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✐♦♥✳