응용통계연구 (The Korean Journal of Applied Statistics)

  • 제31권6호
  • /
  • Pages.801-810
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  • 2018
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  • 1225-066X(pISSN)
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  • 2383-5818(eISSN)
한국통계학회 (The Korean Statistical Society)
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단순 확산과정들에 대한 확률효과 모형

Random effect models for simple diffusions

  • 투고 : 2018.10.29
  • 심사 : 2018.11.07
  • 발행 : 2018.12.31
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초록

확산은 금융이나 물리적 현상의 모형화에 이용되는 확률과정이다. 반복적으로 관측된 확산과정에 대하여 통계적인 모형을 구축할 때, 확률효과를 고려할 필요가 있다. 이 연구에서는 Ornstein-Uhlenbeck 확산모형과 geometric Brownian motion 확산모형에 대하여 확률효과를 도입한다. 모형모수에 대한 최도우도추정법을 적용하기 위하여, 확률효과에 대한 적절한 분포를 가정하여 닫힌 형태로 우도함수를 얻는 방법을 탐색하였다. 1991년부터 2017년까지 27년간 일일 단위로 기록된 다우존스 산업지수에 대하여 확률효과 모형을 적용하였다.

Diffusion is a random process used to model financial and physical phenomena. When we construct statistical models for repeatedly observed diffusion processes, the idea of random effects needs to be considered. In this research, we introduce random parameters for an Ornstein-Uhlenbeck diffusion model and geometric Brownian motion diffusion model. In order to apply the maximum likelihood estimation method, we tried to build likelihoods in closed-forms, by assuming appropriate distributions for random effects. We applied the random effect models to data consisting of Dow Jones Industrial Average indices recorded daily over 27 years from 1991 to 2017.

키워드

Table 4.1. Simulation results for OU models

GCGHDE_2018_v31n6_801_t0001.png 이미지

Table 4.2. Simulation results for GBM models

GCGHDE_2018_v31n6_801_t0002.png 이미지

참고문헌

  1. Ait-Sahalia, Y. (2002). Maximum-likelihood estimation of discretely-sampled diffusions: a closed-form approximation approach, Econometrica, 70, 223-262. https://doi.org/10.1111/1468-0262.00274
  2. Applebaum, D. (2004) Levy processes - from probability theory to finance and quantum groups, Notices of the American Mathematical Society, 51, 1336-1347.
  3. Delattre, M., Genon-Catalot, V., and Samson, A. (2013). Maximum likelihood estimation for stochastic differential equations with random effects, Scandinavian Journal of Statistics, 40, 322-343. https://doi.org/10.1111/j.1467-9469.2012.00813.x
  4. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options, Reviews of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
  5. Hurn, A., Jeisman, J., and Lindsay, K., (2007). Seeing the wood for the trees: a critical evaluation of methods to estimate the parameters of stochastic differential equations, Journal of Financial Econometrics, 5, 390-455. https://doi.org/10.1093/jjfinec/nbm009
  6. Laird, N. M. and Ware, J. H. (1982). Random-effects models for longitudinal data, Biometrics, 38, 963-974. https://doi.org/10.2307/2529876
  7. Lee, Y. D., Song, S., and Lee, E. (2014). The delta expansion for the transition density of diffusion models, Journal of Econometrics, 178, 694-705. https://doi.org/10.1016/j.jeconom.2013.10.008
  8. Picchini, U., De Gaetano, A., and Ditlevsen, S. (2010). Stochastic differential mixed-effects models, Scandinavian Journal of Statistics, 37, 67-90. https://doi.org/10.1111/j.1467-9469.2009.00665.x
  9. Pinheiro, J. C. and Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS, Springer-Verlag, New York.