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Approximation of A Jump-Diffusion Process

  • Sanghoon Lee

We present a weak convergence of a discrete time process to a jump-diffusion process as the length of sampling interval, h, goes to zero. There is an example given for the weak convergency with using GARCH (1,1)-M model by Engle and Bollerslev(1986). It is shown that ARCH type models can be used as discrete time approximations of jump-diffusion processes. We use Exponential ARCH with Poisson Jump component as an example for the approximation. Therefore, we may use a discrete time ARCH process as an approximation of a jump-diffusion process in estimation and forecasting. And we may use the jump-diffusion process as an approximation of ARCH process when there is distributional results available for the jump-diffusion limit of the sequence of ARCH processes

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Paper provided by Econometric Society in its series Econometric Society 2004 Far Eastern Meetings with number 412.

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Date of creation: 11 Aug 2004
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Handle: RePEc:ecm:feam04:412
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  1. Kim, Myung-Jig & Oh, Young-Ho & Brooks, Robert, 1994. "Are Jumps in Stock Returns Diversifiable? Evidence and Implications for Option Pricing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 29(04), pages 609-631, December.
  2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  3. Ball, Clifford A & Torous, Walter N, 1985. " On Jumps in Common Stock Prices and Their Impact on Call Option Pricing," Journal of Finance, American Finance Association, vol. 40(1), pages 155-73, March.
  4. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
  5. Oldfield, George Jr. & Rogalski, Richard J. & Jarrow, Robert A., 1977. "An autoregressive jump process for common stock returns," Journal of Financial Economics, Elsevier, vol. 5(3), pages 389-418, December.
  6. Ball, Clifford A. & Torous, Walter N., 1983. "A Simplified Jump Process for Common Stock Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 18(01), pages 53-65, March.
  7. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
  8. Vlaar, Peter J G & Palm, Franz C, 1993. "The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroscedasticity, and Jumps," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(3), pages 351-60, July.
  9. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-51, July.
  10. Ball, Clifford A. & Roma, Antonio, 1993. "A jump diffusion model for the European monetary system," Journal of International Money and Finance, Elsevier, vol. 12(5), pages 475-492, October.
  11. Amin, Kaushik I, 1993. " Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-63, December.
  12. Ahn, Chang Mo & Thompson, Howard E, 1988. " Jump-Diffusion Processes and the Term Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 43(1), pages 155-74, March.
  13. Bollerslev, Tim & Engle, Robert F. & Nelson, Daniel B., 1986. "Arch models," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 49, pages 2959-3038 Elsevier.
  14. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
  15. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
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