KDI Journal of Economic Policy

Korea Development Institute (한국개발연구원)
권호 표지
DOI QR코드

DOI QR Code

Comparison of the Korean and US Stock Markets Using Continuous-time Stochastic Volatility Models

  • Received : 2018.07.11
  • Published : 2018.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

We estimate three continuous-time stochastic volatility models following the approach by Aït-Sahalia and Kimmel (2007) to compare the Korean and US stock markets. To do this, the Heston, GARCH, and CEV models are applied to the KOSPI 200 and S&P 500 Index. For the latent volatility variable, we generate and use the integrated volatility proxy using the implied volatility of short-dated at-the-money option prices. We conduct MLE in order to estimate the parameters of the stochastic volatility models. To do this we need the transition probability density function (TPDF), but the true TPDF is not available for any of the models in this paper. Therefore, the TPDFs are approximated using the irreducible method introduced in Aït-Sahalia (2008). Among three stochastic volatility models, the Heston model and the CEV model are found to be best for the Korean and US stock markets, respectively. There exist relatively strong leverage effects in both countries. Despite the fact that the long-run mean level of the integrated volatility proxy (IV) was not statistically significant in either market, the speeds of the mean reversion parameters are statistically significant and meaningful in both markets. The IV is found to return to its long-run mean value more rapidly in Korea than in the US. All parameters related to the volatility function of the IV are statistically significant. Although the volatility of the IV is more elastic in the US stock market, the volatility itself is greater in Korea than in the US over the range of the observed IV.

Keywords

GBOGHH_2018_v40n4_1_f0001.png 이미지

FIGURE 1. DAILY KOSPI 200 AND VKOSPI

GBOGHH_2018_v40n4_1_f0002.png 이미지

FIGURE 2. THE TREND OF THE GROWTH RATE OF PRODUCTIVITY

GBOGHH_2018_v40n4_1_f0003.png 이미지

FIGURE 3. DAILY S&P 500 AND VIX

GBOGHH_2018_v40n4_1_f0004.png 이미지

FIGURE 4. DAILY CHANGES IN THE S&P 500 AND THE VIX

GBOGHH_2018_v40n4_1_f0005.png 이미지

FIGURE 5. DAILY OBSERVATIONS OF THE INTEGRATED VOLATILITY AND $\hat{\gamma}$ FOR THE HESTON MODEL OF KOREA

GBOGHH_2018_v40n4_1_f0006.png 이미지

FIGURE 6. DAILY OBSERVATIONS OF THE INTEGRATED VOLATILITY AND $\hat{\gamma}$ FOR THE CEV MODEL OF THE US

GBOGHH_2018_v40n4_1_f0007.png 이미지

FIGURE 7. VOLATILITY FUNCTIONS OF THE INTEGRATED VOLATILITY FOR KOREA AND THE US

TABLE 1—PARAMETER ESTIMATES FOR THE UNIVARIATE CEV MODEL FOR VKOSPI AND VIX

GBOGHH_2018_v40n4_1_t0001.png 이미지

TABLE 2—SUMMARY STATISTICS

GBOGHH_2018_v40n4_1_t0002.png 이미지

TABLE 3—SUMMARY STATISTICS

GBOGHH_2018_v40n4_1_t0003.png 이미지

TABLE 4—MAXIMUM LIKELIHOOD ESTIMATION RESULTS OF STOCHASTIC VOLATILITY MODELS FOR KOREA AND THE US

GBOGHH_2018_v40n4_1_t0004.png 이미지

References

  1. Ait-Sahalia, Y. 1999. "Transition Densities for Interest Rate and Other Nonlinear Diffusions," Journal of Finance, 54: 1361-1395. https://doi.org/10.1111/0022-1082.00149
  2. Ait-Sahalia, Y. 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approximation Approach," Econometrica, 70(1): 223-262. https://doi.org/10.1111/1468-0262.00274
  3. Ait-Sahalia, Y. 2008. "Closed-Form Likelihood Expansions for Multivariate Diffusions," Annals of Statistics, 36(2): 906-937 https://doi.org/10.1214/009053607000000622
  4. Ait-Sahalia, Y. and R. Kimmel. 2007. "Maximum likelihood estimation of stochastic volatility models," Journal of Financial Economics, 83: 413-452. https://doi.org/10.1016/j.jfineco.2005.10.006
  5. Ait-Sahalia, Y. and A. Lo. 1998. "Nonparametric estimation of state-price-densities implicit in financial asset prices," Journal of Finance, 53: 499-547. https://doi.org/10.1111/0022-1082.215228
  6. Bakshi, G., N. Ju, and H. Ou-Yang. 2006. "Estimation of Continuous-time Models with an Application to Equity Volatility," Journal of Financial Economics, 82: 227-249. https://doi.org/10.1016/j.jfineco.2005.09.005
  7. Black, F., and M. Scholes. 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, 81(3): 637-654. https://doi.org/10.1086/260062
  8. Chacko, G., and L. M. Viceira. 2003. "Spectral GMM Estimation of Continuous-Time Processes," Journal of Econometrics, 116: 259-292. https://doi.org/10.1016/S0304-4076(03)00109-X
  9. Chan, K., Karolyi, F. Longstaff, and A. Sanders. 1992. "An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," The Journal of Finance, 47(3), 1209-1227. https://doi.org/10.1111/j.1540-6261.1992.tb04011.x
  10. Chang, J., and S. X. Chen. 2011. "On the Approximate Maximum Likelihood Estimation for Diffusion Processes," Annals of Statistics, 39(6): 2820-2851. https://doi.org/10.1214/11-AOS922
  11. Cho, J. K., W. S. Choi, and K. S. Choe. 2015. "An Asymmetric Nature of the VKOSPI Index," The Journal of Eurasian Studies, 12(4): 21-43. https://doi.org/10.31203/aepa.2015.12.4.002
  12. Choi, H., and S. Han. 2009. "Explanation of the VKOSPI and empirical analysis," KRX Market.
  13. Choi, K., and D. Cho. 2017. "Time-varying Co-movement and Dynamic Spillover Effect among Korean, Chinese, and U.S. Stock Markets," Review of International Money and Finance, 7(2): 5-31.
  14. Choi, S. 2013. "Closed-Form Likelihood Expansions for Multivariate Time-Inhomogeneous Diffusions," Journal of Econometrics, 174(2): 45-65. https://doi.org/10.1016/j.jeconom.2011.12.007
  15. Choi, S. 2015a. "Approximate Transition Probability Density Function of a Multivariate Timeinhomogeneous Jump Diffusion Process in a Closed-Form Expression," Working paper, University of Seoul.
  16. Choi, S. 2015b. "Explicit Form of Approximate Transition Probability Density Functions of Diffusion Processes," Journal of Econometrics, 187: 57-73. https://doi.org/10.1016/j.jeconom.2015.02.003
  17. Choi, S. and D. Yuan. 2018. "Maximum Likelihood Estimation of Continuous-Time Stochastic Volatility Models with Regime Shifts," Working paper, University of Seoul.
  18. Egorov, A. A., H. Li, and Y. Xu. 2003. "Maximum Likelihood Estimation of Time Inhomogeneous Diffusions," Journal of Econometrics, 114: 107-139. https://doi.org/10.1016/S0304-4076(02)00221-X
  19. Feller, W. 1952. "The Parabolic Differential Equations and the Associated Semi-Groups of Transformations," Annals of Mathematics 55(3): 468-519. https://doi.org/10.2307/1969644
  20. Han, H., A. M. Kutan, and D. Ryu. 2015. "Effects of the US Stock Market Return and Volatility on the VKOSPI," Economics: The Open-Access, Open-Assessment E-Journal, 9: 1-34.
  21. Heston, S. L. 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, 6(2): 327-343. https://doi.org/10.1093/rfs/6.2.327
  22. Hull, J., and A. White. 1987. "The pricing of options on assets with stochastic volatilities," Journal of Finance, 42: 281-300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x
  23. Jones, C. S. 2003. "The dynamics of stochastic volatility: evidence from underlying and options markets," Journal of Econometrics, 116: 181-224. https://doi.org/10.1016/S0304-4076(03)00107-6
  24. Kim, S. W. 2010. "A Study on Developing a VKOSPI Forecasting Model via GARCH Class Models for Intelligent Volatility Trading Systems," Journal of Intelligent Information Systems, 16(2): 19-32.
  25. Lee, B. S., and D. Ryu. 2013. "Stock returns and implied volatility: A new VAR approach," Economics: The Open-Access, Open-Assessment E-Journal, 7: 1-20.
  26. Lee, J., and B. K. Yu. 2018. "What Drives the Stock Market Co-movements between Korea and China, Japan and the US?" Working paper, Bank of Korea.
  27. Lewis, A. 2000. Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach.
  28. Li, M. 2010. "A damped diffusion framework for financial modeling and closed-form maximum likelihood estimation," Journal of Economic Dynamics and Central, 34: 132-157. https://doi.org/10.1016/j.jedc.2009.08.001
  29. Meddahi, N. 2001. "An eigenfunction approach for volatility modeling," Working paper, Universite de Montreal.
  30. Merton, R. C. 1973. "Theory of Rational Option Pricing," Bell Journal of Economics and Management Science, 4: 141-183. https://doi.org/10.2307/3003143
  31. Nelson, D. 1990. "ARCH models as diffusion approximations," Journal of Econometrics, 45: 7-38. https://doi.org/10.1016/0304-4076(90)90092-8
  32. Stein, E., and J. Stein. 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, 4: 727-752. https://doi.org/10.1093/rfs/4.4.727
  33. Stein, J. 1989. "Overreactions in the options market," Journal of Finance, 44: 1011-1023. https://doi.org/10.1111/j.1540-6261.1989.tb02635.x
  34. Stramer, O., M. Bognar, and P. Schneider. 2010. "Bayesian Inference for Discretely Sampled Markov Processes with Closed-Form Likelihood Expansions," Journal of Financial Econometrics, 8(4): 450-480. https://doi.org/10.1093/jjfinec/nbp027
  35. Yoon, J. I. 2007. "The Evaluations and the International Comparisons of the Integration with U.S. Stock Market," Journal of Money and Finance, 21(1): 55-92.
  36. Yu, J. 2007. "Closed-Form Likelihood Approximation and Estimation of Jump-Diffusions with an Application to the Realignment Risk of the Chinese Yuan," Journal of Econometrics, 141: 1245-1280. https://doi.org/10.1016/j.jeconom.2007.02.003