Abstract
Bu makalede, stokastik diferansiyel denklemlere Lie
simetri analizinin bir uygulaması olarak asıl çözümün nasıl elde edileceğini
göstereceğiz. Yapacağımız bu analizler stokastik faiz oranı modellerinden Hull-White
modeli özelinde yapılacaktır. İlk olarak Hull-White stokastik modeline karşılık
gelen Hull-White (1+1) lineer parabolik kısmi türevli denklemini elde edeceğiz.
Daha sonra, elde ettiğimiz bu denklemin Lie simetri analiz yöntemleriyle özellikle
de değişmezlik kriterleri altında klasik anlamdaki ısı denklemine
dönüşebileceğini göstereceğiz ve ilgili dönüşümleri bulacağız. Son olarak da,
Hull-White kısmi türevli diferansiyel denkleminin asıl çözümünü, bulduğumuz bu
dönüşümlerle ve ısı denkleminin literatürdeki özelliklerini kullanarak elde
edeceğiz.
Keywords
Isı denklemi Kismi Diferansiyel Denklemler Lie simetri analizi Stokastik diferansiyel denklemler
References
- Hull, J. ve White, A. (1993). One-Factor Interest Rate Models and the Valuation of Interest-Rate Derivative Securities, The Journal of Financial and Quantitative Analysis, 28, 2, 235-254.
- Gazizov, R.K. ve Ibragimov N.H. (1998). Lie Symmetry Analysis of Differential Equations in Finance, Nonlinear Dynam. 17(4), 387-407.
- Goard, J. (2000). New Solutions to the Bond-Pricing Equation via Lie’s Classical Method, Math. Comput. Model., 32, 299-313.
- Lie, S. (1881). On Integration of a Class of Linear Partial Differential Equations by means of Definite Integrals Archiv for Mathematik ıg Naturvidenskab, VI(3) 328-368.
- Mahomed, F.M., Mahomed, K.S., Naz, R. ve Momoniat, E., (2013). Invariant Approaches to Equations of Finance, Math. Comput. Appl., 18(3), 244-250.
- Mahomed, F.M. (2008). Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations, J. Nonlinear Math. Phys., 15, 112-123.
- Merton, R. (1976). Option Pricing when Underlying Stock Returns are Discontinuous, J. Financial Economics, 3, 125-144.
- Pooe, C.A., Mahomed, F.M. ve Wafo Soh, C. (2004). Fundamental Solutions for Zero-Coupon Bond Pricing Models, Nonlinear Dynam., 36, 69-76.
- Bakkaloğlu, A., Aziz, T., Fatima, A., Mahomed, F.M. ve Khalique, C.M., (2016). Invariant Approach to Optimal Investment-Consumption Problem: the constant elasticity of variance (CEV) model, Mathematical Methods in the Applied Sciences, 40, 5, 1382-1395.
- Bakkaloğlu, A., Mahomed, ve F.M., Aziz, T. (2017). Invariant Criteria for the Zero-Coupon Bond Pricing Vasicek and Cox-Ingersoll-Ross Models, New Trends in Mathematical Sciences, 2, 29-46.
- İzgi, B. ve Bakkaloğlu, A. (2017). Deterministic Solutions of the Stochastic Differential Equations Using Invariant Criteria, Proceedings of ICPAS 2017, ISBN: 978-605-9546-02-7, 323-326.
- İzgi, B. ve Bakkaloğlu, A. (2017). Fundamental Solution of Bond Pricing in the Ho-Lee Stochastic Interest Rate Model Under the Invariant Criteria, New Trends in Mathematical Sciences, 5, 1, 196-203.
- İzgi, B. ve Bakkaloğlu, A. (2017). Invariant Approaches for the Analytic Solution of the Stochastic Black-Derman Toy Model, gönderildi.
Abstract
References
- Hull, J. ve White, A. (1993). One-Factor Interest Rate Models and the Valuation of Interest-Rate Derivative Securities, The Journal of Financial and Quantitative Analysis, 28, 2, 235-254.
- Gazizov, R.K. ve Ibragimov N.H. (1998). Lie Symmetry Analysis of Differential Equations in Finance, Nonlinear Dynam. 17(4), 387-407.
- Goard, J. (2000). New Solutions to the Bond-Pricing Equation via Lie’s Classical Method, Math. Comput. Model., 32, 299-313.
- Lie, S. (1881). On Integration of a Class of Linear Partial Differential Equations by means of Definite Integrals Archiv for Mathematik ıg Naturvidenskab, VI(3) 328-368.
- Mahomed, F.M., Mahomed, K.S., Naz, R. ve Momoniat, E., (2013). Invariant Approaches to Equations of Finance, Math. Comput. Appl., 18(3), 244-250.
- Mahomed, F.M. (2008). Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations, J. Nonlinear Math. Phys., 15, 112-123.
- Merton, R. (1976). Option Pricing when Underlying Stock Returns are Discontinuous, J. Financial Economics, 3, 125-144.
- Pooe, C.A., Mahomed, F.M. ve Wafo Soh, C. (2004). Fundamental Solutions for Zero-Coupon Bond Pricing Models, Nonlinear Dynam., 36, 69-76.
- Bakkaloğlu, A., Aziz, T., Fatima, A., Mahomed, F.M. ve Khalique, C.M., (2016). Invariant Approach to Optimal Investment-Consumption Problem: the constant elasticity of variance (CEV) model, Mathematical Methods in the Applied Sciences, 40, 5, 1382-1395.
- Bakkaloğlu, A., Mahomed, ve F.M., Aziz, T. (2017). Invariant Criteria for the Zero-Coupon Bond Pricing Vasicek and Cox-Ingersoll-Ross Models, New Trends in Mathematical Sciences, 2, 29-46.
- İzgi, B. ve Bakkaloğlu, A. (2017). Deterministic Solutions of the Stochastic Differential Equations Using Invariant Criteria, Proceedings of ICPAS 2017, ISBN: 978-605-9546-02-7, 323-326.
- İzgi, B. ve Bakkaloğlu, A. (2017). Fundamental Solution of Bond Pricing in the Ho-Lee Stochastic Interest Rate Model Under the Invariant Criteria, New Trends in Mathematical Sciences, 5, 1, 196-203.
- İzgi, B. ve Bakkaloğlu, A. (2017). Invariant Approaches for the Analytic Solution of the Stochastic Black-Derman Toy Model, gönderildi.
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Engineering |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | June 30, 2018 |
| Acceptance Date | May 20, 2018 |
| Published in Issue | Year 2018 Volume: 30 Issue: 2 |
Cite
Marmara Fen Bilimleri Dergisi
e-ISSN : 2146-5150
MU Fen Bilimleri Enstitüsü
Göztepe Yerleşkesi, 34722 Kadıköy, İstanbul
E-posta: fbedergi@marmara.edu.tr