s✳ ❋✐♥❛❧❧②✱ ✇❡ ♦❜t❛✐♥✿ ∂wk wk δ = ∂Li 1 + δLi 1i>k − i−p+1 k=1 B(t, p + k) q−p k=1 B(t, p + k) ✻✺ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❋✐♥❛❧❧② ✇❡ ❝❛♥ ✉s❡ t❤❡ ❝♦♥✈❡♥✐❡♥t ❡①♣r❡ss✐♦♥ ♦❢ ❘❡❜♦♥❛t♦ ❣✐✈❡♥ ✐♥ ✷✳✷✹ ✇❤❡r❡ ✇❡ s✇✐t❝❤ wk ❜② w¯ k ❞❡✜♥❡❞ ❜②✿ q−1 w ¯ k = wk + Lk (t) k=p ∂wk ∂Li ✷✳✹✳✸ ❆♥❞❡rs❡♥ ❛♥❞ ❆♥❞❡r❡❛s❡♥ ❋♦r♠✉❧❛ ❆ t❤✐r❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♣♦ss✐❜❧❡ ✐s t❤❡ ♦♥❡ ❣✐✈❡♥ ❜② ❆♥❞❡rs❡♥ ❡t ❆♥❞r❡❛s❡♥ ✐♥ ❬✷✵❪✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ s✇❛♣ r❛t❡ Sp,q ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❋♦r✇❛r❞ ❘❛t❡s Li ✳✭❇❛s✐❝❛❧❧②✱ ✉s✐♥❣ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ∂Sp,q ∂Li r❡❧❡✈❛♥t ✇✐t❤ t❤❡ ♠❛t✉r✐t② ✇❡ ❝♦♥s✐❞❡r t❤❛t ✐s ❢r♦♠ Tp t♦ Tq ✮ q−1 dSp,q = k=p dSp,q 1 = Sp,q Sp,q ∂Sp,q dLk ∂Lk q−1 k=p ∂Sp,q Lk γk dWtk + Xk dt ∂Lk ❖♥❝❡ ❛❣❛✐♥ ✇❡ ❞♦ ♥♦t ❝♦♠♣✉t❡ t❤❡ ❞r✐❢t Xk ❛s ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ♦♥❧② ✐♥ t❤❡ q✉❛❞r❛t✐❝ ✈❛r✐❛t✐♦♥✿ dSp,q Sp,q Tp 0 dSp,q Sp,q 2 = 2 dt ≈ 1 2 Sp,q 1 2 Sp,q q−1 j,k=p q−1 j,k=p ∂Sp,q ∂Sp,q Lk Lj γk γj dt ∂Lk ∂Lj ∂Sp,q (0) ∂Sp,q (0) Lk (0)Lj (0) ∂Lk ∂Lj Tp γk γj dt 0 ❲❤❡r❡ ✇❡ ✉s❡ t❤❡ s❛♠❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛s ♣r❡✈✐♦✉s❧② t❛❦✐♥❣ ❢♦r ❝♦♥st❛♥t t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s✱ t❤❡ ❢♦r✇❛r❞ r❛t❡s ❛♥❞ t❤❡ ✐♥st❛♥t❛♥❡♦✉s ❝♦rr❡❧❛t✐♦♥ ❛t 0✳ ❙♦ ✜♥❛❧❧② ✇❡ ❣❡t ❢♦r t❤❡ s✇❛♣t✐♦♥ ♣r✐❝❡✿ Black σp,q = 1 2 (0) Tp Sp,q q−1 j,k=p ∂Sp,q (0) ∂Sp,q (0) Lk (0)Lj (0)ρjk (0) ∂Lk ∂Lj Tp σk σj dt 0 ✭✷✳✷✺✮ ✻✻ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✇❤❡r❡ ✇❡ ❤❛✈❡ ❞❡r✐✈❡❞ t❤❡ t❡r♠ ❢♦r ❡❛❝❤ k✿ 1 ∂Sp,q δ = Sp,q ∂Lk 1 + δLk B(t, Tq ) + B(t, Tp ) − B(t, Tq ) q−p j=k−p+1 δB(t, Tp+j ) q−p j=1 δB(t, Tp+j ) j ■♥ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥✱ ✇❡ ✜♥❛❧❧② ♦♥❧② ❤❛✈❡ ❝❤❛♥❣❡❞ t❤❡ wp,q ❜② ✷✳✺ ∂Sp,q ∂Lj ✳ ▼♦♥t❡ ❈❛r❧♦ ❙✐♠✉❧❛t✐♦♥ ❛♥❞ ❘❡s✉❧ts ♦♥ ✸ ❋❛❝✲ t♦rs ❇●▼ ❚❤✐s s❡❝t✐♦♥ ✐s ❣♦✐♥❣ t♦ ❝♦♠♣❛r❡ t❤❡ ❞✐✛❡r❡♥t ❢♦r♠✉❧❛s ✐♥ t❡r♠ ♦❢ t❤❡✐r ❛❜✐❧✐t② t♦ ✜t t❤❡ s✇❛♣t✐♦♥ ♠❛r❦❡t s✐♠✉❧❛t❡❞ ❜② ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞s ❛♥❞ ❣✐✈❡♥ t❤❡ s❛♠❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs✳ ✷✳✺✳✶ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞ ❚❤❡ ✐❞❡❛ ♦❢ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ✐s t♦ ❝♦♠♣✉t❡ ✈❛❧✉❡s ♦❢ ❛♥② ❦✐♥❞ ♦❢ ❞❡r✐✈❛t✐✈❡s ✐♥str✉♠❡♥ts ❢r♦♠ s✐♠✉❧❛t❡❞ tr❛❥❡❝t♦r✐❡s ❛♥❞ ❡✈❛❧✉❛t❡ t❤❡ r❡s✉❧t ❛s t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤✐s ✈❛❧✉❡s✳ ■♥ ❣❡♥❡r❛❧✱ ▼♦♥t❡ ❈❛r❧♦ ❝♦♠♣✉t❛t✐♦♥ ❛r❡ ✉s❡❞ ❢♦r s✐♠✉❧❛t✐♦♥ ❛♥❞ ♦♣t✐♠✐③❛✲ t✐♦♥ ♣r♦❜❧❡♠s✳ ■♥ ▲✐❜♦r ▼❛r❦❡t ♠♦❞❡❧✱ ✇❡ ❤❛✈❡ t♦ ❝♦♠♣✉t❡ ❡①♣❡❝t❛t✐♦♥s ❛♥❞ t❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ✉s❡ t❤✐s ♣r♦❝❡ss✳ ■♥ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❝♦♥s✐❞❡r ❛ sq✉❛r❡✲✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥ f ∈ L2 (0, 1) ❛♥❞ ❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ x ∈ U[0, 1]✳ ▼❈ ♣❡r♠✐ts ✉s t♦ ❝♦♠♣✉t❡ ❡①♣❡❝t❛t✐♦♥s ❛s ✇❡ ❦♥♦✇ t❤❛t✿ 1 E[f (x)] = f (x)dx, 0 ❈♦♥s✐❞❡r ❛ s❡q✉❡♥❝❡ xin s❛♠♣❧❡❞ ❢r♦♠ U[0, 1]✳ ❆♥ ❡♠♣✐r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡①♣❡❝t❛t✐♦♥ ✐s t❤❡♥✿ E[f (x)] ≈ 1 n n f (xi ) i=1 ✻✼ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❚❤❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❙tr♦♥❣ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs✳ ❚❤✐s ❧❛✇ ✐♠♣❧✐❡s t❤❛t t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❝♦♥✈❡r❣❡♥t ✇✐t❤ ♣r♦❜✲ ❛❜✐❧✐t② ♦♥❡✱ ✐✳❡✳ 1 lim n→∞ n n 1 f (xi ) = ✭✷✳✷✻✮ f (x)dx 0 i=1 ❚❤❡ ❡rr♦r ✇❡ ♠❛❦❡ ✉s✐♥❣ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ❤❡♥❝❡ ✐s✿ 1 n 1 f (x)dx − n = 0 n ✭✷✳✷✼✮ f (xi ) i=1 ❚❤✐s ❡rr♦r ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ ❛ st❛t✐st✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇ ✉s✐♥❣ t❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✳ ❆s n → ∞✱ √ n n (f ) ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥ t♦ σν ✇❤❡r❡ ν ✐s ❛ st❛♥❞❛r❞ ♥♦r♠❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✭✇✐t❤ ♠❡❛♥ ♥✐❧ ❛♥❞ ✈❛r✐❛♥❝❡ ♦❢ ✶✮ ❛♥❞ σ ✐s t❤❡ sq✉❛r❡✲r♦♦t ♦❢ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ f ✿ 1 1 (f (t) − σ(f ) = 0 1/2 f (x)dx)dt 0 ✷✳✺✳✷ ◆✉♠❡r✐❝❛❧ ❘❡s✉❧ts ■♥ ♦r❞❡r t♦ s✐♠✉❧❛t❡ t❤❡ ❋♦r✇❛r❞ ▲✐❜♦r ✉s✐♥❣ ▼♦♥t❡ ❈❛r❧♦✱ ✇❡ ♥❡❡❞ ❛ ✉♥✐q✉❡ ♠❡❛s✉r❡✳ ❆s ♣r❡✈✐♦✉s❧② ❡①♣❧❛✐♥❡❞ ✐♥ ✶✳✶✳✻ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ s♣♦t ♠❛rt✐♥❣❛❧❡ ♠❡❛s✉r❡ P∗ ❛♥❞ ✐ts ♥✉♠❡r❛✐r❡ Bspot (t)✳ ❲❡ ❤❛✈❡ ❞✐s❝r❡t✐s❡❞ t❤❡ {Li } ✉♥❞❡r t❤❡✐r ❡①♣♦♥❡♥t✐❛❧ ❢♦r♠ ✉s✐♥❣ ✶✳✷✺ ♦♥ ❛ t❡♥♦r t❤❛t ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ r❡s❡t ❞❛t❡s Ti ❢♦r ♣r❛❝t✐❝❛❧ r❡❛s♦♥s✿  k d(ln Li (t)) =  δ −1 T δLj (t)(γk (t) γj (t)) γ 2 (t) − i 1 + δLj (t) 2 2   dt + Lk (t)γk (t) dWt∗ ✻✽ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❚❤❡r❡❢♦r❡✱ ∀k ∈ [0, n − 1]✿ k δLj (Tk )(γj (Tk ) γi (Tk )) γ 2 (Tk ) − i 1 + δLj (Tk ) 2 Li (Tk+1 ) = Li (Tk ) exp j=δ −1 Tk 2 ∆Tk d + γi (Tk ) bij jk ∆Tk j=1 ✇❤❡r❡ jk Nd (0, 1)✳ ❚♦ ❝♦♠♣❛r❡ ✜❣✉r❡s ❝♦♠♣❛r❛❜❧❡✱ t❤❡ s❛♠❡ ❝❤❛♥❣❡ ♦❢ ♥✉♠❡r❛✐r❡ ♠✉st ❜❡ ❞♦♥❡ ❢♦r t❤❡ s✇❛♣t✐♦♥ ♣❛②♦✛✿ Swaptionp,q (0) = Bspot (0)E∗  q = E∗  j=p Swaptionp,q (Tp ) |F0 Bspot (Tp )  (Sp,q (Tp ) − κ)+  δB(t, Tp+j ) ) Bspot (Tp ) ❆s Bspot (0) = 1 ❇❛❝❦ t♦ t❤❡ s✇❛♣t✐♦♥s✱ ✇❡ ❡①♣r❡ss t❤❡ ✐♥t❡❣r❛❧ ♦❢ t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t② ❛❝❝♦r❞✐♥❣ t♦ s❡❝t✐♦♥ ✷✳✷ ✭t❤❡ ❧❛st t❡r♠ ♦❢ t❤❡ ❣❡♥❡r✐❝ s✇❛♣t✐♦♥ ❢♦r♠✉❧❛✮✿ Tp Tp ci cj η(Ti − s)η(Tj − s)ds σi (s)σj (s)ds = 0 0 1 Ti c2 γi2 = i Ti ❛♥❞ ❛s γi2 = Ti σi2 (s)ds 0 Ti η(Ti − s)2 ds 0 Tp 0 Tp σi (s)σj (s)ds = γi γj Ti 0 Tj Ti 0 η(Ti − s)η(Tj − s)ds η(Ti − s)2 ds Tj 0 η(Tj − s)2 ds ❚❤✐s ♣❡r♠✐ts ✉s t♦ ❣✐✈❡ ❛♥ ❡①♣❧✐❝✐t ❣❡♥❡r✐❝ ❢♦r♠✉❧❛ ❢♦r t❤❡ s✇❛♣t✐♦♥s✿ 1 (γp,q ) ≈ Tp q−1 2 j,i=1 i wj L L wp,q p,q i j β,b,η∞ γi γj ρij ςi,j,p 2 Sp,q ✭✷✳✷✽✮ ✻✾ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ β,b,η∞ ✇✐t❤ ςi,j,p t❤❡ ✐♥t❡❣r❛❧ t❡r♠ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❞❡r✐✈❛t✐♦♥✿ β,b,η∞ ςi,j,p = Ti Tj Tp 0 Ti 0 η(Ti − s)η(Tj − s)ds η(Ti − s)2 ds Tj 0 η(Tj − s)2 ds ❚❤❛♥❦s t♦ t❤❡ ❘♦②❛❧ ❇❛♥❦ ♦❢ ❙❝♦t❧❛♥❞✱ ■ ❝♦✉❧❞ r✉♥ t❡sts ♦♥ ❛ s✇❛♣t✐♦♥ ♠❛tr✐① 10 × 10 ✇✐t❤ t❤❡s❡ ❢♦r♠✉❧❛s ♦♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ✇✐t❤ ♠❛r❦❡t ♣❛r❛♠❡t❡rs ✐♥ ❞❛t❡ ♦❢ ❖❝t♦❜❡r 30th ✷✵✵✻ ❛♥❞ ❝♦♠♣❛r❡ t❤❡♠ t♦ ❛ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥✳ ❇② s✇❛♣t✐♦♥ ♠❛tr✐①✱ ✇❡ ♠❡❛♥ t❤❡ ❇❧❛❝❦ ✈♦❧❛t✐❧✐t✐❡s ♦❢ t❤❡ s✇❛♣t✐♦♥s ♣✉t ✐♥ ❛♥ ❛rr❛② ✇✐t❤ ♦♥ t❤❡ ①✲❛①✐s t❤❡ t❡♥♦r ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ s✇❛♣ ❛♥❞ ♦♥ t❤❡ ②✲❛①✐s t❤❡ ♠❛t✉r✐t② ♦❢ t❤❡ s✇❛♣t✐♦♥✳ ❍❡♥❝❡ ❛ N × M s✇❛♣t✐♦♥ ✐s ❛ s✇❛♣t✐♦♥ ♦❢ ♠❛t✉r✐t② ◆ ❨❡❛rs ♦♥ ❛ ▼ ❨❡❛rs s✇❛♣✳ ❲❡ r❛♥ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥s ♦✈❡r ✶ ♠✐❧❧✐♦♥ ♣❛t❤s ♦♥ ❛ ✶✵❨ t❡♥♦r s✇❛♣ ❛t ♠❛①✐♠✉♠ ✭✇❤✐❝❤ ✐s ❛ ✈❡r② ❝♦♠♠♦♥ t❡♥♦r ❢♦r str✉❝t✉r❡❞ ♣r♦❞✉❝ts ✐♥ ❆s✐❛✮ ✇✐t❤ ❛ ♠❛①✐♠✉♠ ♦♣t✐♦♥ ♠❛t✉r✐t② ♦❢ ✶✵❨✳ ✭❍❡♥❝❡✱ ✇❡ ❤❛❞ t♦ ✉s❡ t❤❡ ◆♦rt❤✲ ❲❡st ♣❛rt ♦❢ ❛ ❝♦rr❡❧❛t✐♦♥ s✉r❢❛❝❡ 20 × 20✮✳ ❲❡ ❝♦✉❧❞ ❡st✐♠❛t❡ t❤❡ ❛✈❡r❛❣❡ ❡rr♦r ❜❡t✇❡❡♥ t❤❡ ♣r❡✈✐♦✉s ❢♦r♠✉❧❛ ❛♣♣❧✐❡❞ t♦ ❍✉❧❧✲❲❤✐t❡✱ ❘❡❜♦♥❛t♦ ❛♥❞ ❆♥❞❡rs❡♥ ❛♥❞ ❆♥❞r❡❛s❡♥ ❛♥❞ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❜② ▼♦♥t❡✲❈❛r❧♦ s✐♠✉❧❛t✐♦♥ ❜② ✉s✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ ✿ 1 10×10 M onte−Carlo −γ F ormula |✳ ❲❡ ❝❛♥ ❝♦♥❝❧✉❞❡ |γp,q p,q ✐t ✐s ♥♦♥ r❡❧❡✈❛♥t✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❆❝❝✉r❛❝② ▼❛①✐♠✉♠ ❉✐s❝r❡♣❛♥❝② ❆✈❡r❛❣❡ ❉✐s❝r❡♣❛♥❝② ❘❡❜♦♥❛t♦ 0.34% (1 × 2) 0.18% ❍✉❧❧ ❛♥❞ ❲❤✐t❡ 0.17% (5 × 2) 0.10% ❆♥❞❡rs❡♥ ❛♥❞ ❆♥❞r❡❛s❡♥ 0.22% (3 × 2) 0.08% ❚❛❜❧❡ ✷✳✺✿ ❙✇❛♣t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❝❝✉r❛❝② ❢♦r ❞✐✛❡r❡♥t ❢♦r♠✉❧❛s ❙♦♠❡ ❝♦♠♠❡♥ts ❛❜♦✉t t❤❡ ❣❡♥❡r❛❧ ❜❡❤❛✈✐♦✉r ♦❢ ❡❛❝❤ ❢♦r♠✉❧❛✳ ❘❡❜♦♥❛t♦ ❛♥❞ ❍✉❧❧ ❲❤✐t❡ ❢♦r♠✉❧❛ s❡❡♠ t♦ ❜❡ q✉✐❡t ♦✛ ♦♥ t❤❡ s❤♦rt ♠❛t✉r✐t② ❛♥❞ s❤♦rt t❡♥♦r ✭❋✐rst ❧✐♥❡ ❛♥❞ ✜rst ❝♦❧✉♠♥✮ ❛♥❞ ♦t❤❡r✇✐s❡ ✇✐t❤ ❛ ❝♦♥st❛♥t ❞✐s❝r❡♣✲ ❛♥❝② ❛❧♦♥❣ t❤❡ ♠❛tr✐①✳ ❆♥❞❡rs❡♥ ❛♥❞ ❆♥❞r❡❛s❡♥ ❢♦r♠✉❧❛ ✐s ❜❡❤❛✈✐♥❣ t❤❡ ✼✵ ❈❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ♦♣♣♦s✐t❡ ❛s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ q✉❛❧✐t② ❞❡❝r❡❛s❡ ✇❤❡♥ t❤❡ ♠❛t✉r✐t② ❛♥❞ t❤❡ t❡♥♦r ✐♥❝r❡❛s❡ ✭●♦✐♥❣ ❙♦✉t❤ ❊❛st ✐♥ t❤❡ ♠❛tr✐①✮✳ ❍❡♥❝❡ ❛ ❣♦♦❞ str❛t❡❣② ❢♦r ❛ ❝❛❧✐❜r❛t✐♦♥ ✇♦✉❧❞ ❜❡ t♦ ✉s❡ ❍✉❧❧ ❲❤✐t❡ ❢♦r t❤❡ s❤♦rt ❞❛t❡❞ s✇❛♣t✐♦♥ ✭✐♥❢❡r✐♦r t♦ ✺ ②❡❛rs✮ ❛♥❞ t❤❡♥ ❆♥❞❡rs❡♥ ❛♥❞ ❆♥❞r❡❛s❡♥ ❢♦r♠✉❧❛✱ t❤✐s ✐s st✐❧❧ ✇♦r❦ ✐♥ ♣r♦❣r❡ss ❛s ✐t ✐s ✈❡r② ✐♥✈♦❧✈❡❞ t♦ ❣❡t ❝♦♥s✐st❡♥t r❡s✉❧ts ✇✐t❤ t❤✐s ♠❡t❤♦❞ ❛❧❧ ❛❧♦♥❣ t❤❡ s✇❛♣t✐♦♥ ♠❛tr✐①✳ ❋r♦♠ ❛ r✐s❦ ♠❛♥❛❣❡♠❡♥t ♣♦✐♥t ♦❢ ✈✐❡✇✱ s♦♠❡ ♣r♦❞✉❝ts ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ s♦♠❡ t❡♥♦rs ♦r ♠❛t✉r✐t✐❡s✱ ✇❡ ❝❛♥ ❞❡❝✐❞❡ t♦ ❡❧✐♠✐♥❛t❡ t❤❡s❡ ✐rr❡❧❡✈❛♥t s✇❛♣t✐♦♥s ♦r r❡❞✉❝❡ t❤❡✐r ✐♥✢✉❡♥❝❡ ✐♥ t❤❡ ❝❛❧✐❜r❛t✐♦♥ ♣r♦❝❡ss ✭❋♦r ✐♥st❛♥❝❡ ❜② ❝❤❛♥❣✐♥❣ t❤❡ ✇❡✐❣❤t ♠❛tr✐① ✐♥ t❤❡ ❋r♦❜❡♥✐✉s ♥♦r♠✮✳ ❚❤✐s ✐s ✈❡r② ✉s❡❢✉❧ ❢♦r ♣r✐❝✐♥❣ ❛❝❝✉r❛t❡❧② ❇❡r♠✉❞❛♥ s✇❛♣t✐♦♥s ✇❤❡r❡ t❤❡ ❝♦✲t❡r♠✐♥❛❧ s✇❛♣t✐♦♥s✷ ❛r❡ ✈❡r② ✐♠♣♦rt❛♥t✳ ❙❡✈❡r❛❧ ♣r♦❝❡❞✉r❡s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞✱ s❡❡ ❬✶✻❪ ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧s✳ ❋✐♥❛❧❧② t♦ ♣✉t t❤✐s ✐♥ ♣❡rs♣❡❝t✐✈❡ ❛ t②♣✐❝❛❧ ❜✐❞✲♦✛❡r s♣r❡❛❞ ✐♥ ❯❙❉ ✇♦✉❧❞ ❜❡ 0.50% ❤✐❣❤❧✐❣❤t✐♥❣ ❤♦✇ ❣♦♦❞ ❛r❡ t❤♦s❡ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ✷ ❈♦✲t❡r♠✐♥❛❧ s✇❛♣t✐♦♥s ❛r❡ t❤❡ s✇❛♣t✐♦♥ ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ❙❲✲◆❊ ♦❢ t❤❡ ♠❛tr✐① ❈❤❛♣t❡r ✸ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ✸✳✶ ❙t♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ❛♣♣❧✐❡❞ t♦ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❚❤❡ ✇♦r❦ ✇❡ ❤❛✈❡ ♣r♦❞✉❝❡❞ ✉♥t✐❧ ♥♦✇ ✇❛s ❛ss✉♠✐♥❣ ❛ ❞❡t❡r♠✐♥✐st✐❝ ✈♦❧❛t✐❧✐t②✳ ▲✐❦❡ ❢♦r t❤❡ ❡q✉✐t✐❡s✱ ✈♦❧❛t✐❧✐t② ♠❛♣♣✐♥❣s s✉✛❡r ❢r♦♠ ❛ s♠✐❧❡ ✭❤❡r❡ ❛ s❦❡✇✮ t❤❛t ♠❛❦❡s t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ✇❤❡♥ ♠♦✈✐♥❣ ❛✇❛② ❢r♦♠ ❛t t❤❡ ♠♦♥❡② ♣♦✐♥t✳ ❙❡✈❡r❛❧ ♣r♦♣♦s✐t✐♦♥s ❤❛✈❡ ❜❡❡♥ ✇♦r❦❡❞ ♦✉t t♦ ✜t t❤❡ ✈❡r② ♦✉t ♦r ✐♥ t❤❡ ♠♦♥❡② ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ❛♥❞ t❤✐s ✐s st✐❧❧ ✇♦r❦ ✐♥ ♣r♦❣r❡ss✳ ❍❡r❡ ✐s t❤❡ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ t❤❡ ♠♦st ✉s❡❞ ♥♦✇❛❞❛②s ✐♥ t❤❡ ✇♦r❧❞ ♦❢ r❛t❡s✳ ✸✳✶✳✶ ❙t♦❝❤❛st✐❝ α β ρ ♠♦❞❡❧ ✲ ❙❆❇❘ ❖♣❡r❛t♦rs ❤❛✈❡ ✜❣✉r❡❞ ♦✉t s✐♥❝❡ ❛ ❧♦♥❣ t✐♠❡ t❤❛t ✐♥t❡r❡st r❛t❡s ♣r♦❞✉❝ts ✇❡r❡ ♥♦t ✇❡❧❧ q✉♦t❡❞ ✉s✐♥❣ ❞❡t❡r♠✐♥✐st✐❝ ✈♦❧❛t✐❧✐t② ✭❡✈❡♥ t❤❡ ♣r❡✈✐♦✉s ♣✐❡❝❡✇✐s❡ ♦r ▲❛❣✉❡rr❡ t②♣❡ ✈♦❧❛t✐❧✐t②✮✳ ❍❛❣❛♥ ✐♥ ❬✷✶❪ ❤❛s ✐♥tr♦❞✉❝❡❞ ❛ ❧♦❝❛❧ ✈♦❧❛t✐❧✲ ✐t② ♠♦❞❡❧ s❡❧❢✲❝♦♥s✐st❡♥t✱ ❛r❜✐tr❛❣❡✲❢r❡❡ ❛♥❞ ✇❤✐❝❤ ♠❛t❝❤ ♦❜s❡r✈❡❞ ♠❛r❦❡t s❦❡✇s✳ ❲❡ ✇✐❧❧ ♣r❡s❡♥t ✐ts ♠❛✐♥ ❢❡❛t✉r❡s ❛♥❞ ❤♦✇ ✐t ✐s ❤❛♥❞❧❡❞ ✐♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✳ ✼✷ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ▼❛✐♥ ❛ss✉♠♣t✐♦♥ ✐s t❤❛t t❤❡ ✈♦❧❛t✐❧✐t② ❢♦❧❧♦✇s ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ❝♦rr❡✲ ❧❛t❡❞ t♦ t❤❡ ❢♦r✇❛r❞ ♣r✐❝❡ Li (t) ✐♥ ✐ts ❢♦r✇❛r❞ ♠❡❛s✉r❡✿ dLi (t) = ΣB Lβi (t)dW1 dΣB = νΣB dW2 , ΣB (0) = σB ✇❤❡r❡ ν ✐s ♥❛♠❡❞ t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ✈♦❧❛t✐❧✐t②✱ ♥❛♠❡❧② ✈♦❧✈♦❧✳ ❚❤❡ t✇♦ ♣r♦❝❡ss❡s W1 ❛♥❞ W2 ❛r❡ ❝♦rr❡❧❛t❡❞ ❜②✿ dW1 dW2 = ρdt ▼❛♥② ♦t❤❡r ❢♦r♠s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ❢♦r t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss ❢♦r t❤❡ ✈♦❧❛t✐❧✐t②✱ ✇✐t❤ ❛ ❞r✐❢t✱ ✇✐t❤ ❛ ♠❡❛♥ r❡✈❡rs✐♦♥ ❡t❝ ❜✉t t❤✐s ♦r✐❣✐♥❛❧ ❢♦r♠ ❣✐✈❡s t❤❡ ♠❡❛♥s t♦ ♠❛♥❛❣❡ t❤❡ s❦❡✇ r✐s❦ ✐♥ ♠❛r❦❡ts ✇✐t❤ ♦♥❧② ❡①❡r❝✐s❡ ❞❛t❡ ✇❤✐❝❤ ✐s ♦✉r ❝❛s❡ ✇✐t❤ t❤❡ ❝❛♣❧❡ts ❛♥❞ t❤❡ s✇❛♣t✐♦♥s ♠❛r❦❡ts✳ ■♥ t❤❡ ♦♣❡r❛t♦r ♣♦✐♥t ♦❢ ✈✐❡✇✱ ♠❛♥❛❣✐♥❣ t❤❡ ✈❡❣❛ r✐s❦ ❜❡❝♦♠❡s ❧✐❦❡ ❞❡❧t❛✲ ❤❡❞❣✐♥❣ ❛s t❤❡ tr❛❞❡r ✇✐❧❧ ❤❛✈❡ t♦ ❜✉② ❛♥❞ s❡❧❧ ♦♣t✐♦♥s t♦ ❜❡❝♦♠❡ ✈❡❣❛ ♥❡✉tr❛❧✳ ❯s✐♥❣ s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ t❡❝❤♥✐q✉❡s ✇❡ ❝❛♥ ❞❡r✐✈❡ ❛ ♣r✐❝❡ ❢♦r ❊✉r♦✲ ♣❡❛♥s ♦♣t✐♦♥s✱ ✇❡ ✇✐❧❧ ❧❡t t❤❡ r❡❛❞❡r r❡❢❡r t♦ ❬✷✶❪ ❢♦r ❛ ❝♦♠♣❧❡t❡ ♣r♦♦❢✳ ❊✉r♦✲ ♣❡❛♥ ♣r✐❝❡s ❛r❡ ❣✐✈❡♥ ✉s✐♥❣ t❤❡ ❇❧❛❝❦ ❢♦r♠✉❧❛ ✇✐t❤ ❛♥ ♦t❤❡r ❇❧❛❝❦ ✈♦❧❛t✐❧✐t② ΣB (Li (t), κ)✳ ❯s✐♥❣ t❤❡ s❛♠❡ ♥♦t❛t✐♦♥s ❛s ✐♥ ✶✳✸✳✶✿ CapletSABR (t) = 1δB(t, Ti+1 )[Li (t)N (d1 (t, Ti )) − κN (d2 (t, Ti ))], ✇✐t❤✱ d1 = ln(Li (t)/κ) + Σ2B (Ti2−t) ΣB (Ti − t) d 2 = d 1 − ΣB (Ti − t) ✼✸ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ❛♥❞ ✇❤❡r❡ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ✐s ❣✐✈❡♥ ❡①♦❣❡♥♦✉s❧②✿ ΣB (Li (t), κ) = 1 σB 4 (1−β)2 2 Li (0) 1 + 24 ln κ + (1−β) (Li (0)κ) 1920 2 σB 1 ρβνσB (1 − β)2 + + 1−β 24 (Li (0)κ) 4 (Li (0)κ) 1−β 2 1−β 2 ln4 Liκ(0) + + ... z x(z) 2 − 3ρ2 2 ν (Ti − t) + . . . 24 ✇❤❡r❡ ✇❡ r❡❢❡r t♦ z ❛s✿ z= 1−β ν (Li (0)κ) 2 ln(Li (t)/κ), σB ❛♥❞ t♦ x(z) ❛s✿ x(z) = ln 1 − 2ρz + z 2 + z − ρ 1−ρ ❚❤❡s❡ ❢♦r♠✉❧❛s ❣✐✈❡ ❛♥ ❡①♣❧✐❝✐t✶ ❢♦r♠ ❢♦r t❤❡ ✈♦❧❛t✐❧✐t② ✐♥ t❤❡ ❊✉r♦♣❡❛♥ ❝❛s❡ ❛♥❞ t❤✐s ❝❛♥ ❜❡ ❤✐❣❤❧✐❣❤t❡❞ ❛s ✐t ❜❡❝♦♠❡s ❡❛s✐❧② ✐♠♣❧❡♠❡♥t❛❜❧❡ ✐♥ t❤✐s ♠♦❞❡❧✱ ✇❤✐❝❤ ✐s ❣❡♥❡r❛❧❧② ♥♦t t❤❡ ❝❛s❡ ✐♥ t❤❡ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧✳ ■♥ ♦r❞❡r t♦ ✜t t❤❡ ♠❛r❦❡t✱ ✇❡ ❝❛♥ ♣❧❛② ♦♥ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧✳ ❚❤❡ β ❝♦♥tr♦❧s t❤❡ ❜❛❝❦❜♦♥❡ ♦❢ t❤❡ s❦❡✇ t❤❛t ♠❡❛♥s t❤❡ ❆❚▼ ✈♦❧❛t✐❧✐t② ΣB (Li (t), Li (t)) ❡st✐♠❛t❡❞ ✇✐t❤ ❛ ❤✐st♦r✐❝❛❧ ❧♦❣✲❧♦❣ ♣❧♦t ♦❢ t❤❡ ❆❚▼ ✈♦❧❛t✐❧✐t✐❡s✳ ■♥ ❣❡♥❡r❛❧ ✇❡ ✉s❡ β = 0.5 ❢♦r t❤❡ ❯❙❉ ■♥t❡r❡st r❛t❡ ♠❛r❦❡t ✭❧✐❦❡ ✐♥ t❤❡ ❈■❘ ▼♦❞❡❧✮✳ ❚❤❡ α ♣❛r❛♠❡t❡r ✐s ❝♦♥✈❡♥✐❡♥t❧② r❡♣❧❛❝❡❞ ❜② t❤❡ ❆❚▼ ✈♦❧❛t✐❧✐t② ✭❖♥❡ ❝❛♥ ♥✉♠❡r✐❝❛❧❧② ✐♥✈❡rt t❤❡ ❢♦r♠✉❧❛✮ ❛♥❞ ✐s ❝❤❛♥❣❡❞ ❛❧♠♦st ❡✈❡r② ❤♦✉rs✳ ρ ❛♥❞ ν ❝♦♥tr♦❧ t❤❡ s❦❡✇✳ ν ✐s ✈❡r② ❤✐❣❤ ❢♦r s❤♦rt✲❞❛t❡❞ ♦♣t✐♦♥s✱ ❛♥❞ ❞❡✲ ❝r❡❛s❡ ❛s t❤❡ t✐♠❡✲t♦ ❡①❡r❝✐s❡ ✐♥❝r❡❛s❡s✱ ✇❤✐❧❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ρ st❛rts ♥❡❛r 0 ❛♥❞ ❜❡❝♦♠❡s s✉❜st❛♥t✐❛❧❧② ♥❡❣❛t✐✈❡ ❛❧♦♥❣ t✐♠❡✲t♦ ❡①❡r❝✐s❡✳ ■t s❤♦✉❧❞ ❜❡ ♥♦t✐❝❡❞ t❤❛t t❤❡r❡ ✐s ❛ ✇❡❛❦ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ♠❛r❦❡t s❦❡✇ ✶ ❚❤❡ ♦♠✐tt❡❞ t❡r♠s ✐♥ . . . ❛r❡ ♠✉❝❤ s♠❛❧❧❡r ✼✹ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ♦♥ t❤❡ t❡♥♦r ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ s✇❛♣ ❤❡♥❝❡ t❤♦s❡ ♣❛r❛♠❡t❡rs ❛r❡ ❢❛✐r❧② ❝♦♥st❛♥t ❛❧♦♥❣ ♠❛r❦❡t ♠♦✈❡s ❢♦r ❡❛❝❤ t❡♥♦r✳ ■♥ ❣❡♥❡r❛❧✱ t❤❡② ❛r❡ ✉♣✲ ❞❛t❡❞ ♦♥ ❛ ♠♦♥t❤❧② ❜❛s✐s✳ ❖♥❡ s❤♦✉❧❞ ♥♦t✐❝❡ t❤❛t t❤❡ ❝❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡s❡s ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ✐s ♠❛❞❡ ❤❛r❞ ❜② t❤❡ ❛❜s❡♥❝❡ ♦❢ ❧✐q✉✐❞✐t② ♦❢ s♦♠❡ ♣❛rts ♦❢ t❤❡ s❦❡✇ ✐♥ t❤❡ ♠❛r❦❡t ✿ ✈❡r② ♦✉t ♦❢ t❤❡ ♠♦♥❡② ♦r ❞❡❡♣❧② ✐♥ t❤❡ ♠♦♥❡② s✇❛♣t✐♦♥s ❛r❡ ❧❡ss ❧✐❦❡❧② t♦ ❜❡ tr❛❞❡❞ ❛♥❞ ❝♦♥s✐st❡♥❝② ❜❡t✇❡❡♥ ♣r✐❝❡s ✐s q✉✐❡t ❤❛r❞ t♦ ❜❡ ❢♦✉♥❞✳ ❊①t❡♥s✐♦♥s ❝❛♥ ❜❡ ♠❛❞❡ ✇✐t❤ ❛ ✈♦❧❛t✐❧✐t② ♠♦❞❡❧ t❤❛t ❤❛♥❞❧❡ ♠❛r❦❡t ❥✉♠♣s ♦r ✉s❡s ✐♥st❛♥t❛♥❡♦✉s st♦❝❤❛st✐❝ ❝♦rr❡❧❛t✐♦♥✳ ❚❤✐s ✐s ♦❜✈✐♦✉s❧② ✈❡r② ✇♦r❦✲✐♥✲♣r♦❣r❡ss✳ ✸✳✷ ❍②❜r✐❞s Pr♦❞✉❝ts ❚❤✐s s❡❝t✐♦♥ ✐s ♠✉❝❤ ♠♦r❡ q✉❛❧✐t❛t✐✈❡ ❛s t❤✐s t♦♣✐❝ ✐s ❛ ✈❡r② ♥❡✇ ❛♥❞ ❝♦♥✜✲ ❞❡♥t✐❛❧ ♦♥❡ ❛♥❞ ❛ ✈❡r② ❢❡✇ ❛❝❛❞❡♠✐❝ ♣❛♣❡r ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ❆❢t❡r ❞✐s❝✉ss✐♦♥s ❛♥❞ ❛tt❡♥❞❛♥❝❡ t♦ ♠❡❡t✐♥❣s ✇✐t❤ ♠❛r❦❡t ♦♣❡r❛t♦rs✱ ■ ❛♠ ❣♦✐♥❣ t♦ ♣r❡s❡♥t s♦♠❡ ❣❡♥❡r❛❧ ✈✐❡✇s ♦✈❡r t❤❡s❡ ♥❡✇ ❞❡r✐✈❛t✐✈❡s✳ ❆ ❞❡r✐✈❛t✐✈❡ ✐s ❛♥ ❤②❜r✐❞ ✇❤❡♥ t❤❡ ✇❤♦❧❡ ♦r ♣❛rt ♦❢ t❤❡ tr❛❞❡ ❤❛s r✐s❦ ❛❝r♦ss t✇♦ ♦r ♠♦r❡ ❛ss❡t ❝❧❛ss❡s t❤❛t ❝❛♥♥♦t ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ s♣❡❝✐✜❝ ❛ss❡t ❝❧❛ss❡s✷ ✳ ■t ❝❛♥ ❜❡ ❜♦t❤ ❝♦♥s✐❞❡r❡❞ ❛s ❛ ♣r♦❞✉❝t ♦r ❛♥ ❛ss❡t ❝❧❛ss s✐♥❝❡ ❞✉❡ t♦ ❝r♦ss ❝♦♥✈❡①✐t② ♦♥❡ ❛ss❡t ❝❧❛ss ❝❛♥♥♦t ❜❡ r✐s❦ ♠❛♥❛❣❡❞ ✇✐t❤♦✉t ❝♦♥s✐❞❡r✐♥❣ ♦t❤❡r ❛ss❡t ❝❧❛ss❡s ✐♥ ❛ ❣✐✈❡♥ tr❛❞❡✳ ■♥ ❛ ♣r✐❝✐♥❣ ♣❡rs♣❡❝t✐✈❡ t❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ✇✐t❤ s✐♥❣❧❡ ❛ss❡t str✉❝t✉r❡❞ ♣r♦❞✉❝ts ✐s t❤❡ ✐♠♣♦rt❛♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥s✱ ❝♦rr❡❧❛t✐♦♥ ❛♥❞ ❝r♦ss ❝♦♥✈❡①✐t②✳ ❏♦✐♥t ❞✐str✐❜✉t✐♦♥ ❚✇♦ ❞✐✛❡r❡♥t ✇❛②s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ♣❛②♦✛ ✭✐♥ ♦t❤❡r ✇♦r❞s t❤❡ ✐♥t❡❣r❛❧ ❛♥❞ t❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t✇♦ ❛ss❡ts✮ ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ✉s✐♥❣ t❤❡ ✇♦r❦ ❞♦♥❡ ♦♥ s✐♥❣❧❡ ❛ss❡t ❡①♦t✐❝s✿ ■♠♣❧✐❡❞ ❞✐str✐❜✉t✐♦♥s ✭■♥t❡r❡st ❘❛t❡s✮ ❛♥❞ ❈♦♣✉❧❛s ✭❈r❡❞✐t ❉❡r✐✈❛t✐✈❡s✮✳ ✷ ▼❛✐♥ ❛ss❡t ❝❧❛ss❡s ❛r❡✿ ❊q✉✐t②✱ ❘❛t❡s✱ ❋❳✱ ❈r❡❞✐t✱ ❈♦♠♠♦❞✐t✐❡s✱ ■♥✢❛t✐♦♥✳ ✼✺ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ❯s✐♥❣ t❤❡ ✐♠♣❧✐❡❞ ❞✐str✐❜✉t✐♦♥ ♠❡❛♥s t❤❛t ❢r♦♠ t❤❡ ❝❛♣❧❡t✴✢♦♦r❧❡t ♣r✐❝❡s ✇❡ ❜✉✐❧❞ ❛♥ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ❢♦r ❡❛❝❤ ❛ss❡t ❝❧❛ss ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ tr❛❞❡✳ ❆ ❈♦♣✉❧❛ ✐s ❛ r❡❛❧ ❢✉♥❝t✐♦♥ C s✉❝❤ t❤❛t ✐♥ ❛ ✷ ❞✐♠❡♥s✐♦♥s ❝❛s❡ ✐s ❞❡✜♥❡❞ ♦♥ I 2 = [0, 1]2 ❛♥❞✿ C(x, 0) = C(0, x) = 0 ❛♥❞ C(x, 1) = x, C(1, z) = z ✭✸✳✶✮ ❱❡r② ❜❛s✐❝❛❧❧②✱ ✉s✐♥❣ t❤❡ ❙❦❧❛r t❤❡♦r❡♠ t❤❛t s❡ts t❤❛t ❢♦r ❡❛❝❤ ❏♦✐♥t ❞✐str✐✲ ❜✉t✐♦♥ F (X1 , X2 ) t❤❡r❡ ❡①✐st ❛ ❢✉♥❝t✐♦♥ C ❞❡♣❡♥❞s ♦♥ C(F1 (X1 ), F2 (X2 )) ✇❤❡r❡ t❤❡ Fi ❛r❡ t❤❡ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦✉r ❛ss❡ts✱ ✇❡ ❝❛♥ ❞❡t❡r♠✐♥❡ t❤❡ ❏♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✷ ❛ss❡ts✳ ❈♦rr❡❧❛t✐♦♥ ❚❤✐s ✐s ❛♥ ✐ss✉❡ ❢♦r r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛♥❞ ❢♦r ♣r✐❝✐♥❣✳ ❲❡ s❛✇ ✐♥ t❤✐s t❤❡s✐s t❤❛t ♣r✐❝✐♥❣ ✇❛s ❛❧❧ ❛❜♦✉t ❝♦rr❡❧❛t✐♦♥ ❛♥❞ ♠❛r❦❡t ❞❛t❛ ❛r❡ ❛ ❝r✉❝✐❛❧ ♣♦✐♥t ❢♦r ❛ ❣♦♦❞ ❝❛❧✐❜r❛t✐♦♥✳ ❖♥❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤❛t ✇❤❡♥ t✇♦ ❝❧❛ss❡s ♦❢ ❛ss❡ts ❛r❡ ✐♥✈♦❧✈❡❞ t❤❡ ✐ss✉❡ ✐s ❡✈❡♥ ❜✐❣❣❡r t❤❛♥ ✇❤❡♥ t❛❧❦✐♥❣ ❛❜♦✉t ❥✉st t✇♦ ❋♦r✇❛r❞ ▲✐❜♦r r❛t❡s✳ ❚❤✐s ✐s st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠ ❢♦r ♠❛♥② ❤♦✉s❡s✿ ♦♣❡r❛t♦rs ❛r❡ t❛❧❦✐♥❣ ❛❜♦✉t st♦❝❤❛st✐❝ ❝♦rr❡❧❛t✐♦♥ ❜✉t ♠♦st ♦❢ ❛❧❧ r❡❢❡r t♦ t❤❡ ❝♦♠♠♦♥ s❡♥s❡ ❜❡❢♦r❡ ❣✐✈✐♥❣ ❛ ♣r✐❝❡✳ ❈r♦ss✲❝♦♥✈❡①✐t② ❈♦♥✈❡①✐t② ♣r♦❜❧❡♠s ❛r❡ ♥♦t ♥❡✇ t♦ ❛♥②♦♥❡ ✇❤♦ ❛❧r❡❛❞② ❞❡❛❧t ✇✐t❤ ❈♦♥st❛♥t ▼❛t✉r✐t② ❙✇❛♣ ❛♥❞ ✐♥ ❣❡♥❡r❛❧ ✐♥t❡r❡st r❛t❡s✳ ❇❛s✐❝❛❧❧②✱ ✐♥ ❤②❜r✐❞s✱ ♠❛♥❛❣✐♥❣ t❤❡ r✐s❦ ✐♥ t❡r♠s ♦❢ ❞❡❧t❛ ❛♥❞ ❣❛♠♠❛ ✐s ♠✉❝❤ ♠♦r❡ ✐♥✈♦❧✈❡❞ ❞✉❡ t♦ t❤✐s t❡r♠ ♦❢ ❝♦♥✈❡①✐t② ❛❝r♦ss t❤❡ ❛ss❡t ❝❧❛ss❡s✳ ❙✉♠♠❛r② ❍②❜r✐❞s ❛r❡ ❛ ❤♦t t♦♣✐❝ ❛♥❞ ✇❡ ❤❛✈❡ s❡❡♥ ❛ ❣r♦✇✐♥❣ ❞❡♠❛♥❞ ❢♦r t❤♦s❡ ❦✐♥❞s ♦❢ ♣r♦❞✉❝ts ❛❧❧ ❛r♦✉♥❞ ❆s✐❛✳ Pr✐❝✐♥❣ ✐s ✈❡r② ✐♥✈♦❧✈❡❞ ❛♥❞ r✐s❦ ♠❛♥❛❣❡♠❡♥t ❝❛♥ ❜❡ ❛ ♥✐❣❤t♠❛r❡✿ ❢♦r ✐♥st❛♥❝❡✱ ✈♦❧❛t✐❧✐t② ❥✉♠♣s ✐♥ ♦♥❡ ❛ss❡t ❝❧❛ss ✈❡r② ♦❢t❡♥ ❜r✐♥❣s ❛ ❥✉♠♣ ✐♥ ♦t❤❡r ❛ss❡t ❝❧❛ss❡s❀ t❤❡♥✱ t❤❡ ♠❛r❦❡t ♠✐❣❤t ♣r♦❜❛❜❧② ❣❡t ✉♣s❡t ❛♥❞ ❛❧❧ ❛ss✉♠♣t✐♦♥s ♣r❡✈✐♦✉s❧② ♠❛❞❡ ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ r❡❝♦♥s✐❞❡r❡❞✳ ✼✻ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ✸✳✸ ■ss✉❡s r❛✐s❡❞ ✸✳✸✳✶ ❈❤♦✐❝❡ ❜❡t✇❡❡♥ ❍✐st♦r✐❝❛❧ ❛♥❞ ■♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ❚❤❡② ❛r❡ t✇♦ ❛♣♣r♦❛❝❤❡s t♦ t❤❡ ❝❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ s✇❛♣t✐♦♥✳ ❲❤❡t❤❡r ✇❡ ❞❡❝✐❞❡ t♦ s♠♦♦t❤ t❤❡ ❤✐st♦r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ✇✐t❤ ❛ ♣❛r❛♠❡tr✐❝ ❢♦r♠✳ ❚❤❡♥ ❜② ✉s✐♥❣ t❤✐s ❢♦r♠ ✐♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛ ✇❡ ✜t t❤❡ s✇❛♣t✐♦♥ ♣r✐❝❡s ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡r ς ✳ ❖r ✇❡ ✐❣♥♦r❡ t❤❡ ❤✐st♦r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ❛♥❞ ✇❡ ♦♥❧② ❛❞❛♣t t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡ t♦ ❝❛❧✐❜r❛t❡ t❤❡ ♠♦❞❡❧ ♦♥ t❤❡ s✇❛♣t✐♦♥ ♣r✐❝❡s✳ ■♥❞❡❡❞✱ ♦♥❡ ✇♦✉❧❞ s❛② t❤❛t t❤♦s❡ t✇♦ ♠❡t❤♦❞s s❤♦✉❧❞ ♣r♦❞✉❝❡ s✐♠✐❧❛r r❡s✉❧ts✳ ■t ✐s ♥♦t t❤❡ ❝❛s❡ ❛s t❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❡✈❡♥ ❛❢t❡r s♠♦♦t❤✐♥❣ ❜② ❛ ♣❛r❛♠❡tr✐❝ ❢♦r♠ ❣✐✈❡s ❞✐✛❡r❡♥t r❡s✉❧ts ❢r♦♠ t❤❡ s✇❛♣t✐♦♥ ♣r✐❝❡s q✉♦t❡❞ ✐♥ t❤❡ ♠❛r❦❡t✳ ❚❤✐s ❡①♣❧❛✐♥s ❛❧s♦ ✇❤② t❤❡ ✐♠♣❧✐❝✐t ❝♦rr❡❧❛t✐♦♥ s✉r❢❛❝❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♦♥❡ ♦❜t❛✐♥❡❞ ✉s✐♥❣ ❤✐st♦r✐❝❛❧ ❞❛t❛✳ ◆❡✈❡rt❤❡❧❡ss✱ ♦♣❡r❛t♦rs ❤❛✈❡ tr✐❡❞ t♦ ✐♥t❡❣r❛t❡ ❜♦t❤ ❤✐st♦r✐❝❛❧ ❛♥❞ ✐♠♣❧✐❡❞ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤✐s ❞♦❡s ♥♦t s❡❡♠ t♦ ✇♦r❦ ♣r♦♣❡r❧②✳ ❍❡♥❝❡✱ ❛s t❤❡ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ♣❡r♠✐t t♦ ✜♥❞ t❤❡ s✇❛♣t✐♦♥ ♣r✐❝❡s ❛♥❞ ❤❛s ❧❡ss ✈❛❧✉❡ t❤❛t t❤❡ ✐♠♣❧✐❡❞ ✈❛❧✉❡ ✭✇❤✐❝❤ ❜❛s✐❝❛❧❧② ♣r✐❝❡ ✇❤❛t ✐s ❣♦✐♥❣ t♦ ❜❡ t❤❡ ♠❛r❦❡t✮ ✇❡ ♣r❡❢❡r t♦ ❝❤♦♦s❡ t♦ ✉s❡ t❤❡ ✐♠♣❧✐❡❞ ❝♦rr❡❧❛t✐♦♥✳ ✸✳✸✳✷ ■♥t❡r❡st✲r❛t❡s s❦❡✇ ❊①❝❡♣t ✐♥ t❤✐s s❡❝t✐♦♥ ✸✳✶✱ ✇❡ ❤❛✈❡ s✉♣♣♦s❡❞ t❤❡ ✈♦❧❛t✐❧✐t② t♦ ❜❡ ❞❡t❡r♠✐♥✐st✐❝ ❛♥❞ ❛t ♠♦st t✐♠❡ ❞❡♣❡♥❞❡♥t✳ ●r❡❛t ✐♠♣r♦✈❡♠❡♥ts t♦ t❤❡ ❝❛❧✐❜r❛t✐♦♥ ♦❢ t❤❡ ▲▼▼ ❝❛♥ ❜❡ ❞♦♥❡ ❜② ✉s✐♥❣ st♦❝❤❛st✐❝ ✈♦❧❛t✐❧✐t② t♦ ♠♦❞❡❧ ✐♥t❡r❡st r❛t❡ s❦❡✇✳ ❆s ❞❡s❝r✐❜❡❞ ❜❡❢♦r❡✱ ❙❆❇❘ ▼♦❞❡❧ ❞❡✈❡❧♦♣❡❞ ❜② ❍❛❣❛♥ ✐♥ ❬✷✶❪ ✐s t❤❡ ♠♦st ✉s❡❞ ✭❛♥❞ t❤❡ ♦♥❡ ✉s❡❞ ❛t t❤❡ ❘♦②❛❧ ❇❛♥❦ ♦❢ ❙❝♦t❧❛♥❞✮✳ ✼✼ P❡rs♣❡❝t✐✈❡s ❛♥❞ ✐ss✉❡s ✸✳✸✳✸ ❆♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛ ❇② ♥❛t✉r❡ ✉s✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s ❜r✐♥❣s ②♦✉ ✐ss✉❡s✳ ■♥ ♦✉r ❝❛s❡ ✇❡ ❤❛✈❡ ❢♦✉♥❞ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ s✇❛♣t✐♦♥ ♣r✐❝❡s✳ ❚❤♦s❡ ❛r❡ t❤❡ st❛t❡✲♦❢✲t❤❡✲❛rt ♦❢ t❤✐s t♦♣✐❝ ❜✉t st✐❧❧ t❤❡② ❞♦ ♥♦t ♣❡r♠✐t t♦ ♣r✐❝❡ ❛❝❝✉r❛t❡❧② s✇❛♣t✐♦♥s ❛❧❧ ❛❧♦♥❣ t❤❡ ♠❛tr✐① ❜✉t st✐❧❧✱ ❣✐✈❡s ❛♥ ❛❧♠♦st ❧♦❣✲♥♦r♠❛❧ ❜❡❤❛✈✐♦✉r t♦ s✇❛♣ r❛t❡s✳ ✸✳✸✳✹ ▼❛r❦❡t ❧✐q✉✐❞✐t② ■♥ ♦r❞❡r t♦ ♣r✐❝❡ ❧♦♥❣ tr❛❞❡s✱ ✇❡ ♥❡❡❞ t♦ ❝❛❧✐❜r❛t❡ ❛ r❛t❤❡r ❜✐❣ s✇❛♣t✐♦♥ ♠❛tr✐①✳ ❆❢t❡r s❡✈❡r❛❧ ❞✐s❝✉ss✐♦♥s ✇✐t❤ tr❛❞❡rs✱ ■ ❤❛♣♣❡♥❡❞ t♦ r❡❛❧✐③❡ t❤❛t s♦♠❡ ❛r❡ ✈❡r② ✐❧❧✐q✉✐❞ ✭◗✉♦t❡s ❛r❡ ❡✈❡♥ ✇♦rst ✐♥ ♥♦♥ ❯❙❉ ♦r ❊❯❘ ♠❛r❦❡t ❧✐❦❡ ❡♠❡r❣✐♥❣ ❝✉rr❡♥❝✐❡s✿ ❑❘❲✱ ❚❍❇✱ ❚❲❉✱ ❙●❉✱ ❍❑❉✮ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ q✉♦t❡s ❣✐✈❡♥ ❜② ❜r♦❦❡rs ❝❛♥ ❜❡ str❛♥❣❡ ❧❡❛❞✐♥❣ t♦ ❛ ❜❛❞ ❝❛❧✐❜r❛t✐♦♥✳ ❈❤❛♣t❡r ✹ ●❡♥❡r❛❧ ▼❡t❤♦❞♦❧♦❣② ♣r♦♣♦s❡❞ ❢♦r ❝❛❧✐❜r❛t✐♦♥ ❚❤✐s ✐s ❛ s❤♦rt s✉♠♠❛r② ♦❢ ✇❤❛t ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ✐♥ t❤✐s t❤❡s✐s ❛s ♠❡t❤♦❞✲ ♦❧♦❣② t♦ ❝❛❧✐❜r❛t❡ t❤❡ ▲✐❜♦r ▼❛r❦❡t ♠♦❞❡❧ t♦ t❤❡ s✇❛♣t✐♦♥ ♣r✐❝❡s✳ ✹✳✶ ❆ss✉♠♣t✐♦♥s ❼ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✿ ▲♦❣♥♦r♠❛❧✐t② ♦❢ ❢♦r✇❛r❞ r❛t❡s ❼ ❱♦❧❛t✐❧✐t②✿ ❉❡t❡r♠✐♥✐st✐❝ ❼ ❈♦rr❡❧❛t✐♦♥✿ ❉❡t❡r♠✐♥✐st✐❝ ✹✳✷ ▼♦❞❡❧✐♥❣ ❝❤♦✐❝❡s ❼ ❱♦❧❛t✐❧✐t② str✉❝t✉r❡✿ ▲❛❣✉❡rr❡ t②♣❡ γi (t) = σi (t) = ci η(Ti − t) η(s) = ηa,β,η∞ (s) = η∞ + (1 − η∞ + bs)e−βs b, β,η∞ ≥ 0 ✼✾ ●❡♥❡r❛❧ ▼❡t❤♦❞♦❧♦❣② ♣r♦♣♦s❡❞ ❢♦r ❝❛❧✐❜r❛t✐♦♥ ❼ ❈♦rr❡❧❛t✐♦♥ str✉❝t✉r❡✿ ❙❝❤♦❡♥♠❛❦❡rs ❛♥❞ ❈♦✛❡② ρij = exp − |i − j| ln ρ∞ m−1 i2 + j 2 + ij − 3mi − 3mj + 3i + 3j + 2m2 − m − 4 + η1 + (m − 2)(m − 3) i2 + j 2 + ij − mi − mj − 3i − 3j + 3m + 2 − η2 , (m − 2)(m − 3) (i, j) ∈ [1, m]2 , 3η1 ≤ η2 ≤ 0, 0 ≤ η1 + η2 ≤ −lnρ∞ ❼ ❆♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛✿ ❘❡❜♦♥❛t♦✱ ❍✉❧❧ ❲❤✐t❡ ♦r ❆♥❞❡rs❡♥ & ❆♥✲ ❞r❡❛s❡♥ ✹✳✸ ▼❛r❦❡t ❞❛t❛ ❼ ❚❤❡ ②✐❡❧❞ ❝✉r✈❡ ✭❈✉rr❡♥t ♣r✐❝❡ B(0, t) ♦❢ t❤❡ ❜♦♥❞s ♠❛t✉r✐♥❣ ❛t t✐♠❡ t✮ ❼ ❈❛♣❧❡t ✈♦❧❛t✐❧✐t✐❡s✿ σiBlack,LM M 2 = c2i Ti Ti η 2 (Ti − s)ds 0 ❼ ❆t✲❚❤❡✲▼♦♥❡② ❙✇❛♣t✐♦♥s q✉♦t❛t✐♦♥s ✐♥ ✈♦❧❛t✐❧✐t✐❡s ✹✳✹ ❈❛❧✐❜r❛t✐♦♥ ♣r♦❝❡ss ❼ ❋✐t r♦✉❣❤❧② t❤❡ s✇❛♣t✐♦♥ ♠❛tr✐① γp,q ✇✐t❤ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r♠✉❧❛ ❛♥❞ t❤❡ ♠❛r❦❡t ❞❛t❛✱ ❼ ❘✉♥ ❛ Pr✐♥❝✐♣❛❧ ❈♦♠♣♦♥❡♥t ❆♥❛❧②s✐s ♦♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ♣r❡✈✐✲ ♦✉s❧② ✉s❡❞ ❛♥❞ ❦❡❡♣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❛❝t♦rs✱ ❼ ❯s❡ t❤❡ r❛♥❦ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞ ✇✐t❤ ❘❡❜♦♥❛t♦ ❛♥❣❧❡s t♦ ♦❜t❛✐♥ ❛ ❝❧♦s❡r odel ✱ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ρM ij ✽✵ ●❡♥❡r❛❧ ▼❡t❤♦❞♦❧♦❣② ♣r♦♣♦s❡❞ ❢♦r ❝❛❧✐❜r❛t✐♦♥ ❼ ❘❡✲r✉♥ t❤❡ ✜rst ✸ st❡♣s ✇✐t❤ t❤❡ ♣❛r❛♠❡t❡rs ❛❧r❡❛❞② ❢♦✉♥❞ ❛♥❞ ✉s✐♥❣ ♦♥❧② ❛s r❡❢❡r❡♥❝❡ t❤❡ s✇❛♣t✐♦♥s ✉s❡❢✉❧ ❢♦r t❤❡ ♣r✐❝✐♥❣ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡✱ ❼ ❋✐♥❛❧❧②✱ t❤❡ ♠♦❞❡❧ ✐s ✇❡❧❧ ❝❛❧✐❜r❛t❡❞ ♦♥ ❝❛♣❧❡ts ❛♥❞ ♦♥ t❤❡ s✇❛♣t✐♦♥s ✇❡ ♥❡❡❞✳ ❼ ❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ♣r✐❝❡ ■♥t❡r❡st r❛t❡s ❞❡r✐✈❛t✐✈❡s ✇✐t❤ t❤✐s ❝❛❧✐❜r❛t❡❞ ♠♦❞❡❧✿ ❢r♦♠ t❤✐s ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✱ t❤❡ ✈♦❧❛t✐❧✐t② ♠❛♣♣✐♥❣ ❛♥❞ t❤❡ ❋♦r✇❛r❞ r❛t❡s ❛t t✐♠❡ 0✱ r✉♥ ❛ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥ ♦♥ t❤❡ ❞✐s✲ ❝r❡t✐③❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❋♦r✇❛r❞ r❛t❡s ✐♥ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ t♦ ♦❜t❛✐♥ t❤❡✐r ❞✐✛✉s✐♦♥ t❤r♦✉❣❤ t❤❡ t✐♠❡✳ ✹✳✺ ❈♦♥❝❧✉s✐♦♥ ❚❤✐s t❤❡s✐s ❤❛s ❞❡s❝r✐❜❡❞ ❡①t❡♥s✐✈❡❧② t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧ ❛♥❞ ❤♦✇ ✐t ✐s ❛♥ ✐♠♣♦rt❛♥t st❡♣ ✐♥ ■♥t❡r❡st ❘❛t❡s ♠♦❞❡❧✳ ❆❢t❡r t❤✐s t❤❡♦r❡t✐❝❛❧ ❞❡s❝r✐♣t✐♦♥✱ ✇❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ❞✐✛❡r❡♥t ♣❛r❛♠❡tr✐❝ ❢♦r♠s ❢♦r t❤❡ ✐♥st❛♥t❛♥❡♦✉s ✈♦❧❛t✐❧✐t② ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ❛♥❞ ❝❤♦s❡♥ ❛ s❡t ♦❢ ♣❛r❛♠❡t❡rs✿ ▲❛❣✉❡rr❡ t②♣❡ ✈♦❧❛t✐❧✐t② ❛♥❞ ❙❝❤♦❡♥♠❛❦❡rs✲❈♦✛❡② s❡♠✐✲♣❛r❛♠❡tr✐❝ ❝♦rr❡❧❛t✐♦♥✳ ❚❤❡♥✱ ❛ ✸✲❢❛❝t♦r ❝❛s❡ ❝❛❧✐❜r❛t✐♦♥ ♣r♦❝❡ss ♦❢ t❤✐s ♠♦❞❡❧ ✇❛s s❡❧❡❝t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ r❡s✉❧ts ♦❢ ❛ Pr✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s ❞♦♥❡ ♦♥ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❝❤♦s❡♥ ❜❡❢♦r❡✳ ❋r♦♠ s❡✈❡r❛❧ ♠❛r❦❡t ✐♥♣✉ts ❛♥❞ ❞✐✛❡r❡♥t ❥✉st✐✜❡❞ ❛ss✉♠♣t✐♦♥s✱ ✇❡ ❝♦✉❧❞ ❝❛❧✐❜r❛t❡ t❤❡ ♠♦❞❡❧ t♦ ❝❛♣❧❡ts ❛♥❞ s✇❛♣t✐♦♥s ✐♥ ❛ r❡❛s♦♥❛❜❧❡ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ❛♥❞ ✇✐t❤ ❛❝❝❡♣t❛❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥s t❤❛♥❦s t♦ ❝❧♦s❡❞ ❢♦r♠✉❧❛ ❢♦r s✇❛♣t✐♦♥ ♣r✐❝❡s✳ ❚❤✐s ❢♦r♠✉❧❛ ♣❡r♠✐tt❡❞ ✉s t♦ ❛✈♦✐❞ r✉♥♥✐♥❣ s❡✈❡r❛❧ ▼♦♥t❡✲❈❛r❧♦ s✐♠✉❧❛t✐♦♥s✳ ❆s ❤✐❣❤❧✐❣❤t❡❞✱ t❤✐s ♣r♦❝❡ss ✐s st✐❧❧ ❛♥ ♦♣❡♥ ♣r♦❜❧❡♠ ❡s♣❡❝✐❛❧❧② ❢♦r s❦❡✇ ✐ss✉❡s ❛♥❞ ♣r✐❝✐♥❣ ♦❢ ❝r♦ss✲❛ss❡t ♣r♦❞✉❝ts✳ ❇✐❜❧✐♦❣r❛♣❤② ❬✶❪ ❇❧❛❝❦ ❛♥❞ ❙❝❤♦❧❡s✱ ✧❚❤❡ ♣r✐❝✐♥❣ ♦❢ ♦♣t✐♦♥s ❛♥❞ ❝♦r♣♦r❛t❡ ❧✐❛❜✐❧✐t✐❡s✧✱ ❏✳ ♦❢ P♦❧✐t✐❝❛❧ ❊❝♦♥♦♠②✱ ✽✶ ✭✶✾✼✸✮✱ ✻✸✼✲✻✺✾✳ ❬✷❪ ❇❧❛❝❦ ❋✳✱ ✧Pr✐❝✐♥❣ ♦❢ ❛ ❝♦♠♠♦❞✐t② ❝♦♥tr❛❝t✧✱ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊❝♦♥♦♠✐❝s✱✸ ✭✶✾✼✻✮✱ ✶✻✼✲✶✼✾✳ ❬✸❪ ❍✉❧❧ ❏✳✱ ✧❖♣t✐♦♥s✱ ❋✉t✉r❡s ❛♥❞ ♦t❤❡r ❞❡r✐✈❛t✐✈❡s✱ ❙✐①t❤ ❊❞✐t✐♦♥✧✱ P❡❛r✲ s♦♥ ❊❞✉❝❛t✐♦♥✱ ✭✷✵✵✻✮✳ ❬✹❪ ❘❡❜♦♥❛t♦ ❘✳✱ ✧❚❡r♠✲❙tr✉❝t✉r❡ ▼♦❞❡❧s✿ ❆ r❡✈✐❡✇✧✱ ◗❯❆❘❈✱ ❚❤❡ ❘♦②❛❧ ❇❛♥❦ ♦❢ ❙❝♦t❧❛♥❞ ✭✷✵✵✸✮✳ ❬✺❪ ❖❦s❡♥❞❛❧✱ ❇✳✱ ✧❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✲ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✧✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✭✶✾✾✽✮ ❇❡r❧✐♥✳ ❬✻❪ ❈♦①✱ ■♥❣❡rs♦❧❧ ❛♥❞ ❘♦ss✱ ✧❆ ❚❤❡♦r② ♦❢ t❤❡ t❡r♠✲str✉❝t✉r❡ ♦❢ ■♥t❡r❡st ❘❛t❡s✧✱ ❊❝♦♥♦♠❡tr✐❝❛✱ ✺✸ ✭✶✾✽✺✮✱ ✸✽✺✲✹✵✼✳ ❬✼❪ ❍♦ ❛♥❞ ▲❡❡✱ ✧❚❡r♠ ❙tr✉❝t✉r❡ ♠♦✈❡♠❡♥ts ❛♥❞ Pr✐❝✐♥❣ ■♥t❡r❡st ❘❛t❡ ❈♦♥t✐♥❣❡♥t ❈❧❛✐♠s✧✱ ❏✳ ❋✐♥❛♥✳✱ ✹✶ ✭✶✾✽✻✮✱ ✶✵✶✶✲✷✾✳ ❬✽❪ ❇❧❛❝❦ ❛♥❞ ❑❛r❛s✐♥s❦✐✱ ✧❇♦♥❞ ❛♥❞ ❖♣t✐♦♥ Pr✐❝✐♥❣ ✇❤❡♥ ■♥t❡r❡st ❘❛t❡s ❛r❡ ❧♦❣✲♥♦r♠❛❧✧✱ ❋✐♥❛♥✳ ❆♥❛❧②s✐t ❏✳✱ ✭✶✾✾✶✮✱ ✺✷✲✺✾✳ ❬✾❪ ❱❛s✐❝❡❦ ❖✳✱ ✧❆♥ ❡q✉✐❧✐❜r✐✉♠ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ t❡r♠ str✉❝t✉r❡✧✱ ❏✳ ❋✐♥❛♥✳ ❊❝♦♥♦♠✳ ✺✱✭✶✾✼✼✮✱ ✶✼✼✲✶✽✽✳ ✽✷ ❇■❇▲■❖●❘❆P❍❨ ❬✶✵❪ ❘❡♥❞❧❡♠❛♥ ❛♥❞ ❇❛rtt❡r✱ ✧❚❤❡ ♣r✐❝✐♥❣ ♦❢ ❖♣t✐♦♥s ♦♥ ❉❡❜t ❙❡❝✉r✐t✐❡s✧✱ ❏✳ ❋✐♥❛♥✳ ❛♥❞ ◗✉❛♥t✳ ❆♥❛✳✱ ✶✺ ✭✶✾✽✵✮✱ ✶✶✲✷✹✳ ❬✶✶❪ ❉✳ ❍❡❛t❤✱ ❘✳ ❏❛rr♦✇✱ ❆✳ ▼♦rt♦♥✱ ✧❇♦♥❞ ♣r✐❝✐♥❣ ❛♥❞ t❤❡ t❡r♠ str✉❝✲ t✉r❡ ♦❢ ■♥t❡r❡st r❛❡s✿ ❆ ♥❡✇ ♠❡t❤♦❞♦❧♦❣② ❢♦r ❝♦♥t✐♥❣❡♥t ❝❧❛✐♠s ✈❛❧✉✲ ❛t✐♦♥✧ ✱ ❊❝♦♥♦♠❡tr✐❝❛✱ ✻✵ ✭✶✾✾✷✮✱ ✼✼✲✶✵✺✳ ❬✶✷❪ ❆✳ ❇r❛❝❡✱ ❉✳ ●➺❛t❛r❡❦✱ ❉✳ ▼✉s✐❡❧❛✱ ✧❚❤❡ ▼❛r❦❡t ▼♦❞❡❧ ♦❢ ■♥t❡r❡st ❘❛t❡ ❉②♥❛♠✐❝s✧✱ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡ ✼ ✭✶✾✾✼✮✱ ✶✷✼✲✶✺✺✳ ❬✶✸❪ ❋✳ ❏❛♠❛s❤✐❞✐❛♥✱ ✧▲■❇❖❘ ❛♥❞ s✇❛♣ ♠❛r❦❡t ♠♦❞❡❧s ❛♥❞ ♠❡❛s✉r❡s✧✱ ❋✐♥❛♥❝❡ ❛♥❞ ❙t♦❝❤❛st✐❝s✱ ❙♣r✐♥❣❡r✱ ✶ ✭✶✾✾✼✮✱ ✷✾✸✲✸✸✵✳ ❬✶✹❪ ❇r♦②❞❡♥ ❡t ❛❧✳✱ ✧❖♥ t❤❡ ▲♦❝❛❧ ❛♥❞ ❙✉♣❡r❧✐♥❡❛r ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ◗✉❛s✐✲ ◆❡✇t♦♥ ▼❡t❤♦❞s✧✱ ■▼❆ ❏✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✶✷ ✭✶✾✼✸✮✱ ✷✷✸✲✷✹✺✳ ❬✶✺❪ ❑✳ ▼✐❧t❡rs❡♥✱ ❑✳ ❙❛♥❞♠❛♥♥✱ ❉✳ ❙♦♥❞❡r♠❛♥♥✱ ✧❈❧♦s❡❞✲❢♦r♠ s♦❧✉t✐♦♥ ❢♦r t❡r♠ str✉❝✲ t✉r❡ ❞❡r✐✈❛t✐✈❡s ✇✐t❤ ❧♦❣ ♥♦r♠❛❧ ✐♥t❡r❡st r❛t❡s✧✱ ❏♦✉r✲ ♥❛❧ ♦❢ ❋✐♥❛♥❝❡✱ ✭✶✾✾✼✮✱ ✹✵✾✲✹✸✵✳ ❬✶✻❪ ❘❡❜♦♥❛t♦ ❘✳✱ ✧❱♦❧❛t✐❧✐t② ❛♥❞ ❈♦rr❡❧❛t✐♦♥✱ ❚❤❡ ♣❡r❢❡❝t ❍❡❞❣❡r ❛♥❞ ❚❤❡ ❋♦①✱ ❙❡❝♦♥❞ ❊❞✐t✐♦♥✧✱ ❊❞t ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s ✭✷✵✵✹✮✳ ❬✶✼❪ ❇r✐❣♦ ❉✳✱ ▼❡r❝✉r✐♦ ❋✳✱ ✧■♥t❡r❡st ❘❛t❡s ▼♦❞❡❧s✱ ❚❤❡♦r② ❛♥❞ Pr❛❝t✐❝❡✧✱ ✭✷✵✵✶✮✱ ❙♣r✐♥❣❡r ❋✐♥❛♥❝❡✱ ❇❡r❧✐♥✳ ❬✶✽❪ ❈♦✛❡② ❇✳✱ ❙❝❤♦❡♥♠❛❦❡rs ❏✳✱ ✧❙②st❡♠❛t✐❝ ❣❡♥❡r❛t✐♦♥ ♦❢ ♣❛r❛♠❡tr✐❝ ❝♦rr❡❧❛t✐♦♥ str✉❝t✉r❡s ❢♦r t❤❡ ▲✐❜♦r ♠❛r❦❡t ♠♦❞❡❧✧✱ ❲❡✐❡rstr❛ss ■♥✲ st✐t✉t❡ ❇❡r❧✐♥✱ ❲♦r❦✐♥❣ ♣❛♣❡r ✭✷✵✵✷✮ ❬✶✾❪ ❍✉❧❧ ❛♥❞ ❲❤✐t❡✱ ✧❋♦r✇❛r❞ ❘❛t❡ ❱♦❧❛t✐❧✐t✐❡s✱ ❙✇❛♣ ❘❛t❡ ❱♦❧❛t✐❧✐t✐❡s✱ ❛♥❞ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ▲■❇❖❘ ▼❛r❦❡t ▼♦❞❡❧✧✱ ❏♦✉r♥❛❧ ♦❢ ❋✐①❡❞ ■♥❝♦♠❡ ✼✺ ✭✶✾✾✺✮✱ ✶✺✲✸✶✳ ✽✸ ❇■❇▲■❖●❘❆P❍❨ ❬✷✵❪ ❆♥❞❡rs❡♥ ▲✳✱ ❆♥❞r❡❛s❡♥ ❏✳✱ ✧❱♦❧❛t✐❧✐t② ❙❦❡✇s ❛♥❞ ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▲✐❜♦r ▼❛r❦❡t ▼♦❞❡❧✧✱ ●❡♥❡r❛❧ ❘❡ ❋✐♥❛♥❝✐❛❧ Pr♦❞✉❝ts✱ ❲♦r❦✐♥❣ ♣❛✲ ♣❡r✱ ✭✶✾✾✾✮✳ ❬✷✶❪ ❍❛❣❛♥ P✳✱ ❑✉♠❛r ❉✳✱ ▲❡s♥✐❡✇s❦✐ ❆✳✱ ❲♦♦❞✇❛r❞ ❉✳ ✿ ▼❛♥❛❣✐♥❣ s♠✐❧❡ r✐s❦✱ ❲✐❧❧♠♦tt ▼❛❣❛③✐♥❡✱ ✭✷✵✵✷✮✱ ✽✹✲✶✵✽✳ [...]... ✐s✿ n−p F ixedp,n (t) = Sp,n (t)δB(t, Tp+i ) i=0 ❆♥❞ t❤❡ ✢♦❛t✐♥❣ ❧❡❣ ✐s✿ n−p F loatingp,n (t) = B(t, Ti+p )δL(t, Ti−1+p ) i=1 n−p = B(t, Ti+p ) i=1 n−p B(t, Ti−1+p ) −1 B(t, Ti+p ) B(t, Ti−1+p ) − B(t, Ti+p ) = i=1 = B(t, Tp ) − B(t, Tn ) ❚❤❡ s✇❛♣ r❛t❡ ✐s ❜② ❞❡✜♥✐t✐♦♥ t❤❡ ♦♥❡ t❤❛t ❡q✉❛❧✐③❡ ❜♦t❤ ❧❡❣s✿ F ixedp,n (t) = F loatingp,n (t) Sp,n (t) = B(t, Tp ) − B(t, Tn ) n−p i=0 δB(t, Tp+i ) ❚❤✐s s✇❛♣ ✇❛s ♠♦r❡ ... ❚❤❡ ✜①❡❞ ❧❡❣ ✐s✿ n−p F ixedp,n (t) = Sp,n (t)δB(t, Tp+i ) i=0 ❆♥❞ t❤❡ ✢♦❛t✐♥❣ ❧❡❣ ✐s✿ n−p F loatingp,n (t) = B(t, Ti+p )δL(t, Ti−1+p ) i=1 n−p = B(t, Ti+p ) i=1 n−p B(t, Ti−1+p ) −1 B(t, Ti+p... B(t, Tn ) ❚❤❡ s✇❛♣ r❛t❡ ✐s ❜② ❞❡✜♥✐t✐♦♥ t❤❡ ♦♥❡ t❤❛t ❡q✉❛❧✐③❡ ❜♦t❤ ❧❡❣s✿ F ixedp,n (t) = F loatingp,n (t) Sp,n (t) = B(t, Tp ) − B(t, Tn ) n−p i=0 δB(t, Tp+i ) ❚❤✐s s✇❛♣ ✇❛s ♠♦r❡ ♣r❡❝✐s❡❧② ❛... ❍❡♥❝❡✱ ❛♥ ❡①❛❝t ♣r✐❝❡ ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❛s ❢♦r t❤❡ ❝❛♣❧❡ts ✐♥ ▲▼▼✳ ❲✐t❤ str❛✐❣❤t❢♦r✇❛r❞ ♥♦t❛t✐♦♥s✿ q M SwaptionSM (t) p,q B(t, Ti )[Sp,q (t)N (d1 ) − κN (d2 )] = i=p+1 ✇✐t❤ d1,2 (t, Ti ) = ln((Sp,q (t)/κ)