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(77) İÇİNDEKİLER Sayfa ÖZET................................................................................................................... i ABSTRACT ....................................................................................................... ii İÇİNDEKİLER ................................................................................................iii ŞEKİL LİSTESİ ................................................................................................ v TABLO LİSTESİ ............................................................................................. vi SEMBOL LİSTESİ ......................................................................................... vii KISALTMALAR LİSTESİ ...........................................................................viii ÖNSÖZ .............................................................................................................. ix 1. GİRİŞ ............................................................................................................. 1 2. TEMEL TANIM VE TEOREMLER .......................................................... 4 2.1 Kesirli Kalkülüsün Temel Fonksiyonları ............................................. 4 2.2 Kesirli Türevler ve Kesirli İntegraller .................................................. 6 2.2.1 Grünwald-Letnikov Kesirli Türev ve İntegral Operatörü .............. 7 2.2.2 Riemann-Liouville Kesirli İntegral ve Türev Operatörü ............... 9 2.2.3 Caputo Kesirli Türev Operatörü .................................................. 10 2.2.4 Kesirli Diferansiyel Denklemler .................................................. 15 3. KESİRLİ BLACK-SCHOLES OPSİYON FİYATLAMA MODELİ .... 19 3.1 Opsiyonun Tanımı ve Opsiyonla İlgili Temel Kavramlar.................. 19 3.2 Opsiyon Fiyatlama Modelleri ............................................................. 23 3.2.1 Black-Scholes Opsiyon Fiyatlama Modeli (BSOFM) ................. 23 3.2.2 Kesirli Black-Scholes Denklemi (KBSD) ................................... 28 4. KESİRLİ KISMİ DİFERANSİYEL DENKLEMLERİN ANALİTİKNÜMERİK ÇÖZÜMLERİ VE KESİRLİ BLACK-SCHOLES DENKLEMİNE UYGULANMASI........................................................... 29 4.1 Adomian Ayrıştırma Yöntemi ............................................................ 29 4.2 Kesirli Black-Scholes Denkleminin Adomian Ayrıştırma Metodu ile Çözümü ......................................................................................... 36 4.3 Genelleştirilmiş Kesirli Black-Scholes Denkleminin Adomian Ayrışım Metodu ile Çözümü .............................................................. 39 4.4 Varyasyonel İterasyon Metodu........................................................... 42 4.4.1 Varyasyonel İterasyon Metodunda Temel Enstrümanlar ............ 42 4.4.2 Kesirli Varyasyonel İterasyon Metodu ........................................ 56 4.5 Kesirli Black-Scholes Denkleminin Kesirli Varyasyonel İterasyon Metodu ile Çözümü ............................................................................ 57 4.6 Kesirli Laplace Homotopi Pertürbasyon Metodu ............................... 59 4.7 Kesirli Black-Scholes Denkleminin Laplace Homotopi Pertürbasyon Metodu ile Çözümü ............................................................................ 61 4.8 Genelleştirilmiş Diferansiyel Dönüşüm Metodu ................................ 62 4.8.1 Diferansiyel Dönüşüm Metodu .................................................... 63 4.8.2 Genelleştirilmiş Taylor Açılımı ................................................... 66 4.8.3 Genelleştirilmiş Diferansiyel Dönüşüm Yöntemi........................ 67 4.9 Kesirli Black-Scholes Denkleminin Genelleştirilmiş Diferansiyel Dönüşüm Metodu ile Çözümü ........................................................... 76. iii.

(78) 4.10 Padé Yaklaşımı ................................................................................... 81 4.10.1 Tek Değişkenli Padé Yaklaşımı ................................................... 81 4.10.2 Çok Değişkenli Padé Yaklaşımı .................................................. 86 4.11 Kesirli Black-Scholes Denkleminin Çok Değişkenli Padé Yaklaşımı ile Çözümü ......................................................................................... 90 4.12 Sonlu Fark Yaklaşımı (SFY) ............................................................ 106 4.13 Kesirli Black-Scholes Denkleminin Sonlu Fark Yaklaşımı ile Çözümü ............................................................................................ 109 5. SONUÇ VE ÖNERİLER .......................................................................... 114 6. KAYNAKLAR .......................................................................................... 116. iv.

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