Journal of the Korean Operations Research and Management Science Society (한국경영과학회지)

The Korean Operations Research and Management Science Society (한국경영과학회)
권호 표지
DOI QR코드

DOI QR Code

An Approximation of the Cumulant Generating Functions of Diffusion Models and the Pseudo-likelihood Estimation Method

확산모형에 대한 누율생성함수의 근사와 가우도 추정법

  • Received : 2013.01.23
  • Accepted : 2013.02.21
  • Published : 2013.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Diffusion is a basic mathematical tool for modern financial engineering. The theory of the estimation methods for diffusion models is an important topic of the financial engineering. Many researches have been tried to apply the likelihood estimation method for estimating diffusion models. However, the likelihood estimation method for diffusion is complicated and needs much amount of computing. In this paper we develop the estimation methods which are simple enough to be compared to the Euler approximation method, and efficient enough statistically to be compared to the likelihood estimation method. We devise pseudo-likelihood and propose the maximum pseudo-likelihood estimation methods. The pseudo-likelihoods are obtained by approximating the transition density with normal distributions. The means and the variances of the distributions are obtained from the delta expansion suggested by Lee, Song and Lee (2012). We compare the newly suggested estimators with other existing estimators by simulation study. From the simulation study we find the maximum pseudo-likelihood estimator has very similar properties with the maximum likelihood estimator. Also the maximum pseudo-likelihood estimator is easy to apply to general diffusion models, and can be obtained by simple numerical steps.

Keywords

References

  1. 이은경, 최영수, 이윤동, "확산모형 전이확률밀도의 급수전개법과 그 계수", 응용통계연구 , 제23권, 제2호(2010), pp.383-392.
  2. Ait-Sahalia, Y., "Transition densities for interest rate and other nonlinear diffusions," Journal of Finance, Vol.54(1999), pp.1361-1395. https://doi.org/10.1111/0022-1082.00149
  3. Ait-Sahalia, Y., "Maximum-likelihood estimation of discretely-sampled diffusions : A closedform approximation approach," Econometrica, Vol.70(2002), pp.223-262. https://doi.org/10.1111/1468-0262.00274
  4. Ait-Sahalia, Y., "Closed-form Likelihood Expansions for Multivariate Diffusions," The Annals of Statistics, Vol.36, No.2(2008), pp. 906-937. https://doi.org/10.1214/009053607000000622
  5. Ait-Sahalia, Y. and J. Yu, "Saddlepoint approximations for continuous-Time Markov processes," Journal of Econometrics, Vol. 134(2006), pp.507-551. https://doi.org/10.1016/j.jeconom.2005.07.004
  6. Barndorff-Nielsen, O.E. and D.R. Cox, Asymptotic Techniques for Use in Statistics, Chapman and Hall, New York, (1989).
  7. Beskos, A., O. Papaspiliopoulos, G.O. Robert, and P. Fearnhead, "Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes," The Journal of the Royal Statistical Society, series B., Vol.68(2006), pp.333-383. https://doi.org/10.1111/j.1467-9868.2006.00552.x
  8. Chan, K.C., G.A. Karolyi, F.A. Longstaff, and A.B. Sanders, "An empirical comparison of alternative models of the short-term interest rate," Journal of Finance, Vol.47(1992), pp.1209-1227. https://doi.org/10.1111/j.1540-6261.1992.tb04011.x
  9. Cox, J., "Notes on option pricing I : Constant elasticity of variance diffusions," Working paper, Stanford University reprinted in Journal of Portfolio Management, Vol.22(1996), pp.15-17.
  10. Cox, J., J. Ingersoll, and S. Ross,. "A theory of the term structure of interest rates," Econometrica, Vol.53(1985), pp.385-407. https://doi.org/10.2307/1911242
  11. Dacunha-Castelle, D. and D. Florens-Zmirou, "Estimation of the Coefficients of a Diffusion from Discrete Observations," Stochastics, Vol.19(1986), pp.263-284. https://doi.org/10.1080/17442508608833428
  12. Daniels, H., Saddlepoint approximations in statistics, Annals of Mathematical Statistics, Vol.25(1954), pp.631-650. https://doi.org/10.1214/aoms/1177728652
  13. Durham, G. and R. Gallant, Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes, Technical report, (2001).
  14. Easton, G.S. and E. Ronchetti, "General Saddlepoint Approximations with Applications to L Statistics," Journal of the American Statistical Association, Vol.81(1986), pp.420-430.
  15. Elerian, O., S. Chib, and N. Shephard, "Likelihood Inference for Discretely Observed Nonlinear Diffusions," Econometrika, Vol.69(2001), pp.959-993. https://doi.org/10.1111/1468-0262.00226
  16. Eraker, B., "MCMC Analysis of Diffusion Models with Application to Finance," Journal of Business and Economic Statistics, Vol.19(2001), pp.177-191. https://doi.org/10.1198/073500101316970403
  17. Goutis, C. and G. Casella, "Explaining the Saddlepoint Approximation," The American Statistician, Vol.53(1999), pp.216-224.
  18. Hurn, A., J. Jeisman, and K. Lindsay, "Seeing the wood for the trees : A critical evaluation of methods to estimate the parameters of stochastic differential equations," Journal of Financial Econometrics, Vol.5(2007), pp. 390-455. https://doi.org/10.1093/jjfinec/nbm009
  19. Kloeden, P. and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, (1999).
  20. Ko, Y., S.P. Hong, and C. Jun, "On Parameter Estimation of Growth Curves for Technological Forecasting by Using Non-linear Least Squares," International Journal of Management Science, Vol.14, No.2(2008), pp.89-104.
  21. Kwon, O., "Uncertainty, View, and Hedging : Optimal Choice of Instrument and Strike for Value Maximization," International Journal of Management Science, Vol.17, No.2(2011), pp.99-129.
  22. Lee, Y.D., S. Song, and E. Lee, "The Delta Expansion for the Transition Density of Diffusion Models," Technical Report, (2012).
  23. Nicolau, J., "A new technique for simulating the likelihood of stochastic differential equations," Econometrics Journal, Vol.5(2002), pp.91-102. https://doi.org/10.1111/1368-423X.t01-1-00075
  24. Oksendall, B., Stochastic Differential Equations : An Introduction with Applications, Springer, (2003).
  25. Park, S. and B. Jang, "Stock Returns and Market Making with Inventory," Management Science and Financial Engineering, Vol.18, No.2(2012), pp.1-4.
  26. Pederson, A.R., "A New Approach to Maximum Likelihood Estimation for Stochastic Differential Equations based on Discrete Observations," Scandinavian Journal of Statistics, Vol.22(1995), pp.55-71.
  27. Rogers, L., "Smooth Transitional Densities for One-Dimensional Diffusions," Bulletin of the London Mathematical Society, Vol.17(1985), pp.157-161. https://doi.org/10.1112/blms/17.2.157
  28. Shoji, I. and T. Ozaki, "Estimation for nonlinear stochastic differential equations by a local linearization method," Stochastic Analysis and Applications, Vol.16(1998), pp.733-752, https://doi.org/10.1080/07362999808809559
  29. Vasicek, O., "An equilibrium characterization of the term structure," Journal of Financial Economics, Vol.5(1977), pp.177-188. https://doi.org/10.1016/0304-405X(77)90016-2