Blek-sholec yoki Blek-Sholec-Merton modeli

Doimiy ravishda qo'yish 

Amerikalik put variantlari uchun umumiy analitik echim yo'qligiga qaramay, doimiy optsion holati uchun bunday formulani olish mumkin, ya'ni variant hech qachon tugamaydi (ya'ni, {\ displaystyle T \ rightarrow \ infty} ). Bunday holda, opsiyaning parchalanishi nolga teng, bu esa Blek-Skoulz PDE ning ODE bo'lishiga olib keladi:

{\ displaystyle {1 \ over {2}} \ sigma ^ {2} S ^ {2} {d ^ {2} V \ over {dS ^ {2}}} + (rq) S {dV \ over {dS }} - rV = 0}

Ruxsat bering {\ displaystyle S _ {-}} mashqning quyi chegarasini belgilang, quyida variantni bajarish uchun maqbuldir. Chegara shartlari:

{\ displaystyle V (S _ {-}) = K-S _ {-}, \ quad V_ {S} (S _ {-}) = - 1, \ quad V (S) \ leq K}

ODE echimlari har qanday ikkita chiziqli mustaqil echimlarning chiziqli birikmasi:

{\ displaystyle V (S) = A_ {1} S ^ {\ lambda _ {1}} + A_ {2} S ^ {\ lambda _ {2}}}

Uchun {\ displaystyle S _ {-} \ leq S} , ushbu eritmani ODE ga almashtirish {\ displaystyle i = {1,2}}  hosil:

{\ displaystyle \ left [{1 \ over {2}} \ sigma ^ {2} \ lambda _ {i} (\ lambda _ {i} -1) + (rq) \ lambda _ {i} -r \ right ] S ^ {\ lambda _ {i}} = 0}

Shartlarni qayta tuzish quyidagilar:

{\ displaystyle {1 \ over {2}} \ sigma ^ {2} \ lambda _ {i} ^ {2} + \ left (rq- {1 \ over {2}} \ sigma ^ {2} \ right) \ lambda _ {i} -r = 0}

Foydalanish kvadrat formulasini echimlar uchun,{\ displaystyle \ lambda _ {i}}  ular:

{\ displaystyle {\ begin {aligned} \ lambda _ {1} & = {- \ left (rq- {1 \ over {2}} \ sigma ^ {2} \ right) + {\ sqrt {\ left (rq) - {1 \ over {2}} \ sigma ^ {2} \ right) ^ {2} +2 \ sigma ^ {2} r}} \ over {\ sigma ^ {2}}} \\\ lambda _ { 2} & = {- \ chap (rq- {1 \ ustidan {2}} \ sigma ^ {2} \ o'ng) - {\ sqrt {\ chap (rq- {1 \ ustidan {2}} \ sigma ^ { 2} \ o'ng) ^ {2} +2 \ sigma ^ {2} r}} \ over {\ sigma ^ {2}}} \ end {aligned}}}

Doimiy qo'yilish uchun cheklangan echimga ega bo'lish uchun chegara shartlari put qiymatiga yuqori va pastki chegaralarni nazarda tutganligi sababli, qo'yish kerak {\ displaystyle A_ {1} = 0} , hal qilishga olib keladi {\ displaystyle V (S) = A_ {2} S ^ {\ lambda _ {2}}} . Birinchi chegara shartidan ma'lumki:

{\ displaystyle V (S _ {-}) = A_ {2} S _ {-} ^ {\ lambda _ {2}} = K-S _ {-} \ A_ {2} = {K-S _ {-} \ ni anglatadi {S _ ​​{-}}}} dan yuqori

Shuning uchun abadiy qo'yishning qiymati quyidagicha bo'ladi:

{\ displaystyle V (S) = (K-S _ {-}) \ chap ({S \ over {S _ {-}}} \ right) ^ {\ lambda _ {2}}}

Ikkinchi chegara sharti pastki mashq chegarasining joylashishini keltirib chiqaradi:

{\ displaystyle V_ {S} (S _ {-}) = \ lambda _ {2} {K-S _ {-} \ over {S _ {-}}} = - 1 \ S _ {-} = {\ lambda _ {2} K \ over {\ lambda _ {2} -1}}}

Xulosa qilish uchun, uchun {\ textstyle S \ geq S _ {-} = {\ lambda _ {2} K \ over {\ lambda _ {2} -1}}} , Amerikaning doimiy put opsiyasi quyidagicha:

{\ displaystyle V (S) = {K \ over {1- \ lambda _ {2}}} \ chap ({\ lambda _ {2} -1 \ over {\ lambda _ {2}}} \ right) ^ {\ lambda _ {2}} \ chap ({S \ over {K}} \ right) ^ {\ lambda _ {2}}}