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First Order Strong Approximations of Jump Diffusions

Author

Listed:
  • Bruti-Liberati Nicola
  • Nikitopoulos-Sklibosios Christina
  • Platen Eckhard

    (University of Technology Sydney, School of Finance & Economics and Department of Mathematical Sciences, PO Box 123, Broadway, NSW, 2007, Australia)

Abstract

This paper presents new results on strong numerical schemes, which are appropriate for scenario analysis, filtering and hedge simulation, for stochastic differential equations (SDEs) of jump-diffusion type. It provides first order strong approximations for jump-diffusion SDEs driven by Wiener processes and Poisson random measures. The paper covers first order derivative-free, drift-implicit and jump-adapted strong approximations. Moreover, it provides a commutativity condition under which the computational effort of first order strong schemes is independent of the total intensity of the jump measure. Finally, a numerical study on the accuracy of several strong schemes applied to the Merton model is presented.

Suggested Citation

  • Bruti-Liberati Nicola & Nikitopoulos-Sklibosios Christina & Platen Eckhard, 2006. "First Order Strong Approximations of Jump Diffusions," Monte Carlo Methods and Applications, De Gruyter, vol. 12(3), pages 191-209, October.
    • Handle: RePEc:bpj:mcmeap:v:12:y:2006:i:3:p:191-209:n:6
    DOI: 10.1515/156939606778705191
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    References listed on IDEAS

    as
    1. Das, Sanjiv R., 2002. "The surprise element: jumps in interest rates," Journal of Econometrics, Elsevier, vol. 106(1), pages 27-65, January.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Pan, Jun, 2002. "The jump-risk premia implicit in options: evidence from an integrated time-series study," Journal of Financial Economics, Elsevier, vol. 63(1), pages 3-50, January.
    4. Carl Chiarella & Christina Sklibosios, 2003. "A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 10(2), pages 87-127, September.
    5. Eckhard Platen, 1999. "An Introduction to Numerical Methods for Stochastic Differential Equations," Research Paper Series 6, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    8. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," The Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
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    Citations

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    Cited by:

    1. Carl Chiarella & Christina Nikitopoulos Sklibosios & Erik Schlogl, 2007. "A Control Variate Method for Monte Carlo Simulations of Heath-Jarrow-Morton Models with Jumps," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(5), pages 365-399.
    2. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007, March.
    3. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.

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