❋❛✉st♦ ❏✉♥✐♦r ▼❛rt✐♥s ❋❡rr❡✐r❛
❋❆❚❖❘❊❙ ❉❊ ❘■❙❈❖ ❆❉❆P❚❆❉❖❙ ❉❊ ❚❆❳❆ ❉❊
❈➶▼❇■❖ ◆❖ ▼❖❉❊▲❖ ❉❊ ❇▲❆❈❑ ❊ ❙❈❍❖▲❊❙
❋❆❯❙❚❖ ❏❯◆■❖❘ ▼❆❘❚■◆❙ ❋❊❘❘❊■❘❆
❋❆❚❖❘❊❙ ❉❊ ❘■❙❈❖ ❆❉❆P❚❆❉❖❙ ❉❊ ❚❆❳❆ ❉❊
❈➶▼❇■❖ ◆❖ ▼❖❉❊▲❖ ❉❊ ❇▲❆❈❑ ❊ ❙❈❍❖▲❊❙
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡s✲ tr❛❞♦ ❡♠ ❙✐st❡♠❛s ❊❧❡trô♥✐❝♦s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜✲ t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛✳
➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❙✐st❡♠❛s ❊❧❡trô♥✐❝♦s✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❋❧á✈✐♦ ❆❧♠❡✐❞❛ ❞❡ ▼❛❣❛❧❤ã❡s ❈✐♣♣❛rr♦♥❡ ✲ ❯❙P
❋❖▲❍❆ ❉❊ ❆P❘❖❱❆➬➹❖
❋❛✉st♦ ❏✉♥✐♦r ▼❛rt✐♥s ❋❡rr❡✐r❛
❋❛t♦r❡s ❞❡ ❘✐s❝♦ ❆❞❛♣t❛❞♦s ❞❡ ❚❛①❛ ❞❡ ❈â♠❜✐♦ ♥♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡s✲ tr❛❞♦ ❡♠ ❙✐st❡♠❛s ❊❧❡trô♥✐❝♦s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜✲ t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛✳
➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❙✐st❡♠❛s ❊❧❡trô♥✐❝♦s✳
❆♣r♦✈❛❞♦ ❡♠✿ ❏❯▲❍❖✴✷✵✶✺
❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛
Pr♦❢✳ ❋❧á✈✐♦ ❞❡ ❆❧♠❡✐❞❛ ❈✐♣♣❛rr♦♥❡ ❆ss✐♥❛t✉r❛✿ ■♥t✐t✉✐çã♦✿ ❯❙P
Pr♦❢✳ ▼❛r❝♦s ❊✉❣ê♥✐♦ ❞❛ ❙✐❧✈❛ ❆ss✐♥❛t✉r❛✿ ■♥t✐t✉✐çã♦✿ ❯❙P
Este exemplar foi revisado e corrigido em relação à versão original, sob responsabilidade única do autor e com a anuência de seu orientador. São Paulo, ______ de ____________________ de __________
Assinatura do autor: ________________________
Assinatura do orientador: ________________________
Catalogação-na-publicação
Ferreira, Fausto
FATORES DE RISCO ADAPTADOS DE TAXA DE CÂMBIO NO MODELO DE BLACK E SCHOLES / F. Ferreira -- versão corr. -- São Paulo, 2015.
32 p.
Dissertação (Mestrado) - Escola Politécnica da Universidade de São Paulo. Departamento de Engenharia de Sistemas Eletrônicos.
1.Engenharia Financeira 2.Finanças 3.Derivativos 4.Volatilidade Implícita I.Universidade de São Paulo. Escola Politécnica. Departamento de
❘❡s✉♠♦
❋❊❘❘❊■❘❆✱ ❋❛✉st♦ ❏ ▼❛rt✐♥s✳ ❋❛t♦r❡s ❞❡ ❘✐s❝♦ ❆❞❛♣t❛❞♦s ❞❡ ❚❛①❛ ❞❡ ❈â♠❜✐♦ ♥♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✳ ✷✵✶✺✳ ✸✶ ❢✳ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳
❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❝á❧❝✉❧♦ ❞❡ s❡♥s✐❜✐❧✐❞❛❞❡s ✉t✐❧✐③❛♥❞♦ ❡q✉❛✲ çõ❡s ❛♥❛❧ít✐❝❛s✱ ❧❡✈❛♥❞♦ ❡♠ ❛ ❝♦♥t❛ ❛ ❝♦rr❡çã♦ ❞❡ s♠✐❧❡ ♥❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡✱ q✉❡ ♥ã♦ é ❝♦♥t❡♠♣❧❛❞❛ ♥♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✳ ❉❛❞❛ ❛ ❞✐❢❡r❡♥ç❛ s✐❣♥✐✜❝❛t✐✈❛ ♥❛ ♠❡♥✲ s✉r❛çã♦ ❞♦ r✐s❝♦ ❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ❝❛❧❝✉❧❛♠ s✉❛s s❡♥s✐❜✐❧✐❞❛❞❡s ✐♥❝♦r♣♦r❛♥❞♦ ❡st❛ ❝♦rr❡çã♦✱ ♠❛s t❛❧ ❞❡t❡r♠✐♥❛çã♦ t❡♠ s✐❞♦ r❡❛❧✐③❛❞❛ ♣♦r ♠ét♦❞♦s ♥✉♠ér✐❝♦s✱ q✉❡ ❛❝❛❜❛♠ s❡♥❞♦ ♠❛✐s ❧❡♥t♦s q✉❡ ❛ ❛❜♦r❞❛❣❡♠ ❛q✉✐ ♣r♦♣♦st❛✳ ❙ã♦ ❛♣r❡s❡♥t❛❞❛s ❡q✉❛çõ❡s ❛♥❛❧ít✐✲ ❝❛s ♣❛r❛ ❛s ♣r✐♥❝✐♣❛✐s s❡♥s✐❜✐❧✐❞❛❞❡s ❞♦ ♠♦❞❡❧♦ ❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ❞❡ ♠❡r❝❛❞♦ ✉s❛❞♦s ♥❛ ❝♦♥st✉çã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛✳ ■❧✉str❛♠♦s ❛ ❝♦♠♣❛r❛çã♦ ❞❛ té❝♥✐❝❛ ♣r♦♣♦st❛ ❝♦♠ ♦ ♠ét♦❞♦ ♥✉♠ér✐❝♦ ❝♦♠ ❜❛s❡ ♥♦ ♠❡r❝❛❞♦ ❞❡ ♦♣çõ❡s s♦❜r❡ t❛①❛ ❞❡ ❝â♠❜✐♦ ❇r❛s✐❧❡✐r♦✳
❆❜str❛❝t
❋❊❘❘❊■❘❆✱ ❋❛✉st♦ ❏ ▼❛rt✐♥s✳ ❋♦r❡✐❣♥ ❊①❝❤❛♥❣❡ ❆❞❛♣t❡❞ ❘✐s❦ ❋❛❝t♦rs ♦♥ ❛ ❇❧❛❝❦ ❛♥❞ ❙❝❤♦❧❡s ▼♦❞❡❧✳ ✷✵✶✺✳ ✸✶ ♣✳ ❉✐ss❡rt❛t✐♦♥ ✭▼❛st❡rs❤✐♣✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✺✳
❚❤✐s ✇♦r❦ ♣r❡s❡♥ts ❛ st✉❞② ♦♥ ❤♦✇ ✇❡ s❤♦✉❧❞ ❛❞❛♣t❡❞ t❤❡ ●r❡❡❦s ♦r r✐s❦ ❢❛❝t♦rs ♦❢ t❤❡ ❇❧❛❝❦ ❛♥❞ ❙❝❤♦❧❡s ♠♦❞❡❧✳ ❲❡ ❝❛♥ ❞❡r✐✈❡ ❛♥❛❧②t✐❝❛❧ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ♠❛✐♥ s❡♥s✐t✐✈✐t✐❡s ♦❢ t❤❡ ♠♦❞❡❧ ❛♥❞ ✉s✐♥❣ t❤❡ ♠❛r❦❡t ❞❛t❛ t♦ ❜✉✐❧❞ ❛♥ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② s✉r❢❛❝❡ ❛♥❞ t♦ ❣❡t ❛❞❞✐t✐♦♥❛❧ t❡r♠s ❢♦r t❤❡ r✐s❦ ❢❛❝t♦rs✳ ❲❡ ♣r♦♣♦s❡ t♦ ✐♠♣❧❡♠❡♥t t❤✐s ♠♦❞❡❧ ✐♥ ❛ s❝❤❡♠❡ ♦❢ ❛♥❛❧②t✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ♣r✐❝✐♥❣ ♠♦❞❡❧ ❛♥❞ ❢r♦♠ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ❢✉♥❝t✐♦♥✳ ❚❤❡ ❜✉✐❧❞✐♥❣ ♦❢ t❤✐s ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ❛♥❞ r✐s❦ ❢❛❝t♦rs ✇❛s ❜❛s❡❞ ♦♥ t❤❡ ❢♦r❡✐❣♥ ❡①❝❤❛♥❣❡ ❇r❛③✐❧✐❛♥ ♠❛r❦❡t✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❱♦❧❛t✐❧✐❞❛❞❡ ■♠♣❧í❝✐t❛Σ ♦❜t✐❞❛ ❛tr❛✈és ❞❡ ❞❛❞♦s ❞❡ ♠❡r❝❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✷ ❈♦rt❡ ♥❛ ❙✉♣❡r❢í❝✐❡ ❞❡ ❱♦❧❛t✐❧✐❞❛❞❡ ■♠♣❧í❝✐t❛ Σ ❡♠ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✼
✷ ❈❛r❛❝t❡ríst✐❝❛s ❞❡ ❖♣çõ❡s ❱❛♥✐❧❧❛ ❡ ▼♦❞❡❧♦s ❆❞♦t❛❞♦s ✾
✷✳✶ Pr❡ss✉♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ ●r❡❣❛s ●❡r❛✐s s❡♠ ♦ ❙♠✐❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷✳✶ ❉❡❧t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷✳✷ ●❛♠♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷✳✸ ❱❡❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷✳✹ ❱❛♥♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷✳✺ ❱♦❧❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ❱♦❧❛t✐❧✐❞❛❞❡ ■♠♣❧í❝✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✹ ❈♦♥❞✐çõ❡s ❞❡ ◆ã♦ ❆r❜✐tr❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✺ ❈♦♥✈❡♥çõ❡s ❡ ❈♦t❛çõ❡s ❞❡ ▼❡r❝❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✻ ❉❡❧t❛✲❙tr✐❦❡ ❈♦♥✈❡rsã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✼ P❛r❛♠❡tr✐③❛çã♦ ❞❛ ❙✉♣❡r❢í❝✐❡ ❞❡ ❱♦❧❛t✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✼✳✶ P♦❧✐♥ô♠✐♦ ❞❡ ◗✉❛rt♦ ●r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✼✳✷ ❙❆❇❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✼✳✸ ❱❛♥♥❛✲❱♦❧❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✸ ❉✐♥â♠✐❝❛ ❞❡ ❱♦❧❛t✐❧✐❞❛❞❡ ✷✵
✸✳✶ ❊str✉t✉r❛ ❛ t❡r♠♦ ❞♦ ❙❦❡✇ ✭❆ss✐♠❡tr✐❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✹ ●r❡❣❛s ❆❞❛♣t❛❞❛s ♣❛r❛ ♦♣çõ❡s ❱❛♥✐❧❧❛ s♦❜r❡ ❚❛①❛ ❞❡ ❈â♠❜✐♦ ✷✷ ✹✳✶ ❉❡❧t❛ ❆❞❛♣t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✷ ❱❡❣❛ ❆❞❛♣t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✸ ●❛♠♠❛ ❆❞❛♣t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✹ ❱❛♥♥❛ ❆❞❛♣t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✺ ❱♦❧❣❛ ❆❞❛♣t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✺ ❆♥á❧✐s❡ ◆✉♠ér✐❝❛ ✷✻
✻ ❈♦♥❝❧✉sã♦ ✷✾
✼
✶ ■♥tr♦❞✉çã♦
■♥st✐t✉✐çõ❡s ❋✐♥❛♥❝❡✐r❛s ♠❛♥té♠ ❝❛rt❡✐r❛s ❞❡ ❞❡r✐✈❛t✐✈♦s q✉❡ ♣r❡❝✐s❛♠ s❡r ❤❡❞❣❡❛❞❛s ❝♦♥✲ tr❛ ♦s r✐s❝♦s ❞❡ ♠❡r❝❛❞♦✳ P❛r❛ ✐st♦ sã♦ ♥❡❝❡ssár✐❛s ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ♣r✐♥❝✐♣❛✐s s❡♥✲ s✐❜✐❧✐❞❛❞❡s ❞❡ ♣r❡ç♦ ❞♦s ❞❡r✐✈❛t✐✈♦s ❡♠ r❡❧❛çã♦ ❛♦s ♣❛râ♠❡tr♦s q✉❡ ✐♥✢✉❡♠ ♥❡st❡ ♣r❡ç♦ ✭✈♦❧❛t✐❧✐❞❛❞❡✱ t❛①❛ ❞❡ ❥✉r♦s✱ ♣r❡ç♦ ❞♦ ✉♥❞❡r❧②✐♥❣✱ t❡♠♣♦ ❡ ♣r❡ç♦ ❞❡ ❡①❡r❝í❝✐♦✮✳
◆♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞❛s s✉♣❡r❢í❝✐❡s ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ é ❢❡✐t❛ ❛ ❝♦❧❡t❛ ❞♦s ♣r❡ç♦s ❞❡ ♠❡r❝❛❞♦ ♣❛r❛ ❛s ♦♣çõ❡s ❞✐s♣♦♥í✈❡✐s ❡♠ ✈ár✐♦s ✈❡♥❝✐♠❡♥t♦s✱ ❛ ♣❛rt✐r ❞❡ss❡s ♣r❡ç♦s✱ s❡❣✉✐♥❞♦ ❝r✐tér✐♦s ❞❡ ♥ã♦ ❛r❜✐tr❛❣❡♠✱ ❝♦♠♦ ❢❛❧❛r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✸ sã♦ ♦❜t✐❞❛s ❛s r❡s♣❡❝✲ t✐✈❛s ✈♦❧❛t✐❧✐❞❛❞❡s ✐♠♣❧í❝✐t❛s ♣❛r❛ ❝❛❞❛ str✐❦❡ ✭♣r❡ç♦ ❞❡ ❡①❡r❝í❝✐♦✮ ❡ ✈❡♥❝✐♠❡♥t♦ ✉t✐❧✐③❛❞♦✳ ❊ss❡ ♣r♦❝❡ss♦ é ❢❡✐t♦ ❞✐❛r✐❛♠❡♥t❡ ♣❛r❛ ♠❛♥t❡r ❛ s✉♣❡r❢í❝✐❡ s❡♠♣r❡ ❛t✉❛❧✐③❛❞❛ ❝♦♠ ♦ ♠❡r✲ ❝❛❞♦ ❝♦rr❡♥t❡✱ ❛ ♣❛rt✐r ❞❛ ❡s❝♦❧❤❛ ❞♦ t✐♣♦ ❞❡ ♣❛r❛♠❡tr✐③❛çã♦ q✉❡ s❡rá ❢❡✐t♦ ✭ ❝❛♣ít✉❧♦ ✸✮ ❞❡✜♥✐♠♦s ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✜♥❛❧ ❡ ❝♦♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ss❡ ♠ét♦❞♦ ❞❡ ♣❛r❛✲ ♠❡tr✐③❛çã♦ ❞❡r✐✈❛♠♦s t❡r♠♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛❣r❡❣❛r ♦s ❡❢❡✐t♦s ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ s♠✐❧❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ♥❛ ❞❡✜♥✐çã♦ ❞♦ ❉❡❧t❛✱ ❱❡❣❛✱ ●❛♠♠❛ ❡ ♦✉tr❛s s❡♥s✐❜✐❧✐❞❛❞❡s ❛♦s ❢❛t♦r❡s ❞❡ r✐s❝♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✳
❆❧❣✉♠❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ❞❡r✐✈❛♠ ❡st❛s s❡♥s✐❜✐❧✐❞❛❞❡s ❞❡ ❢♦r♠❛ s✐♠♣❧✐st❛✱ ✉t✐✲ ❧✐③❛♥❞♦ ♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ♦r✐❣✐♥❛❧✱ s❡♠ ❧❡✈❛r ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❡ss❛s ❛♥♦♠❛❧✐❛s q✉❡ sã♦ ❛♠♣❧❛♠❡♥t❡ ❝♦♥❤❡❝✐❞❛s✳
■♥st✐t✉✐çõ❡s ♠❛✐s s♦✜st✐❝❛❞❛s ❡♠♣r❡❣❛♠ ♠ét♦❞♦s ♥✉♠ér✐❝♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r ❡st❛s s❡♥s✐❜✐❧✐❞❛❞❡s✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ ❞❡✈✐❞❛ ❝♦rr❡çã♦ ❞❡ s♠✐❧❡ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡✳ ◆❡st❡ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈❡♠♦s ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❞❡r✐✈❛❞❛s ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ❡s❝♦❧❤✐❞❛✱ ♣❛r❛ ❛s s❡♥s✐❜✐❧✐❞❛❞❡s ❞❡ ♦♣çõ❡s ❡✉r♦♣❡✐❛s ❛♦s ♣❛râ♠❡tr♦s ♣r❡ç♦ ❞♦ ✉♥❞❡r❧②✐♥❣ ❡ ✈♦❧❛t✐❧✐❞❛❞❡✱ ❞✐s♣❡♥s❛♥❞♦✱ ♣♦r✲ t❛♥t♦✱ ❛ ✉t✐❧✐③❛çã♦ ❞❡st❡s ♠ét♦❞♦s ♥✉♠ér✐❝♦s✱ ❛❝❡❧❡r❛♥❞♦ ❛ss✐♠ t❛✐s ❝á❧❝✉❧♦s✳ ❯t✐❧✐③❛♠♦s ❛s s❡♥s✐❜✐❧✐❞❛❞❡s ❣❡r❛✐s ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ❡ ❛❞✐❝✐♦♥❛♠♦s ♥♦✈❛s ❞❡r✐✈❛❞❛s ❛ ❡ss❛s ❣r❡❣❛s ❞❡ ♠♦❞♦ ❛ t❡r♠♦s s❡♥s✐❜✐❧✐❞❛❞❡s ❛♦s ❢❛t♦r❡s ❞❡ r✐s❝♦ ♠❛✐s ❝♦rr❡t❛s ❝♦♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ❛♣r❡ç❛♠❡♥t♦ ❞❡ ♦♣çõ❡s q✉❡ ✉t✐❧✐③❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛✳
❖❜❥❡t✐✈♦ ❞♦ tr❛❜❛❧❤♦
❖ ♦❜❥❡t✐✈♦ ❞♦ tr❛❜❛❧❤♦ é ❞❡❞✉③✐r ❛s ❡q✉❛çõ❡s ♣❛r❛ ❛s s❡♥s✐❜✐❧✐❞❛❞❡s ❞♦ ♣r❡ç♦ ❞❡ ✉♠❛ ♦♣çã♦ ✈❛♥✐❧❧❛ q✉❡ ♦❜❡❞❡❝❡♠ ❛♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❞❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✱ ♠❛s ♥❡ss❡ ❝❛s♦ ❝♦♥✲ s✐❞❡r❛♥❞♦ ♦ ❡❢❡✐t♦ ❞♦ s♠✐❧❡ ♥❛ ♠❡♥s✉r❛çã♦ ❞♦ r✐s❝♦✳ ❙❡rã♦ ❞❡❞✉③✐❞❛s ❡ ❛♣r❡s❡♥t❛❞❛s ❡①♣r❡ssõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ✉♠❛ ♦♣çã♦ ✈❛♥✐❧❧❛ s♦❜r❡ ♠♦❡❞❛s ✭♦ ❝❛s♦ ❡st✉❞❛❞♦ ❢♦✐ ♦ ❞❡ ♦♣✲ çõ❡s ❞❡ ❉ó❧❛r✲❘❡❛❧✮✱ ❛❞❛♣t❛r❡♠♦s ❛s s❡♥s✐❜✐❧✐❞❛❞❡s ♦❜t✐❞❛s ♣❡❧♦ ♠♦❞❡❧♦ ❝❧áss✐❝♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ ❛❣r❡❣❛♥❞♦ ❝♦♠♣♦♥❡♥t❡s ♥❛s ♠❡s♠❛s✳ ❊ss❛ ❛❜♦r❞❛❣❡♠ ♣♦r s✐ só ♠❛✐s ❡✜❝✐✲ ❡♥t❡ q✉❡ q✉❛❧q✉❡r ♠ét♦❞♦ ♥✉♠ér✐❝♦ ♣❛r❛ ❝á❧❝✉❧♦ ❞❡s❛s s❡♥s✐❜✐❧✐❞❛❞❡s✱ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ✭✶✾✼✸✮✳
✽
❊str✉t✉r❛ ❞♦ tr❛❜❛❧❤♦
❊st❡ tr❛❜❛❧❤♦ ❡stá ❡str✉t✉r❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❆♣r❡s❡♥t❛r❡♠♦s ❝❛r❛❝t❡ríst✐❝❛s ❣❡r❛✐s ❞❡ ♦♣çõ❡s ❱❛♥✐❧❧❛✱ ♦s ♠♦❞❡❧♦s ❛❞♦t❛❞♦s✱ ❝❛r❛❝t❡r✐③❛r❡♠♦s ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❡ ♦ ♣r♦❝❡ss♦ q✉❡ ❡♥✈♦❧✈❡ ❛ s✉❛ ❝♦♥str✉çã♦ ❡ ❛❜♦r❞❛r❡♠♦s ❛s s❡♥s✐❜✐❧✐❞❛❞❡s ✭❣r❡❣❛s✮ ❞♦ ♠♦❞❡❧♦ ❝❧áss✐❝♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✳
◆♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞✐♥â♠✐❝❛ ❛❞♦t❛❞❛ ♣❛r❛ ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ q✉❡ s❡rá ✉s❛❞❛ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ❛ ❞❡❞✉çã♦ ❞❛s s❡♥s✐❜✐❧✐❞❛❞❡s ❛❞❛♣t❛❞❛s✳
◆♦ ❈❛♣ít✉❧♦ ✹✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❢♦r♠✉❧❛çã♦ ❛♥❛❧ít✐❝❛ ♣r♦♣♦st❛ ♣❛r❛ ❛s s❡♥s✐❜✐❧✐❞❛❞❡s ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ✭✶✾✼✸✮✱ ❛ ❞❡✜♥✐çã♦ ❡ ❢♦r♠✉❧❛çã♦ ❞❛ ❞❡r✐✈❛çã♦ ❛ s❡r ❢❡✐t❛ ♥❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ♣❛r❛ t♦❞♦s ♦s ❝❛s♦s ❡st✉❞❛❞♦s✳
◆♦ ❈❛♣ít✉❧♦ ✺✱ ❢❛r❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ♥✉♠ér✐❝❛ ❞❛s s❡♥s✐❜✐❧✐❞❛❞❡s ♦❜t✐❞❛s ❝♦♠ ♦ ♠ét♦❞♦ ❝❧áss✐❝♦ ❡ ♦ ♠ét♦❞♦ ♥✉♠ér✐❝♦ ❞❡ ♦❜t❡♥çã♦ ❞❛s ♠❡s♠❛s✳
✾
✷ ❈❛r❛❝t❡ríst✐❝❛s ❞❡ ❖♣çõ❡s ❱❛♥✐❧❧❛ ❡ ▼♦❞❡❧♦s ❆❞♦t❛✲
❞♦s
Pr✐♠❡✐r♦ ✈❛♠♦s ❞❡✜♥✐r ♦s t❡r♠♦s ❣❡r❛✐s ❡ ❛ ♥♦t❛çã♦ q✉❡ ✉s❛♠♦s✳ ❖ ♣r❛③♦ ❞❡ ✈❛❧✐❞❛❞❡ ❞♦ ♦♣çã♦ é ♦ ú❧t✐♠♦ ❞✐❛ ❡♠ q✉❡ ❛ ♦♣çã♦ ♣♦❞❡ s❡r ❡①❡r❝✐❞❛✱ s❡ ♦ t✐t✉❧❛r ❝♦♥s✐❞❡r❛ r❡♥tá✈❡❧ ❢❛③ê✲❧❛✳ ❆ ♦♣çã♦ ❡✉r♦♣❡✐❛ só ♣♦❞❡ s❡r ❡①❡r❝✐❞❛ ♥♦ ❞✐❛ ❞♦ ✈❡♥❝✐♠❡♥t♦✳ ❖ str✐❦❡ ❞❛ ♦♣çã♦✱ K✱ é ♦ ♣r❡ç♦ ♣❡❧♦ q✉❛❧ ♦ ❛t✐✈♦ s✉❜❥❛❝❡♥t❡ é ❝♦♠♣r❛❞♦ ✭♣❛r❛ ✉♠❛ ❝❛❧❧✮ ♦✉ ✈❡♥❞✐❞♦ ✭♣❛r❛ ✉♠ ♣✉t✮✳ ❖ ✈❡♥❝✐♠❡♥t♦ ❞❛ ♦♣çã♦ T é ♦ t❡♠♣♦✱ ♠❡❞✐❞♦ ❡♠ ❛♥♦s✱ ❡♥tr❡ ♦ ✐♥í❝✐♦ ❞❛ ♦♣çã♦ ❡ s❡✉ ✈❡♥❝✐♠❡♥t♦✳ ❯♠❛ ♦♣çã♦ é ❞❡✜♥✐❞❛ ♣❡❧❛ s✉❛ ❢✉♥çã♦ ❞❡ ♣❛②✲♦✛✳ ❖ ♣❛②✲♦✛ ♣❛r❛ ✉♠❛ ♦♣çã♦ ❡✉r♦♣❡✐❛ é s❡✉ ✈❛❧♦r ♥❛ ❞❛t❛ ❞❡ ✈❡♥❝✐♠❡♥t♦✳ ❊st❡ ✈❛❧♦r ❞❡♣❡♥❞❡ ❞❛ ❞❛t❛ ❞❡ ✈❡♥❝✐♠❡♥t♦✱ ❞♦ ❡①❡r❝í❝✐♦ ❞❛ ♦♣çã♦ ❡ q✉❛✐sq✉❡r ♦✉tr❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ♦♣çã♦✱ ❛ss✐♠ ❝♦♠♦ ♦ ♣r❡ç♦ ❞♦ ❛t✐✈♦ s✉❜❥❛❝❡♥t❡ ♥♦ ❞❛t❛ ❞❡ ✈❡♥❝✐♠❡♥t♦✱ST✳ ❖ ♣❛②✲♦✛ ❞❡ ✉♠❛ ♦♣çã♦ ❞❡♣❡♥❞❡ ❞❛ tr❛❥❡tór✐❛ ❞♦ ♣r❡ç♦ ❞♦ ✉♥❞❡r❧②✐♥❣ ❛té ❛ ❞❛t❛ ❞❡ ❡①♣✐r❛çã♦✳
❊♠ q✉❛❧q✉❡r ♠♦♠❡♥t♦t✱ ❝♦♠0≤t≤T✱ ♦ t❡♠♣♦ r❡s✐❞✉❛❧ ♣❛r❛ ❛ ♠❛t✉r✐❞❛❞❡ éT −t✱ ♦✉ s❡❥❛✱ ♦ t❡♠♣♦ r❡st❛♥t❡ ❛té ❛ ❞❛t❛ ❞❡ ✈❡♥❝✐♠❡♥t♦✱ ❞❡♥♦t❛♠♦s ♣♦r τ✳ ❖ ♣r❡ç♦ ❞♦ ❛t✐✈♦ s✉❜❥❛❝❡♥t❡ ❞❡ ✉♠❛ ♦♣çã♦ ♥♦ t❡♠♣♦t é ❞❡♥♦t❛❞♦ St✳ ➚s ✈❡③❡s ♣♦❞❡♠♦s ❡♥❢❛t✐③❛r s❡ ❡st❡ é ✉♠ ♣r❡ç♦ à ✈✐st❛✱ ✉♠ ❢✉t✉r♦ ♦✉ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦s✳ ◆❡ss❡ ❝❛s♦✱ q✉❛♥❞♦ ♦ ✉♥❞❡❧②✐♥❣ é ✉♠ ❢✉t✉r♦ ✉s❛♠♦s F ♣❛r❛ ❞❡♥♦t❛r ♦ ♣r❡ç♦ ❞♦ ❛t✐✈♦ s✉❜❥❛❝❡♥t❡✳
❊st❛ s❡çã♦ ❡①♣❧✐❝❛ ♦ ♠♦❞❡❧♦ ❞❡ ♣r❡❝✐✜❝❛çã♦ ❞❡ ❝❛❧❧s ❡ ♣✉ts ♣❛r❛ ♦♣çõ❡s ❞❡ ♣❛❞rã♦ ❡✉r♦♣❡✉ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ✭✶✾✼✸✮ ❡ ♣♦r ▼❡rt♦♥ ✭✶✾✼✸✮✳ ❖s ♣r❡ss✉♣♦st♦s ❞♦ ♠♦❞❡❧♦ sã♦ ♠♦str❛❞♦s ❡ ❛❧❣✉♠❛ ✐♥t✉✐çã♦ s♦❜r❡ ❛ ❢ór♠✉❧❛ ❞❡ ❛♣r❡ç❛♠❡♥t♦ ❞❡ ♦♣çõ❡s é ❢♦r♥❡❝✐❞❛✳
✷✳✶ Pr❡ss✉♣♦st♦s
❖ ♣r✐♠❡✐r♦ ♣r❡ss✉♣♦st♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥ é s♦❜r❡ ♦ ♣r♦❝❡ss♦ ❡st♦❝ást✐❝♦ ♣❛r❛ ♦ ♣r❡ç♦ ❞♦ ❛t✐✈♦ s✉❜❥❛❝❡♥t❡✳ ❖ ❛rt✐❣♦ ♦r✐❣✐♥❛❧ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ✭✶✾✼✸✮ ❛ss✉♠❡ q✉❡ ♦ ❛t✐✈♦ ♥ã♦ ♣❛❣❛ ❞✐✈✐❞❡♥❞♦s✱ q✉❡ ❛ t❛①❛ ❞❡ ❥✉r♦s ❧✐✈r❡ ❞❡ r✐s❝♦ é ✉♠❛ ❝♦♥st❛♥t❡ ❝♦♥❤❡❝✐❞❛ ❡ q✉❡ ❛ ❞✐♥â♠✐❝❛ ❞♦s ♣r❡ç♦s sã♦ r❡❣✐❞❛s ♣♦r ✉♠ ♠♦✈✐♠❡♥t♦ ❜r♦✇♥✐❛♥♦ ❣❡♦♠étr✐❝♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ ♣r❡ç♦ s✉❜❥❛❝❡♥t❡S s❡❣✉❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡st♦❝ást✐❝❛ ❞❛ ❢♦r♠❛
dS(t)
S(t) =rdt+σdW(t), ✭✶✮
♦♥❞❡W(t)é ✉♠ ♣r♦❝❡ss♦ ❞❡ ❲✐❡♥❡r✳
❯s❛♥❞♦ ♦ ❧❡♠❛ ❞❡ ■tô✱ ✈❡r ❑✐②♦s❤✐✱■ ✭✶✾✺✶✮✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ✭✷✮ ♥❛ ❢♦r♠❛ ❡q✉✐✈❛✲ ❧❡♥t❡
dln(S(t)) = (r−1
2σ
2)dt+σdW(t). ✭✷✮
✶✵
③❡r♦ ❡ ✈❛r✐â♥❝✐❛ dt✱ ✭✸✮ ✐♠♣❧✐❝❛ q✉❡ ♦s ❧♦❣ r❡t♦r♥♦s ❞❡ ♣❡rí♦❞♦s t sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞♦s ❝♦♠ ♠é❞✐❛(r−12σ2)t ❡ ❞❡s✈✐♦ ♣❛❞rã♦σ√t. ❙❡ ♦s ❧♦❣ r❡t♦r♥♦s
❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❛♠❛♥❤♦tsã♦ ♥♦r♠❛❧♠❡♥t❡ ❞✐str✐❜✉í❞♦s ❡♥tã♦ ♦ ♣r❡ç♦ ❢✉t✉r♦ ❡♠t t❡rá ✉♠❛ ❞✐str✐❜✉✐çã♦ ❧♦❣♥♦r♠❛❧✳ ❆ss✐♠✱ ♦ ♣r✐♠❡✐r♦ ♣r❡ss✉♣♦st♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ q✉❡ ♦ ♣r❡ç♦ ❞♦ ✉♥❞❡r❧②✐♥❣ é ❣❡r❛❞♦ ♣♦r ✉♠ ♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ❣❡♦♠étr✐❝♦ ❝♦♠ ✈♦❧❛t✐❧✐❞❛❞❡ ❝♦♥st❛♥t❡✱ ✐♠♣❧✐❝❛ q✉❡ ♦s ♣r❡ç♦s t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❧♦❣♥♦r♠❛❧✳
❆ s❡❣✉♥❞❛ s✉♣♦s✐çã♦ ❢❡✐t❛ ♣♦r ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ❢♦✐ ❛ ❞❡ q✉❡ ♦s ♠❡r❝❛❞♦s ❡♠ q✉❡ ♦ ✉♥❞❡r❧②✐♥❣ ❡ ♦s ❝♦♥tr❛t♦s ❞❡ ♦♣çõ❡s sã♦ ♥❡❣♦❝✐❛❞♦s ♥ã♦ ♣♦ss✉❡♠ ❢r✐❝çõ❡s✱ ♦✉ s❡❥❛✱ é ♣♦ssí✈❡❧ ❝♦♠♣r❛r ❡ ✈❡♥❞❡r q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❛ q✉❛❧q✉❡r ♠♦♠❡♥t♦ s❡♠ ✐♥❝♦rr❡r ❡♠ ❝✉st♦s ❞❡ tr❛♥s❛çã♦✳ ❊ss❛ s✉♣♦s✐çã♦ ❝♦♠❜✐♥❛❞❛ ❝♦♠ ❛ s✉♣♦s✐çã♦ ❞❡ q✉❡ ♦s ♣r❡ç♦s s❡❣✉❡♠ ✉♠ ♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ❣❡♦♠étr✐❝♦ ✭✷✮✱ ❢♦✐ s✉✜❝✐❡♥t❡ ♣❛r❛ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ❛♣❧✐❝❛r❡♠ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ♥ã♦ ❛r❜✐tr❛❣❡♠ ♣❛r❛ ❞❡r✐✈❛r ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ♣❛r❛ ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ♦♣çã♦✱ q✉❡ ❡❧❡s ♠♦str❛r❛♠ t❡r ✉♠❛ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ q✉❛♥❞♦ ❛ ♦♣çã♦ é ✉♠❛ ♣✉t ♦✉ ❝❛❧❧ ❡✉r♦♣❡✐❛✳
◆♦ ♠❡s♠♦ ❛♥♦ ✭✶✾✼✸✮ ▼❡rt♦♥ ❝❤❡❣♦✉ ❛ ✉♠❛ ❞❡r✐✈❛çã♦ ❛❧t❡r♥❛t✐✈❛ ❞❡ss❛ ❡q✉❛çã♦ ❞✐✲ ❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❡ ❛ ❡st❡♥❞❡✉ ❞❡ ✈ár✐❛s ❢♦r♠❛s✿ ❛♣❧✐❝❛❞❛ ❛ ❛t✐✈♦s q✉❡ ♣❛❣❛♠ ❞✐✈✐❞❡♥❞♦s❀ ♣❛r❛ ✐♥❝❧✉✐r ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ t❛①❛s ❞❡ ❥✉r♦s ❡st♦❝ást✐❝❛s❀ ❡ ♣❛r❛ ✈❛❧♦r❛r ♦♣çõ❡s ❆♠❡r✐❝❛✲ ♥❛s ❡ ♦♣çõ❡s ❝♦♠ ❜❛rr❡✐r❛✳ ❊♥tr❡t❛♥t♦✱ ♥❡ss❡s ❝❛s♦s ♥ã♦ s❡r ♣♦ssí✈❡❧ ❞❡r✐✈❛r ✉♠❛ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ♣❛r❛ ❡ss❛s ❡q✉❛çõ❡s ♣❛r❝✐❛✐s✱ ❡①❝❡t♦ ♣❛r❛ ♦♣çõ❡s ❡✉r♦♣❡✐❛s✱ ❡ q✉❛♥❞♦ ❛♠❜❛s ❛s t❛①❛s ❞❡ ❥✉r♦s ❡ ♦s ❞✐✈✐❞❡♥❞♦s sã♦ ❝♦♥st❛♥t❡s✳
❇❧❛❝❦✱ ❙❝❤♦❧❡s ❡ ▼❡rt♦♥ ✉s❛r❛♠ ♦ ❛r❣✉♠❡♥t♦ ❞❡ ♥ã♦ ❛r❜✐tr❛❣❡♠ ♣❛r❛ t✐♣♦s ❡s♣❡❝í✜❝♦s ❞❡ ♦♣çõ❡s ❡♠ ✉♠❛ ❡❝♦♥♦♠✐❛ ♠✉✐t♦ r❡str✐t❛✱ ♦♥❞❡ ❤á ❛♣❡♥❛s ✉♠ ✉♥❞❡r❧②✐♥❣ ❡ ❛ ♦♣çã♦✳ ❆té ♦ ♣❛♣❡r ♣✐♦♥❡✐r♦ ❞❡ ❍❛rr✐s♦♥ ❡ ❑r❡♣s ✭✶✾✼✾✮ ❝✉❥❛ t❡♦r✐❛ ❞❡ ❛♣r❡ç❛♠❡♥t♦ ❞❡ ♦♣çõ❡s ❢♦✐ ❛❞✐❝✐♦♥❛❞❛ ❝♦♠ ❢✉♥❞❛♠❡♥t♦s t❡ór✐❝♦s ✜r♠❡s✳ ❍❛rr✐s♦♥ ❡ ❑r❡♣s ✐♥tr♦❞✉③✐r❛♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♠❡r❝❛❞♦s ❝♦♠♣❧❡t♦s ❝♦♠♦ ✉♠ ♠❡r❝❛❞♦ ❡♠ q✉❡ s❡ ♣♦❞❡ r❡♣❧✐❝❛r ♦ ♣r❡ç♦ ❞❡ q✉❛❧q✉❡r tít✉❧♦ ❞❡ ❝ré❞✐t♦ ♥❡❣♦❝✐❛♥❞♦ ♦ ✉♥❞❡r❧②✐♥❣ ❡ ❛ t❛①❛ ❧✐✈r❡ ❞❡ r✐s❝♦✱ ❡❧❡s ♠♦str❛r❛♠ q✉❡ ❡♠ ♠❡r❝❛❞♦s ❝♦♠♣❧❡t♦s t♦❞♦ tít✉❧♦ ❞❡ ❝ré❞✐t♦ t❡♠ ✉♠ ú♥✐❝♦ ♣r❡ç♦ ♥ã♦ ❛r❜✐trá✈❡❧✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ❤á ✉♠❛ ú♥✐❝❛ ♠❡❞✐❞❛ ♠❛rt✐❣❛❧❡ ❡q✉✐✈❛❧❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❤á ✉♠ ú♥✐❝♦ ♣♦rt❢ó❧✐♦ ❛✉t♦✜♥❛♥❝✐❛❞♦ q✉❡ r❡♣❧✐❝❛ ♦ ✈❛❧♦r ❞♦ ❝ré❞✐t♦✳
P❛r❛ ♦ ❛♣r❡ç❛♠❡♥t♦ ❞❛s ♦♣çõ❡s ❡ ♠❡♥s✉r❛çã♦ ❞❛s s❡♥s✐❜✐❧✐❞❛❞❡s✱ s❡rá ❛❞♦t❛❞♦ ❝♦♠♦ ❜❛s❡ ♦ ♠♦❞❡❧♦ ●❛r♠❛♥ ✲ ❑♦❤❧❤❛❣❡♥ ✭✶✾✽✸✮✱ ✉♠❛ ❛❞❛♣t❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ♣❛r❛ ♦♣çõ❡s s♦❜r❡ ♠♦❡❞❛✳
CBS(S, K, t, T, σ, r, rf) =e−rτ[F N(d1)−KN(d2)] ✭✸✮
❖♥❞❡✱
❙✿ ❚❛①❛ ❞❡ ❝â♠❜✐♦ ❙♣♦t ✱
✶✶
rf✿ t❛①❛ ❞❡ ❥✉r♦s ❡①♣♦♥❡♥❝✐❛❧ ❡①t❡r♥❛✱ ♣♦r ❡①❡♠♣❧♦ ❛ t❛①❛ ❞❡ ❝✉♣♦♠ ❝❛♠❜✐❛❧ ♥♦ ❝❛s♦ ❞❡ ♦♣çõ❡s s♦❜r❡ ♠♦❡❞❛ ❞ó❧❛r ✲ r❡❛❧✱
t✿ t❡♠♣♦ ❡♠ ❛♥♦s ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ❞❡s❞❡ ♦ ✐♥✐❝✐♦ ❞❛ ♦♣çã♦ ❛té ♦ s❡✉ ✈❡♥❝✐♠❡♥t♦✱
❑✿ ♣r❡ç♦ ❞❡ ❡①❡r❝í❝✐♦ ❞❛ ♦♣çã♦✱
❋✿ ❋♦r✇❛r❞ ❞❛ ♦♣çã♦✱
F =Seerfrt
d1 =
ln(FK)+0.5σ2t σ√t
d2 =d1−σ
√
t
N(x) =
Rx −∞e−
y2
2 dy
√ 2π
n(x) = e−x 2 2
√ 2π
τ =T −t✱ ❡♥tr❡ ❛ ❞❛t❛ ❞❡ r❡❢❡rê♥❝✐❛ ❡ ❛ ♠❛t✉r✐❞❛❞❡ ❞❛ ♦♣çã♦✳
❖ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✱ q✉❡ ❛❞♦t❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❝♦♥st❛♥t❡ é ❢r❡q✉❡♥t❡♠❡♥t❡ ❡s❝♦❧❤✐❞♦ ❝♦♠♦ ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛✳ ❊♥tr❡t❛♥t♦✱ r❡s✉❧t❛❞♦s ❡♠♣ír✐❝♦s tê♠ r❡✈❡❧❛❞♦ q✉❡ ♦ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛ ❞❡s✈✐♦s ❡♠ r❡❧❛çã♦ à r❡❛❧✐❞❛❞❡ ❞♦ ♠❡r❝❛❞♦ ❞❡ ♦♣çõ❡s✳
✷✳✷ ●r❡❣❛s ●❡r❛✐s s❡♠ ♦ ❙♠✐❧❡
❙ã♦ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s s✐♠♣❧❡s ❞❛ ❢ór♠✉❧❛ ❞❡ ❛♣r❡ç❛♠❡♥t♦ ❞❛s ♦♣çõ❡s ✭●❛r♠❛♥ ✲ ❑♦❤❧❤❛✲ ❣❡♥✮ ❡♠ r❡❧❛çã♦ ❛♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦✱ ✈❡r ❍❛✉❣✱ ❊ ✭✷✵✵✼✮
✷✳✷✳✶ ❉❡❧t❛
❊ss❛ é ❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❞♦ ♣r❡ç♦ ❞❛ ♦♣çã♦ ✭❝❛❧❧ ♦✉ ♣✉t✮ ❡♠ r❡❧❛çã♦ ❛♦ ♣r❡ç♦S ❞♦ s♣♦t✳
V =φF N(φd1)−KN(φd2)
er.τ ✭✹✮
❖♥❞❡✱φ = 1 ♣❛r❛ ✉♠❛ ❝❛❧❧ ❡ φ =−1 ♣❛r❛ ✉♠❛ ♣✉t✱
δ = ∂V
∂S =φe
−rfτN(φd
1) ✭✺✮
✷✳✷✳✷ ●❛♠♠❛
●❛♠♠❛ ✱ ❛ss✉♠✐♥❞♦✲s❡ ✈♦❧❛t✐❧✐❞❛❞❡ ❝♦♥st❛♥t❡✱ é ❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❞♦ ❉❡❧t❛ ❝♦♠ r❡❧❛çã♦ ❛♦ ❙♣♦t✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✿
γ = ∂△
∂S =e
−rftn(d1)
✶✷
✷✳✷✳✸ ❱❡❣❛
❱❡❣❛ ♣❛r❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❝♦♥st❛♥t❡✱ é ❛ ✈❛r✐❛çã♦ ♥♦ ♣r❡ç♦ ❞❡ ✉♠❛ ♦♣çã♦ ♣❛r❛ ✉♠ ❝❤♦q✉❡ ♥♦ ✈❛❧♦r ❞❛ ✈♦❧❛t✐❧✐❞❛❞❡✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✿
V ega= ∂V
∂σ =Se
−rfτ√tn(d
1) ✭✼✮
✷✳✷✳✹ ❱❛♥♥❛
➱ ❛ ❞❡r✐✈❛❞❛ ❞♦ ❱❡❣❛ ❝♦♠ r❡❧❛çã♦ ❛♦ ❙♣♦t✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✿
V anna= ∂
2V
∂σ∂S =−
e−rftn(d
1)d2
σ ✭✽✮
✷✳✷✳✺ ❱♦❧❣❛
➱ ❛ ❞❡r✐✈❛❞❛ ❞♦ ❱❡❣❛ ❝♦♠ r❡❧❛çã♦ à ✈♦❧❛t✐❧✐❞❛❞❡✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✿
V olga= ∂
2V
∂σ2 =S
e−rft√τ n(d
1)d1d2
σ ✭✾✮
✷✳✸ ❱♦❧❛t✐❧✐❞❛❞❡ ■♠♣❧í❝✐t❛
❚r❛❞❡rs ❡ ✐♥✈❡st✐❞♦r❡s ♥ã♦ ❛❝r❡❞✐t❛♠ ♥❛s s✉♣♦s✐çõ❡s ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✳ ❖s ♠❡r❝❛❞♦s ♥ã♦ sã♦ ❝♦♠♣❧❡t♦s✱ ❢r✐❝çõ❡s ❡①✐st❡♠ ❡ ❛ ❞✐str✐❜✉✐çã♦ ❞♦ ♣r❡ç♦ ❞♦ ✉♥❞❡r❧②✐♥❣ ♣♦ss✉✐✱ t✐♣✐❝❛♠❡♥t❡✱ ✉♠❛ ❝❛✉❞❛ ♠✉✐t♦ ♣❡s❛❞❛ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r❡ç♦s t❡r s✐❞♦ ❣❡✲ r❛❞❛ ♣♦r ✉♠ ♠♦✈✐♠❡♥t♦ ❇r♦✇♥✐❛♥♦ ❣❡♦♠étr✐❝♦✳ P❛r❛ ❡①♣❧✐❝❛r ♣♦rq✉❡ s❛❜❡♠♦s q✉❡ ♦s ♣❛rt✐❝✐♣❛♥t❡s ❞❡ ♠❡r❝❛❞♦ ♥ã♦ ❛❝r❡❞✐t❛♠ ♥❛s s✉♣♦s✐çõ❡s ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ ❝♦♥s✐❞❡r❡ t♦❞❛s ❛s ♦♣çõ❡s ❡✉r♦♣❡✐❛s ❞❡ ❝❛❧❧s ❡ ♣✉ts ♥❡❣♦❝✐❛❞❛s ❡♠ ❛♣❡♥❛s ✉♠ ✉♥❞❡r❧②✐♥❣✳ P♦❞❡♠♦s ♦❜s❡r✈❛r ♦ ♣r❡ç♦ ❞♦ ✉♥❞❡r❧②✐♥❣ S ❡ s❛❜❡♠♦s q✉❡ ❝❛❞❛ ♦♣çã♦ é ❞❡t❡r♠✐♥❛❞❛ ♣♦r ✉♠ ❞✐❢❡r❡♥t❡ ✈❛❧♦r ❞❡T ❡K✳ P♦❞❡♠♦s ♦❜s❡r✈❛r t❛♠❜é♠ q✉❡ ❛ t❛①❛ ❧✐✈r❡ ❞❡ r✐s❝♦r s❡rá ❛ ♠❡s♠❛ ♣❛r❛ t♦❞❛s ❛s ♦♣çõ❡s ❝♦♠ ❛ ♠❡s♠❛ ♠❛t✉r✐❞❛❞❡✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ ❛ ✈♦❧❛t✐❧✐❞❛❞❡σ ❞❡✈❡r✐❛ s❡r ❛ ♠❡s♠❛ ♣❛r❛ t♦❞❛s ❛s ♦♣çõ❡s✱ ✉♠❛ ✈❡③ q✉❡ ❡stã♦ ❞❡t❡r♠✐♥❛❞❛s ♣❡❧❛ ♠❡s♠❛ ♠❛t✉r✐❞❛❞❡✳ ❆ ✈♦❧❛t✐❧✐❞❛❞❡ ♥❛ ❢ór♠✉❧❛ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s é ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❞♦ ♣r❡ç♦ ♥♦ ♣r♦❝❡ss♦ ❇r♦✇♥✐❛♥♦ ❣❡♦♠étr✐❝♦ ✭✷✮ ❡ ✉♠❛ ✈❡③ q✉❡ ❤❛❥❛ ✉♠ ú♥✐❝♦ ✉♥❞❡r❧②✐♥❣ ❤❛✈❡rá ✉♠❛ ú♥✐❝❛ ✈♦❧❛t✐❧✐❞❛❞❡σ.❊♥tr❡t❛♥t♦✱ ❤á ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ✈♦❧❛t✐❧✐❞❛❞❡ ❡ ♦✉tr❛s ✈❛r✐á✈❡✐s✿ S ❡r sã♦ ♦❜s❡r✈á✈❡✐s ♥♦ ♠❡r❝❛❞♦✱ ♠❛s ♥ã♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ❛ ✈♦❧❛t✐❧✐❞❛❞❡✳
✶✸
❙❡ tr❛❞❡rs ❡ ✐♥✈❡st✐❞♦r❡s ❛❝r❡❞✐t❛ss❡♠ ♥❛s s✉♣♦s✐çõ❡s ❢❡✐t❛s ♥♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦✲ ❧❡s ❡♥tã♦ ❛ ✈♦❧❛t✐❧✐❞❛❞❡ q✉❡ ❡stá ✐♠♣❧í❝✐t❛ ♥♦s ♣r❡ç♦s ♦❜s❡r✈❛❞♦s ❞❡ t♦❞❛s ❛s ♦♣çõ❡s ❞❡✈❡r✐❛ s❡r ❛ ♠❡s♠❛✳ ❊ ❡ss❛ ✈♦❧❛t✐❧✐❞❛❞❡ ú♥✐❝❛ ❞❡✈❡r✐❛ s❡r ❛ ♠❡s♠❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❞♦ ♣r♦❝❡ss♦ ❇r♦✇✲ ♥✐❛♥♦ ❣❡♦♠étr✐❝♦ ✭✷✮✳ ▼❛s q✉❛♥❞♦ ❝❛❧❝✉❧❛♠♦s ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❞❡ ❞✐❢❡r❡♥t❡s ♦♣çõ❡s ♣❛r❛ ♦ ♠❡s♠♦ ✉♥❞❡r❧②✐♥❣ ❡♥❝♦♥tr❛♠♦s q✉❡ sã♦ ❞✐❢❡r❡♥t❡s✳ ◆❛ ✈❡r❞❛❞❡ ❝❡rt❛s ❝❛r❛❝t❡rís✲ t✐❝❛s ✜❝❛♠ ❡✈✐❞❡♥t❡s q✉❛♥❞♦ ♣❧♦t❛♠♦s ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞♦ str✐❦❡ ✭♦✉ ❞♦ ❞❡❧t❛ ❞❛ ♦♣çã♦ ❝♦♠♦ ✈❡r❡♠♦s ❛❜❛✐①♦✮ ❡ ❞❛ ♠❛t✉r✐❞❛❞❡✳ ❊♥t❡♥✲ ❞❡♥❞♦ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❞❡ ♠❡r❝❛❞♦ ❡st❛❜❡❧❡❝❡♠♦s ♦s ❢✉♥❞❛♠❡♥t♦s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡✳
❆ s❡❣✉✐r t❡♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❝♦♠♦ ♦❜t❡♠♦s ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❛ ♣❛rt✐r ❞♦s ♣r❡ç♦s ❞❡ ♠❡r❝❛❞♦ ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❝❛❞❛ ✈❡♥❝✐♠❡♥t♦✳
VBS(S, K, r, r
f, t, Σ(K, S, T, r, rf, σ, VM K) = VM K ✭✶✵✮
❖♥❞❡✱
VBS é ♦ ♣rê♠✐♦ ❞❛ ♦♣çã♦ ♦❜t✐❞♦ ✈✐❛ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ VM K é ♦ ♣rê♠✐♦ ❞❛ ♦♣çã♦ ❞❛❞♦ ♣❡❧♦ ♠❡r❝❛❞♦✱
Σ(K, S, T, r, rf, σ, VM K)é ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ♦❜t✐❞❛✱ ♥❛ ♥♦ss❛ ❛❜♦r❞❛❣❡♠ ❞❡✜♥✐✲ ♠♦s Σ(δ(K, S, r, rf), t))✳
❆♦ ✐♥✈és ❞❛ ✈♦❧❛t✐❧✐❞❛❞❡ s❡r ❝♦♥st❛♥t❡ ❝♦♠♦ ♣r♦♣õ❡ ♦ ♠♦❞❡❧♦ ♦r✐❣✐♥❛❧✱ t❡♠♦sΣ(δ(K, S, r, rf), t)) ❞❡♣❡♥❞❡♥t❡ ❞♦ ❞❡❧t❛δ ❡ ❞♦ ♣r❛③♦✱ ❛ ✜❣✉r❛ ❛ s❡❣✉✐r ✐❧✉str❛ ❡ss❛ ❞❡♣❡♥❞ê♥❝✐❛✳ ◆♦ ❡✐①♦ ❞❛s
❛❜s❝✐ss❛s t❡♠♦s ♦ ❞❡❧t❛ ❞❡✜♥✐❞♦ ❞❡ ✺✪ ❛té ✾✺✪ ❡ ♥♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ r❡s✉❧t❛♥t❡✳
✶✹
❋✐❣✉r❛ ✷✳✶✿ ❱♦❧❛t✐❧✐❞❛❞❡ ■♠♣❧í❝✐t❛ Σ ♦❜t✐❞❛ ❛tr❛✈és ❞❡ ❞❛❞♦s ❞❡ ♠❡r❝❛❞♦✳
0 0.2
0.4 0.6
0.8
1 0 100
200
300 0.2
0.3 0.4 0.5
Dias úteis(t) Delta
Volatilidade
❋✐❣✉r❛ ✷✳✷✿ ❈♦rt❡ ♥❛ ❙✉♣❡r❢í❝✐❡ ❞❡ ❱♦❧❛t✐❧✐❞❛❞❡ ■♠♣❧í❝✐t❛Σ❡♠ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦ ❉❡❧t❛
✶✺
✷✳✹ ❈♦♥❞✐çõ❡s ❞❡ ◆ã♦ ❆r❜✐tr❛❣❡♠
◆❛ ❝♦♥❝❡♣çã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❞❡✈❡♠♦s t♦♠❛r ♦ ❝✉✐❞❛❞♦ ❞❡ ♥ã♦ ❧❡✈❛r♠♦s ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♣♦♥t♦s ❝♦♠ ❛r❜✐tr❛❣❡♠✱ ❛ss✐♠ ♥ã♦ ❣❡r❛r❡♠♦s ✉♠ ♠❛❧ ❛♣r❡ç❛✲ ♠❡♥t♦ ❞❛s ♦♣çõ❡s ✉t✐❧✐③❛♥❞♦ ❡ss❛ s✉♣❡r❢í❝✐❡✱ ❛ s❡❣✉✐r t❡♠♦s ❛s ❤✐♣ót❡s❡s q✉❡ ❞❡✈❡♠ s❡r ❛t❡♥❞✐❞❛s✳
✶✳ C ≥S−Ke−r(T−t)✱ ❛ss✉♠✐♥❞♦ ❞✐✈✐❞❡♥❞♦s ✐❣✉❛✐s à ③❡r♦ ♣❛r❛ ❝❛❧❧s ❡✉r♦♣❡✐❛s✳
❯♠ ❋♦r✇❛r❞ ✈❛❧❡ ❙✲❑ ♥♦ ✈❡♥❝✐♠❡♥t♦✱ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❛ss✉♠✐♥❞♦ ❞✐✈✐❞❡♥❞♦s ✐❣✉❛✐s ❛ ③❡r♦ ❡ ✉♠❛ t❛①❛ ❞❡ r✐s❝♦ ✐❣✉❛❧ ❛ r✱ ✈❛❧❡ ❛ ✈❛❧♦r ♣r❡s❡♥t❡ S−Ke−r(T−t)✳ ❯♠❛ ♦♣çã♦
s❡♠♣r❡ ✈❛❧❡ ♠❛✐s q✉❡ ✉♠ ❋♦r✇❛r❞✱ ♣♦rq✉❡ t❡♠ ♦ ♠❡s♠♦ ♣❛②♦✛ q✉❛♥❞♦ ST ≥K✱ ❡ ✈❛❧❡ ♠❛✐s q✉❛♥❞♦ ST < K✳
✷✳ P❛r❛ ♦ ♠❡s♠♦ ✈❡♥❝✐♠❡♥t♦✱ ♦s ♣r❡ç♦s ❞❛s ♦♣çõ❡s s❛t✐s❢❛③❡♠ ♣❛r❛ ♦ ❣r❛❞✐❡♥t❡ ❞❡ ♣r✐✲ ♠❡✐r❛ ❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡♠ r❡❧❛çã♦ ❛♦ str✐❦❡✱ ✈❡r ❋❡♥❣❧❡r✱▼ ✭✷✵✵✺✮
∂C(K, T)
∂K ≤0, K >0, T ∈(0, γ]
∂2C(K, T)
∂K2 ≥0, K >0, T ∈(0, γ]
∂P(K, T)
∂K ≥0, K >0, T ∈(0, γ]
∂2P(K, T)
∂K2 ≥0, K >0, T ∈(0, γ]
✶✻
✷✳✺ ❈♦♥✈❡♥çõ❡s ❡ ❈♦t❛çõ❡s ❞❡ ▼❡r❝❛❞♦
✶✳ ❙♣♦t ♦✉ ❋♦r✇❛r❞ ❞❡❧t❛ ✕ ❚r❛♥s❛çõ❡s ❝♦♠ s♣♦t sã♦ ♥♦r♠❛❧♠❡♥t❡ ♠❛✐s ❧✐q✉✐❞❛s ❞♦ q✉❡ ❋♦r✇❛r❞s✱ ♣♦r ✐ss♦ ❛ ♣r❡❢❡rê♥❝✐❛ ♣❡❧♦ ❙♣♦t ❞❡❧t❛✳ ❊♥tr❡t❛♥t♦✱ ♣❛r❛ ♣❛r❡s ❞❡ ♠♦❡❞❛s ♦♥❞❡ t❡♠♦s ✉♠❛ ✐♥❝❡rt❡③❛ ❛ r❡s♣❡✐t♦ ❞❛ ❡✈♦❧✉çã♦ ❞❛ t❛①❛ ❧✐✈r❡ ❞❡ r✐s❝♦ é ❝♦♠✉♠ ❢❛③❡r♠♦s ♦ ❤❡❞❣✐♥❣ ❝♦♠ ❢♦r✇❛r❞s✳ ❈♦♠♦ r❡❣r❛ ❣❡r❛❧✱ ♣❛r❛ ♦♣çõ❡s ❝♦♠ ✈❡♥❝✐♠❡♥t♦ ♠❛✐s ❝✉rt♦ ❡♠ ♠❡r❝❛❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s ❛❞♦t❛♠♦s ❝♦t❛çõ❡s ❡♠ ❙♣♦t ❞❡❧t❛❀ ♣❛r❛ ♦♣çõ❡s ♠❛✐s ❧♦♥❣❛s ❡♠ ♠❡r❝❛❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s ❡ ❡♠❡r❣❡♥t❡s ♦ ♠❛✐s ❝♦♠✉♠ sã♦ ❝♦t❛çõ❡s ❡♠ ❋♦r✇❛r❞ ❞❡❧t❛✳
✷✳ ❆❚▼ ✭❆t t❤❡ ▼♦♥❡②✮ ✕ ❈♦♠♦ ♦ ♣ró♣r✐♦ ♥♦♠❡ s✉❣❡r❡✱ é ✉♠❛ ♦♣çã♦ q✉❡ ❡stá ❛ ♠❡✐♦ ❝❛♠✐♥❤♦ ❡♥tr❡ ✉♠❛ ♦♣çã♦ ✐♥ t❤❡ ♠♦♥❡② ❡ ✉♠❛ ♦✉t ♦❢ t❤❡ ♠♦♥❡②✳ ❍á ❜❛s✐❝❛✲ ♠❡♥t❡ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ ❆❚▼✱ ❆❚▼ ❋♦r✇❛r❞ ♦✉ ❆❚▼ ❙♣♦t✳
✸✳ ❘✐s❦ ❘❡✈❡rs❛❧ ✲ é ✉♠❛ ♣♦s✐çã♦ ❝♦♠♣r❛❞❛ ❡♠ ✉♠❛ ❝❛❧❧ ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ❡ ✉♠❛ ♣♦s✐çã♦ ✈❡♥❞✐❞❛ ❡♠ ✉♠❛ ♣✉t ♦✉t✲♦❢✲t❤❡✲♠♦♥❡② ❝♦♠ ✉♠ ∆ ✭❉❡❧t❛✮ s✐♠étr✐❝♦✳ ❖
♠❛✐s ❝♦♠✉♠ é tr❛❜❛❧❤❛r ❝♦♠ ♦♣çõ❡s ❝♦♠ ∆25 ✭❉❡❧t❛ ✷✺✪✮✳ ❚r❛❞❡rs ✈❡❡♠ ✉♠
r✐s❦✲r❡✈❡rs❛❧ ♣♦s✐t✐✈♦ ❝♦♠♦ ✉♠ ✐♥❞✐❝❛❞♦r ❞❡ ✉♠ ♠❡r❝❛❞♦ ❜✉❧❧✐s❤✱ ✉♠❛ ✈❡③ q✉❡ ❝❛❧❧s ❡st❛r✐❛♠ ❝✉st❛♥❞♦ ♠❛✐s q✉❡ ♣✉ts ❡ ✈✐❝❡✲✈❡rs❛✳
✹✳ ❙tr❛♥❣❧❡ ✲ é ✉♠❛ ♣♦s✐çã♦ ❝♦♠♣r❛❞❛ ❡♠ ✉♠❛ ❝❛❧❧ ❡ ✉♠❛ ♣♦s✐çã♦ ❝♦♠♣r❛❞❛ ❡♠ ✉♠❛ ♣✉t ❝♦♠ ♦ ♠❡s♠♦ str✐❦❡ ❡ ♣r❛③♦ ✕ ♦ tr❛❞❡r ❛♣♦st❛ ❡♠ ❛✉♠❡♥t♦ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡✳ ❖ ♣rê♠✐♦ ❞❡ ✉♠ str❛♥❣❧❡ ♠❛♥tê♠ ❛ ✐♥❢♦r♠❛çã♦ s♦❜r❡ ❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ❞♦ ❛t✐✈♦ ❜❛s❡ ✕ ❛❧t❛ ✈♦❧❛t✐❧✐❞❛❞❡ s✐❣♥✐✜❝❛ ❣❛♥❤♦ ❛❧t♦✱ ❡ ❝♦♠♦ r❡s✉❧t❛❞♦ ✉♠ ♣rê♠✐♦ ❛❧t♦✳
✺✳ ❇✉tt❡r✢② ✲ é ❝♦♥str✉í❞♦ ❝♦♠ ✉♠❛ ♣♦s✐çã♦ ✈❡♥❞✐❞❛ ❡♠ ✉♠ str❛♥❣❧❡ ❛t✲t❤❡✲♠♦♥❡② ❡ ✉♠❛ ♣♦s✐çã♦ ❝♦♠♣r❛❞❛ ❡♠ ✉♠ str❛♥❣❧❡ ∆25✳ ❖ ❝♦♠♣r❛❞♦r ❞❡ ✉♠ ❜✉tt❡r✢② ❡s♣❡r❛
✉♠ ❣❛♥❤♦ ❡stá✈❡❧ s♦❜r❡ ♦ ❛t✐✈♦ ❜❛s❡ ✭❇❋❂✭❈✰P✮✴✷✲❆❚▼✮✳
✷✳✻ ❉❡❧t❛✲❙tr✐❦❡ ❈♦♥✈❡rsã♦
❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❝♦♠✉♠ ❞❛s ♣❛r❛♠❡tr✐③❛çõ❡s é q✉❡ ❛s ♦♣çõ❡s sã♦ ♣❛r❛♠❡tr✐③❛❞❛s ♥❛ ❛❜s❝✐ss❛ ♣❡❧♦ ∆✳ ◆♦ ❡♥t❛♥t♦ ♣♦❞❡♠♦s ❢❛③❡r ❡ss❛ ♣❛r❛♠❡tr✐③❛çã♦ t❛♠❜é♠ ♣❡❧♦ str✐❦❡✱
❜❛st❛♥❞♦ ✉t✐❧✐③❛r ❛ s❡❣✉✐♥t❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❛♠❜♦s ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠❛ ❝❛❧❧ ✳
K =F.exph−N−1(∆)σ√τ+σr
−rf + σ
2
τi ✭✶✶✮
❖♥❞❡✿
✶✼
✷✳✼ P❛r❛♠❡tr✐③❛çã♦ ❞❛ ❙✉♣❡r❢í❝✐❡ ❞❡ ❱♦❧❛t✐❧✐❞❛❞❡
❯♠❛ s✉♣❡r❢í❝✐❡ é ❝♦♥str✉í❞❛ ♣♦r ♦❜s❡r✈❛çõ❡s ❢❡✐t❛s ♥♦ ♠❡r❝❛❞♦ ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦ ♣r❛③♦✳ ❊♠ ♣r✐♥❝í♣✐♦ ✉♠❛ ❝✉r✈❛ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ♣♦❞❡ s❡r ❢❡✐t❛ ❝♦♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♥ú♠❡r♦ ❞❡ ♣r❡ç♦s ♦❜s❡r✈❛❞♦s ❡♠ ✉♠ ❞❛❞♦ ♣❡rí♦❞♦✳ ◆❛ ♣rát✐❝❛✱ ❛ ❝♦♥str✉çã♦ ❡stá ❧✐❣❛❞❛ ❞✐r❡t❛♠❡♥t❡ ❛ ✉♠ ♥ú♠❡r♦ s✉✜❝✐❡♥t❡ ❞❡ ♦♣çõ❡s ❧íq✉✐❞❛s✱ q✉❡ é ♠❛✐♦r ♣❛r❛ ♦♣çõ❡s ❛t✲t❤❡✲♠♦♥❡②✳ ●❡r❛❧✲ ♠❡♥t❡ ❛s ❝✉r✈❛s sã♦ ❝♦♥str✉í❞❛s ✉s❛♥❞♦ ❡st❛s ♦♣çõ❡s✱ ❛❧é♠ ❞❡ ♦♣çõ❡s ❝❛❧❧ ❡ ♣✉t ❝♦♠ ❉❡❧t❛ ❞❡ ✶✵✪ ❡ ❝♦♠ ❉❡❧t❛ ❞❡ ✷✺✪ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ❞❡ ❢♦r♠❛ ♠❛✐s ❛♠♣❧❛ ♦ ♠❡r❝❛❞♦✱ ❡♠❜♦r❛ tr❛❞❡rs ♣♦ss❛♠ ❞❛r ♣r❡ç♦ ♣❛r❛ ♦♣çõ❡s ❝♦♠ ❉❡❧t❛ ❞❡ ✶✵✪ ❝♦♠ ♠❡♥♦r ❝❡rt❡③❛ ❞♦ q✉❡ ♣❛r❛ ♦♣çõ❡s ❝♦♠ ❉❡❧t❛ ❞❡ ✷✺✪✳ ❊ss❡s ♣♦♥t♦s ♣♦❞❡♠ s❡r ♦r❞❡♥❛❞♦s ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡ ❞❡ str✐❦❡✳
◆❛ ❞❡✜♥✐çã♦ ❞❡ss❛ ❝✉r✈❛ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡✱ ✉♠❛ ❡s❝♦❧❤❛ ♣❛r❛ ❛ ❛❜s❝✐ss❛ é ♦ str✐❦❡ ❞❛ ♦♣çã♦✱ ♦✉tr❛ ❡s❝♦❧❤❛ é ✉s❛r ♦ ❉❡❧t❛ ✭❝♦♠♦ ✉s✉❛❧ ♥♦ ♠❡r❝❛❞♦ ❞❡ ❝â♠❜✐♦✮✳ ❯♠❛ ✈❛♥t❛❣❡♠ ❞❡ s❡ ✉s❛r ♦ ❉❡❧t❛ é q✉❡ ❞✐♠❡♥s✐♦♥❛♠♦s ❡ss❡ ❡✐①♦ ♥♦ ✐♥t❡r✈❛❧♦ ♣❛❞r♦♥✐③❛❞♦ ❞❡ [0,1] ♣❛r❛
t♦❞♦s ♦s ♣r❛③♦s✳
❆ ❢♦r♠❛ ❞❡ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❝✉r✈❛ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ é ❛ ❝❛r❛❝t❡ríst✐❝❛ ♣r✐♥❝✐♣❛❧ ❞❛ s✉♣❡r❢í❝✐❡✱ ❞❡✈❡♠♦s ♦❜s❡r✈❛r ♥❛ ❝♦♥str✉çã♦ ❞❡❧❛ ❛s ❝♦♥❞✐çõ❡s ❞❡ ♥ã♦ ❛r❜✐tr❛❣❡♠ ❞❡✜♥✐❞❛s ❡♠ ✸✳✸✳
✷✳✼✳✶ P♦❧✐♥ô♠✐♦ ❞❡ ◗✉❛rt♦ ●r❛✉
❊ss❡ ❢♦✐ ♦ ♠ét♦❞♦ ❛❞♦t❛❞♦ ♣❛r❛ ❢❛③❡r♠♦s ❛s ✐♥t❡r♣♦❧❛çõ❡s ❛♦ ❧♦♥❣♦ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦✲ ❧❛t✐❧✐❞❛❞❡ ❡ ❡♥❝♦♥tr❛r♠♦s ❞❡r✐✈❛♥❞♦ ❞✐r❡t❛♠❡♥t❡ ❡ss❡ ♣♦❧✐♥ô♠✐♦ ❛s ❞❡r✐✈❛❞❛s ❞❡ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞♦ s♠✐❧❡ ❡♠ r❡❧❛çã♦ ❛♦ ❞❡❧t❛ ❞❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ s✉♣❡r❢í❝✐❡✱ ✈✐❞❡ ❏✐♠ ●❛t❤❡r❛❧ ✭✷✵✵✺✮✳
Σ =ax4+bx3+cx2+d+σAT M ✭✶✷✮
❉❡✜♥✐♠♦s✿
˜
d1 =
ln(F K)
σAT M√τ❀
˜
∆ = ˜N( ˜d1) = 1 1+exp√−4
2πd1˜ ❀
σAT M é ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❆❚▼ ❞❛s ♦♣çõ❡s❀ x= ˜∆−∆˜AT M❀
❉❡ ♣♦ss❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡s ❞❡ ♠❡r❝❛❞♦ σloP, σhiP, σAT M, σhiC, σloC✱ ♦✉ s❡❥❛✱ ❞❡✜♥❡✲s❡ ❛ ♣❛r❛♠❡tr✐③❛çã♦ ♣❛r❛ ♦ ❆❚▼✱ ❉❡❧t❛s ✶✵✪ ❡ ✷✺✪ ♣❛r❛ ❝❛❧❧s ❡ ♣✉ts ❡ r❡s♦❧✈❡♠♦s ♦ s✐st❡♠❛ ❛ s❡❣✉✐r ♣❛r❛ ❛✱ ❜✱ ❝ ❡ ❞✳
❖♥❞❡✱
σloP é ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ♣✉t ❝✉❥♦ ❞❡❧t❛ é ✶✵✪❀ σhiP é ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ♣✉t ❝✉❥♦ ❞❡❧t❛ é ✷✺✪❀ σAT M é ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❆❚▼ ❞❛s ♦♣çõ❡s❀
✶✽
σhiC é ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❝❛❧❧ ❝✉❥♦ ❞❡❧t❛ é ✷✺✪❀
σloP σhiP σhiC σloC =
xloP xloP xloP xloP xhiP xhiP xhiP xhiP xhiC xhiC xhiC xhiC xloC xloC xloC xloC
a b c d
❆ ♠❛tr✐③ ♣♦ss✉✐ ✐♥✈❡rs❛ s❡ ❡ s♦♠❡♥t❡ s❡ ♦s str✐❦❡s ❞❛s ♦♣çõ❡s ❞❡ ♠❡r❝❛❞♦ sã♦ ❞✐❢❡r❡♥t❡s✳ ❖s ❡❧❡♠❡♥t♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ♠❛tr✐③ ❞❡♣❡♥❞❡♠ s♦♠❡♥t❡ ❞❛ ♦❜s❡r✈❛çã♦ ❞♦s ❞❛❞♦s ❞❡ ♠❡r❝❛❞♦✳
P❛r❛ ❝♦♥str✉✐r ❡ss❛ s✉♣❡r❢í❝✐❡ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♣♦❞❡♠♦s ✉s❛r ♦✉tr♦s ♠ét♦❞♦s ❝♦♠♦ ♠♦str❛r❡♠♦s ❛ s❡❣✉✐r✳
✷✳✼✳✷ ❙❆❇❘
❯♠❛ ❞❛s ❝r✐t✐❝❛s ❛ ✐♥t❡r♣♦❧❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❡♠ ❉❡❧t❛ ❛❝✐♠❛ é q✉❡ ❡❧❛ ♥ã♦ r❡t♦r♥❛ ✉♠❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❛❞❡q✉❛❞❛ ♥❛s ✏❛s❛s✑✱ ✐st♦ é✱ ♣❛r❛ ❝❛❧❧s ❡ ♣✉ts ❞❡❡♣❧② ✐♥ t❤❡ ♠♦♥❡②✳ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ é ♦ ♠ét♦❞♦ ♣r♦♣♦st♦ ♣♦r ❍❛❣❛♥ ❡t ❛❧✳ ✭✷✵✵✷✮✱ ❡♠ q✉❡ ♦s ❛✉t♦r❡s ❞❡r✐✈❛♠ ✉♠❛ ❢♦r♠✉❧❛ ❢❡❝❤❛❞❛ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ ❛s ✈♦❧❛t✐❧✐❞❛❞❡s ✐♠♣❧í❝✐t❛s ❞❡ ✉♠❛ ♦♣çã♦ ❡✉r♦♣❡✐❛✱ ✉s❛♥❞♦ ✉♠ ♠♦❞❡❧♦ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ❡st♦❝ást✐❝❛ ♣❛rt✐❝✉❧❛r✳
❖ ♠♦❞❡❧♦ t❡♠ ♦ ♣❛râ♠❡tr♦β✱ q✉❡ ❞❡t❡r♠✐♥❛ ✉♠❛ ❡s❝❛❧❛ ❡♥tr❡ ✉♠ ♠♦❞❡❧♦ ❡st♦❝ást✐❝♦ ♥♦r♠❛❧ ✭β = 0✮ ❡ ✉♠ ♠♦❞❡❧♦ ❡st♦❝ást✐❝♦ ❧♦❣♥♦r♠❛❧ ✭β = 1✮✱ ♦ ❙❆❇❘ t❡♠ três ♦✉tr♦s
♣❛râ♠❡tr♦s✿
α=❱♦❧❛t✐❧✐❞❛❞❡ ■♥✐❝✐❛❧❀
υ =❱♦❧❛t✐❧✐❞❛❞❡ ❞❛ ❱♦❧❛t✐❧✐❞❛❞❡❀
ρ=❈♦rr❡❧❛çã♦ ❡♥tr❡ ♦ ❙♣♦t ❡ ❛ ✈♦❧❛t✐❧✐❞❛❞❡❀
Σ(K) = α
(FTK)(1−β)/2{1 + [(1−β)2/24]log2(FT/K) + [(1−β)4/1920]log4(FT/K)} ✭✶✸✮
z χ(z)
n
1 +h(1−24β)2(F α2
TK)(1−β) +
1 4
ρβυα
(FTK)(1−β)/2
2−3ρ2
24 υ2 i
To
❖♥❞❡✿ z = υ
α(FTK)
(1−β)/2log(F
T/K)❀ χ(z) =log
√
1−2ρz+z2+z−ρ
1−ρ
❀
✷✳✼✳✸ ❱❛♥♥❛✲❱♦❧❣❛
✶✾
❖ ♠ét♦❞♦ ❱❛♥♥❛ ✕ ❱♦❧❣❛ é ♥♦r♠❛❧♠❡♥t❡ ✉s❛❞♦ ❡♠ ♠❡r❝❛❞♦s ❋❳✱ ♦♥❞❡ t✐♣✐❝❛♠❡♥t❡ três ❝♦t❛çõ❡s ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡s ❡stã♦ ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ✉♠❛ ❞❛❞❛ ♠❛t✉r✐❞❛❞❡❀ ❆❚▼✱RR25✱BF25✳
❆ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ♥♦s ♣❡r♠✐t❡ ❞❡r✐✈❛r ✈♦❧❛t✐❧✐❞❛❞❡s ✐♠♣❧í❝✐t❛s ♣❛r❛ ❞✐❢❡r❡♥t❡s ❞❡❧t❛s✱ ✈❡r ❈❛st❛❣♥❛ ❛♥❞ ▼❡r❝✉r✐♦ ✭✷✵✵✺✮✳
C(K) =CBS(K)+ P3
i=1
xi(K)
CM KT(K
i)−CBS(Ki)
✱ ♦♥❞❡✿ CM KT(K
i)❂ Pr❡ç♦ ❞❡ ♠❡r❝❛❞♦ ❞❡ ✉♠❛ ❝❛❧❧ ❝♦♠ str✐❦❡ Ki❀ CM KT(K
i)❂ Pr❡ç♦ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ❞❡ ✉♠❛ ❝❛❧❧ ❝♦♠ str✐❦❡ Ki❀ x1(K) = νν((K1K))
lnK2
Kln K3
K
lnK2
K1ln
K3 K1 ❀
x2(K) = νν((K2K))
lnK K1ln
K3 K
lnK2
K1ln K3 K21
❀
x3(K) = νν((K3K))
lnK K1lnKK2
lnK3
K1ln K3 K2
❀
ν(Ki)❂ ❱❡❣❛ ❞❡ ✉♠❛ ❝❛❧❧ ❝♦♠ str✐❦❡ Ki❀
C(K)≈CBS(K)+
3 P
i=1
xi(K)ν(Ki) [σi−σ]
P♦❞❡♠♦s ♣❛r❛♠❡tr✐③❛r ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ❡♠ t❡r♠♦s ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡✿
σ(K) =σ2+ −
σ2+ p
σ2
2 +d1(K)d2(K)(2σ2D1(K) +D2(K))
d1(K)d2(K)
❡✱
D1(K) =
lnK2
K ln K3
K
lnK2
K1ln K3 K1
σ1+
lnK K1ln
K3 K
lnK2
K1ln K3 K21
σ2+
lnK K1lnKK2
lnK3
K1ln K3 K2
σ3−σ2❀
D2(K) =
lnK2
K ln K3
K
lnK2
K1ln K3 K1
d1(K)d2(K)(σ1−σ2)2+
lnK K1lnKK2
lnK3
K1ln K3 K2
✷✵
✸ ❉✐♥â♠✐❝❛ ❞❡ ❱♦❧❛t✐❧✐❞❛❞❡
❉❛❞♦ ♦ ♣r❡s❡♥t❡ s❦❡✇ ✭♦✉ s❡❥❛✱ q✉❛♥t♦ ✈❛r✐❛ ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❡♠ r❡❧❛çã♦ ❛♦ str✐❦❡✱ ♠♦♥❡②✲ ♥❡ss✶ ♦✉ ❉❡❧t❛ ❞❛ ♦♣çã♦✮ ❞❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛✱ s❡r✐❛ ❞❡s❡❥❛❞♦ ❝♦♥❤❡❝❡r ❝♦♠♦ ♦ ♠❡s♠♦ ✐r✐❛ ✈❛r✐❛r ❞❡ ❛❝♦r❞♦ ♦ ♥í✈❡❧ ❞♦ í♥❞✐❝❡ ♥♦ ❢✉t✉r♦✳ ❍á ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❞✐❢❡r❡♥t❡s ✈✐sõ❡s ❞❡ ❝♦♠♦ ♦s ❛s♣❡❝t♦s ❞♦ ♣r❡s❡♥t❡ s❦❡✇ sã♦ st✐❝❦② ❝♦♥❢♦r♠❡ ♦ í♥❞✐❝❡ s❡ ♠♦✈❡ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ♥♦s ❞❡t❡r❡♠♦s ❛ ❡①♣♦r ❛ ❞✐♥â♠✐❝❛ q✉❡ ✉t✐❧✐③❛r❡♠♦s✳✳
✶✳ ❙t✐❝❦② ❉❡❧t❛✴▼♦♥❡②♥❡ss
❙t✐❝❦② ▼♦♥❡②♥❡ss s✐❣♥✐✜❝❛ q✉❡ ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♦♣çã♦ ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞♦ ♠♦♥❡②♥❡ss K/S✱ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ♦♣çã♦ ❝♦♠ ♦ ♥í✈❡❧ ❞♦ í♥❞✐❝❡ ❡ ❞♦ str✐❦❡ ❞❡r✐✈❛ ✐♥t❡✐r❛♠❡♥t❡ ❞❡ss❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♠ ♦ ♠♦♥❡②♥❡ss✱
Σ(δ(K, S, r, rf), t)) = Σ(δ(K, S, r, rf), t = 0))−b(K/S−1)S0 ✭✶✹✮
➱ ✉♠❛ t❡♥t❛t✐✈❛ ❞❡ ♠✉❞❛r ♦ s❦❡✇ ❝♦♥❢♦r♠❡ ♦ ♣r❡ç♦ ❞♦ st♦❝❦ s❡ ♠♦✈❡✱ ❛❥✉st❛♥❞♦ ♦ ♠♦♥❡②♥❡ss✳ P❛r❛ ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❆t✲❚❤❡✲▼♦♥❡② ❡st❛❜❡❧❡❝❡ q✉❡ ❡ss❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❞❡✈❡r✐❛ ♠❛♥t❡r✲s❡ ❝♦♥st❛♥t❡ ❝♦♥❢♦r♠❡ ♦ í♥❞✐❝❡ s❡ ♠♦✈❡✳
❊ss❡ ♠♦❞❡❧♦ ❛ss✉♠❡ q✉❡ ♦ ♠❡r❝❛❞♦ t❡r✐❛ ✉♠❛ r❡✈❡rsã♦ à ♠é❞✐❛ ♣❛r❛ ✉♠❛ ❞❡✜♥✐❞❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❆t✲❚❤❡✲▼♦♥❡② Σ(δ(K = S, S, r, rf), t = 0))✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♥í✈❡❧ ❞♦ ♠❡r❝❛❞♦✳
◆♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✱ ❛ ❡①♣♦s✐çã♦∆BS❞❡♣❡♥❞❡ ❞❡K❡S❛tr❛✈és ❞♦moneyness K/S✱ ❛ss✐♠ st✐❝❦② ♠♦♥❡②♥❡ss é ❡q✉✐✈❛❧❡♥t❡ ❛ st✐❝❦② ❞❡❧t❛✳
C = f(S, σ, t, r, rf, K)✱ ♠❛s σ é ❢✉♥çã♦ ❞❡ S ✭♥❛ ♣rát✐❝❛ ♦❜s❡r✈❛♠♦s ✐ss♦✮✳ ❆ss✐♠✱ ♣❛r❛ ❢❛③❡r♠♦s ❛ ❞❡r✐✈❛❞❛ t♦t❛❧ ❞❡C ❡♠ r❡❧❛çã♦ ❛♦ s♣♦t ❛♣❛r❡❝❡ ✉♠ t❡r♠♦ ❛❞✐❝✐♦♥❛❧✱ ❞❡✈❡♠♦s ❢❛③❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
dΣ
dS =
dΣ d∆
∂∆
∂S ✭✶✺✮
❖♥❞❡ dΣ
d∆ ❛♣❛r❡❝❡ ❝♦♠♦ r❡s✉❧t❛❞♦ ❞❛ ❞❡r✐✈❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡✱ ❝♦♥❢♦r♠❡ ❡q✉❛çã♦ ✭✶✽✮✳ P♦r ✜♠ ✏♦ ♥♦✈♦ ❞❡❧t❛✑ ✜❝❛
dC
dS =
∂C
∂S +
∂C ∂σ
dΣ
dS ✭✶✻✮
❆s ❞❡r✐✈❛❞❛s ∂C ∂S ❡
∂C
∂σ sã♦ ♦❜t✐❞❛s ❞✐r❡t❛♠❡♥t❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ ❝♦♠♦ ♠♦str❛❞♦ ❡♠ ✷✳✸✳
✷✶
✸✳✶ ❊str✉t✉r❛ ❛ t❡r♠♦ ❞♦ ❙❦❡✇ ✭❆ss✐♠❡tr✐❛✮
❖ s❦❡✇ ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❣r❛❞✐❡♥t❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❞❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❡♠ r❡❧❛çã♦ ❛♦ str✐❦❡ é ♥♦r♠❛❧♠❡♥t❡ ♥❡❣❛t✐✈♦ ♣❛r❛ ❡q✉✐t✐❡s✳ ❍á ❛❧❣✉♠❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ✐ss♦✿
• ●r❛♥❞❡s ❏✉♠♣s ♥♦ s♣♦t t❡♥❞❡♠ ❛ s❡r❡♠ ♠❛✐♦r❡s ♣❛r❛ ❜❛✐①♦ ❞♦ q✉❡ ♣❛r❛ ❝✐♠❛✳ ❙❡
❤á ✉♠ ❥✉♠♣ ♥♦ ♣r❡ç♦ ❞♦ st♦❝❦✱ ❡ss❡ é ♥♦r♠❛❧♠❡♥t❡ ♠❛✐s ❛❝❡♥t✉❛❞♦ ♣❛r❛ ❜❛✐①♦ ❡♠ r❛③ã♦ ❞❡ ✉♠ ❡✈❡♥t♦ ✏r✉✐♠✑ s❡r ♠❛✐s ✐♥❡s♣❡r❛❞♦ ❞♦ q✉❡ ♦❝♦rr❡r ✉♠ ❡✈❡♥t♦ ✏❜♦♠✑ ♥ã♦ ❡s♣❡r❛❞♦✳
• ❆ ✈♦❧❛t✐❧✐❞❛❞❡ é ✉♠❛ ♠❡❞✐❞❛ ❞❡ r✐s❝♦ ❝✉❥♦ ✈❛❧♦r ❝♦st✉♠❛ s❡r ✐♥❝r❡♠❡♥t❛❞♦ ❝♦♥❢♦r♠❡
❤á ✉♠ ❞❡❝❧í♥✐♦ ♥♦ ♠❡r❝❛❞♦ ❞❡ ❡q✉✐t✐❡s ✭❙t✐❝❦② ❙tr✐❦❡✮✳ ❙❡ ❛ss✉♠✐r♠♦s q✉❡ ♥ã♦ ❤á ♠✉❞❛♥ç❛ ♥♦ ♥ú♠❡r♦ ❞❡ ❛çõ❡s ❡♠✐t✐❞❛s ♦✉ ♥♦ t❛♠❛♥❤♦ ❞♦ ❞é❜✐t♦✱ ❡♥tã♦ ❝♦♥❢♦r♠❡ ♦ ♣r❡ç♦ ❞♦ st♦❝❦ ❞❛ ❝♦♠♣❛♥❤✐❛ ❞❡❝❧✐♥❛✱ ♦ t❛♠❛♥❤♦ ❞♦ s❡✉ ❞é❜✐t♦ ❛✉♠❡♥t❛✳ ❆❧❛✈❛♥✲ ❝❛❣❡♠ ❡ ✈♦❧❛t✐❧✐❞❛❞❡ sã♦ ♠❡❞✐❞❛s ❞❡ r✐s❝♦ ❝♦rr❡❧❛❝✐♦♥❛❞❛s✱ ❛ss✐♠✱ ❝♦♠ ✈♦❧❛t✐❧✐❞❛❞❡ ❝r❡s❝❡♥❞♦ t❛❧ q✉❛❧ ❤á ✉♠ ❞❡❝❧í♥✐♦ ♥♦ ♠❡r❝❛❞♦ ❞❡ ❡q✉✐t✐❡s✳
• ❈♦♠♦ r❡s✉❧t❛❞♦s ❞✐ss♦✱ t✐♣✐❝❛♠❡♥t❡ ✐♥✈❡st✐❞♦r❡s ❡stã♦ ✐♥t❡r❡ss❛❞♦s ❡♠ ❝♦♠♣r❛r ♣✉ts
♣❛r❛ ♣r♦t❡çã♦ ❡♠ ✈❡③ ❞❡ ✈❡♥❞ê✲❧♦s✳ ■ss♦ ❡❧❡✈❛ ❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ♣❛r❛ str✐❦❡s ♠❡♥♦r❡s✳ ❆❧é♠ ❞✐ss♦✱ ❛❧❣✉♥s ✐♥✈❡st✐❞♦r❡s ♣r❡❢❡r❡♠ tr♦❝❛r ❡ss❛s ♣♦s✐çõ❡s ♣♦r ❝❛❧❧s ❝✉❥♦ r❡✢❡①♦ ♠❛✐♦r s❡ ❞á ♥❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❞♦s str✐❦❡s ♠❛✐♦r❡s✳
❖ ♣❛②♦✉t ❞❡ ✉♠ ♣✉t s♣r❡❛❞ ✭❝❛❧❧ s♣r❡❛❞✮ é s❡♠♣r❡ ♣♦s✐t✐✈♦✱ ♣♦r ✐ss♦ ❞❡✈❡ s❡♠♣r❡ t❡r ✉♠ ❝✉st♦ ♣♦s✐t✐✈♦✳ ❙❡ ❢♦ss❡ ♣♦ssí✈❡❧ ♠♦♥t❛r ✉♠❛ ♣♦s✐çã♦ ❝♦♠♣r❛❞❛ ❡♠ ✉♠ ♣✉t ✭❝❛❧❧✮ s♣r❡❛❞✱ s❡♠ ♥❡♥❤✉♠ ❝✉st♦ ✭♦✉ ♣♦t❡♥❝✐❛❧♠❡♥t❡ ❣❛♥❤❛r ✉♠ ♣rê♠✐♦ ♣❡q✉❡♥♦✮ q✉❛❧q✉❡r ✐♥✈❡st✐❞♦r r❛❝✐♦♥❛❧ ✐r✐❛ ❢❛③❡r ✉♠❛ ♣♦s✐çã♦ ♠❛✐♦r ❡ ♦❜t❡r ❣❛♥❤♦s s❡♠ r✐s❝♦s ✭✉♠❛ ✈❡③ q✉❡ ❛ ♣♦s✐çã♦ ♥ã♦ ♣♦❞❡ s♦❢r❡r ♣❡r❞❛✮✳
✷✷
✹ ●r❡❣❛s ❆❞❛♣t❛❞❛s ♣❛r❛ ♦♣çõ❡s ❱❛♥✐❧❧❛ s♦❜r❡ ❚❛①❛
❞❡ ❈â♠❜✐♦
❆q✉✐ ❢❛r❡♠♦s ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ♦s ❡❢❡✐t♦s ❞♦ s♠✐❧❡ ♥❛s ❞❡r✐✈❛❞❛s ❞❛ ♦♣çã♦C✉t✐❧✐③❛❞❛s ♥♦ ♠♦❞❡❧♦ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛✳ ❘❡❧❛❝✐♦♥❛♥❞♦✲❛s ❝♦♠ ❛s ❞❡r✐✈❛❞❛s ❞❡ ✉♠❛ ♦♣çã♦ ❡✉r♦♣❡✐❛ ❡q✉✐✈❛❧❡♥t❡CBS ♥♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ✭✶✾✼✸✮✳
❆ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ❞❡ ✉♠❛ ❝❛❧❧ ✈❛♥✐❧❧❛ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ❝♦♠♦✿
Σ(δ(K, S, r, rf), t)) =CBS−1(F, K, t, T, Cmkt) ✭✶✼✮
♣❛r❛K >0❡ ❚ǫ✭✵✱γ],♦♥❞❡CBS−1é ❛ ✐♥✈❡rs❛ ❞❡ ✉♠❛ ❝❛❧❧ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ❡♠ ✈♦❧❛t✐❧✐❞❛❞❡
❡Cmkt é ♦ ♣r❡ç♦ ❞❡ ♠❡r❝❛❞♦ ❞❡ ✉♠❛ ❝❛❧❧✳
❆ ❝♦rr❡çã♦ ♥❛s ❞❡r✐✈❛❞❛s ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s ✭✶✾✼✸✮ ♥❛ ♣r❡s❡♥ç❛ ❞♦ s♠✐❧❡ ❞❡ ✈♦❧❛t✐✲ ❧✐❞❛❞❡ sã♦ ♦❜t✐❞❛s ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ♦ ❡❢❡✐t♦ ❞♦ t❡r♠♦ ❛❞✐❝✐♦♥❛❧✱ ✈❡r ✸✱ s❡♥❞♦ ❞❡r✐✈❛❞♦ ♥❛ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛✳
✹✳✶ ❉❡❧t❛ ❆❞❛♣t❛❞♦
❖ ❉❡❧t❛ ❆❞❛♣t❛❞♦ ❞❡ ✉♠❛ ♦♣çã♦ ✈❛♥✐❧❧❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❡❢❡✐t♦ s♠✐❧❡ é ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣♦r ❉❡❧t❛s✱ ♥♦s ❝❛s♦s ❞❡r✐✈❛❞♦s ❛ s❡❣✉✐r ✉t✐❧✐③❛♠♦s ✉♠ s✉♣❡r❢í❝✐❡ ♦❜t✐❞❛ ✉t✐❧✐③❛♥❞♦ ✉♠ ♣♦❧✐♥ô♠✐♦ ❞♦ q✉❛rt♦ ❣r❛✉✱ ✈❡r ✷✳✼✳✶✱ ♠❛s ❛s ♠❡s♠❛s ❞❡r✐✈❛çõ❡s s❡rã♦ t❛♠❜é♠ ♦❜t✐❞❛s ✉t✐❧✐③❛♥❞♦ q✉❛❧q✉❡r ❞♦s ♠ét♦❞♦s ♣r♦♣♦s✲ t♦s ❡♠ ✷✳✼✳ ❆ ♦❜t❡♥çã♦ ❞❛ ❞❡r✐✈❛❞❛ dΣ
d△ é ❢❡✐t❛ ❛ ♣❛rt✐r ❞❡ss❛s ♣❛r❛♠❡tr✐③❛çõ❡s ❛❞♦t❛❞❛s
♣❛r❛ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ✐♠♣❧í❝✐t❛✱ t♦❞❛s ❡ss❛s ♣❛r❛♠❡tr✐③❛çõ❡s ♣r♦♣♦st❛s sã♦ ❢✉♥✲ çõ❡s ❢❛❝✐❧♠❡♥t❡ ❞❡r✐✈á✈❡✐s✱ ❛ss✐♠ ♣❛r❛ ♦ ❝❛s♦ ❞♦ ❉❡❧t❛ t❡♠♦s ✉♠❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞✐r❡t❛ ♥♦ s♣♦t ❡ ♥❛ ✈♦❧❛t✐❧✐❞❛❞❡✱ ❛♦ ❢❛③❡r♠♦s ❡ss❛ ❞❡r✐✈❛çã♦ ♣r✐♠❡✐r♦ ❢❛③❡♠♦s ♥❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ s♣♦t ❡ ❞❡♣♦✐s ♥❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ✈♦❧❛t✐❧✐❞❛❞❡✱ ♥♦ ❝❛s♦ ❣❡r❛❧ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ ✈❡r ✷✳✷✱ ♥ã♦ ❝♦♥s✐❞❡r❛♠♦s ❡ss❛ ❞❡♣❡♥❞ê♥❝✐❛ ♥❛ ✈♦❧❛t✐❧✐❞❛❞❡ ❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ dΣ
dS é ③❡r♦✳ ❖ t❡r♠♦ ∂C
∂S é ♦❜t✐❞♦ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡r✐✈❛❞❛ ❞♦ ♠♦❞❡❧♦✱ ✈❡r ✷✳✷✳✶✱ ❛ ❝♦♠♣♦♥❡♥t❡ dΣ
dS é ♦❜t✐❞❛ ❝♦♠♦ ❞❡s❝r✐t♦ ❛❝✐♠❛ ❛ ♣❛rt✐r ❞❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✈♦❧❛t✐❧✐❞❛❞❡ ❡s❝♦❧❤✐❞❛✱ ♦ t❡r♠♦ ∂△
∂S t❛♠❜é♠ é ♦❜t✐❞♦ ❞✐r❡t❛♠❡♥t❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ ❡ss❡ t❡r♠♦ é ♦ ●❛♠♠❛ ❇❧❛❝❦ ❡ ❙❝❤♦❧❡s✱ ✈❡r ✷✳✷✳✷✱❞❡ ♠♦❞♦ ❣❡r❛❧ t❡r❡♠♦s ♦ s❡❣✉✐♥t❡✿
dΣ
dS =
dΣ d△
∂△
∂S ✭✶✽✮
△AD = dc
dS =
∂C
∂S +
∂C ∂σ
dΣ
