OU, Vasicek, CIR ve Hull-White Modellerine Aktüeryal Uygulamaları Üzerinden Kısa Bir Bakış

Yıl 2021, Cilt: 5 Sayı: 2, 37 - 49, 30.09.2021

Sinem Kozpınar

Öz

Amaç: Orntein-Uhlenbeck (OU), Vasicek, Cox-Ingersoll-Ross (CIR) ve Hull-White afin süreçleri, önemli özelliklerine kısaca değinilerek ele alınmıştır. Bu makalenin temel amacı söz konusu afin süreçlerin, farklı stokastik modeller ve matematiksel metotların kullanıldığı altı yakın dönem aktüeryal uygulamasını tartışmaktır.
Sonuç ve Katkılar: Bu uygulamalar, bir yandan söz konusu afin süreçlerin modelleme sürecine nasıl dahil edildiğini göstermekte, diğer yandan ise matematiksel hesaplamalar/veri analizi yoluyla bu afin süreçleri kullanmanın avantajları hakkında fikir vermektedir.

Anahtar Kelimeler

Ornstein-Uhlenbeck modeli, Vasicek modeli, Cox-Ingersoll-Ross modeli, Hull-White modeli.

A Brief Look at OU, Vasicek, CIR and Hull-White Models Through Their Actuarial Applications

Öz

Aim: A brief overview of the affine processes, namely the Orntein-Uhlenbeck (OU) process, the Vasicek process, the Cox-Ingersoll-Ross (CIR) process and the Hull-White process, is presented through their important features. The main purpose of this paper is to discuss six very recent actuarial applications of these affine processes that focus on different problems with different stochastic models and different mathematical methods.
Conclusion and Contributions: On one hand, these applications show how to incorporate the corresponding affine processes into the modelling framework. On the one hand they give an insight about the advantages of using these affine processes through mathematical calculations/data analysis.

Anahtar Kelimeler

Ornstein-Uhlenbeck model, Vasicek model, Cox-Ingersoll-Ross model, Hull-White model.

Kaynakça

• Beekman, J. A. and Shiu, E. S. (1988). Stochastic models for bond prices, function space integrals and immunization theory. Insurance: Mathematics and Economics, 7(3), 163-173.
• Brigo, D. and Mercurio, F. (2007). Interest rate models-theory and practice: with smile, inflation and credit. Springer Science & Business Media.
• Cox, J. C., Ingersoll Jr, J. E. and Ross, S. A. (2005). A theory of the term structure of interest rates. In Theory of Valuation (pp. 129-164).
• Feller, W. (1951). Two singular diffusion problems. Annals of mathematics, 173-182.
• Grasselli, M. R., & Lipton, A. (2019). On the normality of negative interest rates. Review of Keynesian Economics, 7(2), 201-219.
• Hull, J. and White, A. (1990). Pricing interest-rate-derivative securities. The review of Financial Studies, 3(4), 573-592.
• Hull, J. and White, A. (1994). Numerical procedures for implementing term structure models I: Single-factor models. Journal of Derivatives, 2(1), 7-16.
• Hull, J. (1996). Using Hull-White interest rate trees. Journal of Derivatives, 3(3), 26-36. Jackson, H. (2015). The international experience with negative policy rates (No. 2015-13). Bank of Canada Staff Discussion Paper.
• Jamshidian, F. (1989). An exact bond option formula. The Journal of Finance, 44(1), 205-209.
• Lamberton, D. and Lapeyre, B. (2011). Introduction to stochastic calculus applied to finance. CRC press.
• Li, Y., Mao, X., Song, Y., and Tao, J. (2020). Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization.
• Ma, C., Yue, S., Wu, H., & Ma, Y. (2020). Pricing vulnerable options with stochastic volatility and stochastic interest rate. Computational Economics, 56(2), 391-429.
• Uhlenbeck, G. E. and Ornstein, L. S. (1930). On the theory of the Brownian motion, Physical Review, 36(5), 823-841.
• Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177-188.
• Wang, G., Wang, X., & Zhou, K. (2017). Pricing vulnerable options with stochastic volatility. Physica A: Statistical Mechanics and its Applications, 485, 91-103.
• Yoon, J. H., & Kim, J. H. (2015). The pricing of vulnerable options with double Mellin transforms. Journal of Mathematical Analysis and Applications, 422(2), 838-857.
• Zeddouk, F. and Devolder, P. (2019). Pricing of longevity derivatives and cost of capital. Risks, 7(2), 41.
• Zeddouk, F., & Devolder, P. (2020). Mean reversion in stochastic mortality: why and how?. European Actuarial Journal, 10(2), 499-525.
• Zeytun, S. and Gupta, A. (2007). A comparative study of the Vasicek and the CIR model of the short rate. Technical Report 124, Fraunhofer (ITWM).